UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Representations of Hecke Algebras of finite groups with BN-Pairs of classical type Hoefsmit, Peter Norbert 1974

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Notice for Google Chrome users:
If you are having trouble viewing or searching the PDF with Google Chrome, please download it here instead.

Item Metadata

Download

Media
831-UBC_1974_A1 H63_8.pdf [ 4.67MB ]
Metadata
JSON: 831-1.0080112.json
JSON-LD: 831-1.0080112-ld.json
RDF/XML (Pretty): 831-1.0080112-rdf.xml
RDF/JSON: 831-1.0080112-rdf.json
Turtle: 831-1.0080112-turtle.txt
N-Triples: 831-1.0080112-rdf-ntriples.txt
Original Record: 831-1.0080112-source.json
Full Text
831-1.0080112-fulltext.txt
Citation
831-1.0080112.ris

Full Text

REPRESENTATIONS OF HECKE ALGEBRAS OF FINITE GROUPS WITH BN-PAIRS OF CLASSICAL TYPE BY PETER NORBERT HOEFSI1IT B.A. C a l i f o r n i a S ta te U n i v e r s i t y at F u l l e r t o n , 1968 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY - ' • i n the Department of MATHEMATICS We accept t h i s t h e s i s as conforming to the r equ i r ed s tandard THE UNIVERSITY OF BRITISH COLUMBIA August , 1974 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r equ i r emen t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s tudy . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y pu rposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d tha t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l owed w i thout my w r i t t e n p e r m i s s i o n . Department o f C 1YA . ^ The U n i v e r s i t y o f B r i t i s h Co lumbia Vancouver 8, Canada Date O J U r t ^ V W _ ABSTRACT . Le t G be a f i n i t e group w i t h BN-pair and Coxeter system (17, R) . Le t A be the gener i c r i n g cor respond ing to (W, R) i n the sense of T i t s , de f i ned over the po l ynomia l r i n g D = Qfu^, r e R ] . Le t k be any f i e l d of c h a r a c t e r i s t i c z e r o . For the homomorphism ' <j) : D —> k de f ined by ^(u^) = q^, the index parameters of G, the s p e c i a l z e d a lgeb ra A^ ^ i s i somorphic to the Hecke a l geb ra ^ ( G , B) o f G w i t h respec t to a B o r e l subgroup B of G, w h i l e f o r the s p e c i a l i z a t i o n de f i ned b y . <Ku r) = 1> r e R, A^ ^ i s i somorph ic to the group a l geb ra kW. As G the Hecke a lgeb ra H^(G, B) a f f o rds the induced r e p r e s e n t a t i o n 1^, the G i r r e d u c i b l e r ep resen ta t i ons of G appear ing i n 1 can be obta ined from the r ep resen ta t i ons o f B) . I n t h i s t h e s i s , we o b t a i n a l l the i r r e d u c i b l e r e p r e s e n t a t i o n s , de f i ned over the quo t i en t f i e l d of D, o f the gene r i c r i n g cor respond ing to a Coxeter system of c l a s s i c a l t ype . The method employed i n v o l v e s a g e n e r a l i z a t i o n of Young's c o n s t r u c t i o n of the semi-normal ma t r i x r e p r e s e n -t a t i o n s of the symmetric group. We a l so ob t a i n an e x p l i c i t formula f o r the gener i c degree o f these r ep resen ta t i ons i n terms of the hook lengths of Young diagrams. Thus the degrees of a l l the i r r e d u c i b l e c o n s t i t u e n t s of 1 are ob ta ined f o r the a 1 2 1 2 1 2 f a m i l i e s of Cheva l l ey groups A^(q ) , B £ ( q ) , A 2 £ ( q ) , ^ £ - 1 ^ ) » D £ ^ Q ^ and f o r D^(q), £ odd. A l s o , most o f the degrees are obta ined f o r D 0 (q ) , I even. - ' TABLE OF CONTENTS INTRODUCTION CHAPTER 1 PRELIMINARIES 1.1 P a r t i t i o n s and Tableaux 1.2 The Semi-normal Representa t ions of the Symmetric Group CHAPTER 2 REPRESENTATION OF THE GENERIC RING CORRESPONDING TO A COXETER SYSTEM OF CLASSICAL TYPE 2.1 Hecke A lgebras and F i n i t e Groups w i t h BN-pairs K 2.2 The Representa t ions of O (B ) n K K 2.3 The Representa t ions of 01 ( A n ) a n d 01 (Dn> CHAPTER 3 DEGREES OF THE IRREDUCIBLE" CONSTITUENTS OF L? •' Js 3.1 D e f i n i t i o n s and Characters of P a r a b o l i c Type 3.2 An I nduc t i on Formula 3.3 The E v a l u a t i o n o f E^ c~^ 3.4 Gener i c Degrees BIBLIOGRAPHY i v AGOOWLEDGEXENTS I w i s h to express my s i n c e r e a p p r e c i a t i o n to my t h e s i s s u p e r v i s o r , D r . B. Chang, and to D r . R. Ree, f o r t h e i r cont inued encouragement and adv i se d u r i n g the p r e p a r a t i o n of t h i s t h e s i s . The f i n a n c i a l support of the U n i v e r s i t y of B r i t i s h Columbia and . the N a t i o n a l Research C o u n c i l of Canada i s a l s o g r a t e f u l l y acknowledged. 1 INTRODUCTION The results i n this thesis are concerned with the Irreducible complex representations of a f i n i t e group G with BN-pair and Weyl group W of c l a s s i c a l type, which appear i n the induced permutation representa— G t i o n l g from a Borel subgroup B of G. These representations were constructed by Steinberg i n [16] for GL(n, q), parametrized by p a r t i t i o n -of n and he showed an elegant formula to hold for thei r degrees i n terms of the hook lengths of a Young diagram. Some spec i a l representations of the generic ri n g of a Coxeter system and the degrees of the corresponding representations of the generic ring corresponding to a Coxeter system of c l a s s i c a l type, i . e . of type A n, B n (n >_ 2) and D q (n >^  4) , which spe c i a l i z e to ir r e d u c i b l e representations of the Hecke algebra H (G, B) G affording the induced representation 1 . The method employed involves a generalization of Young's construction of the semi-normal matrix represen-tations of the symmetric group. This construction also enables us to Q compute the degrees of the irre d u c i b l e constituents of L . In the case that the Weyl group of G i s of type B^, we obtain a formula for the degrees determined by the hook lengths of pairs of Young diagrams comparable to Steinberg's formula for GL(n, q). Indeed, Steinberg's formula i s recovered as a special case. i r r e d u c i b l e constituents of were also obtained by Kilmoyer (cf. [ 6 ] ) . In this thesis we construct e x p l i c i t l y a l l the irreducible Here i s a survey of the contents of this thesis. Chapter 1 2 contains the necessary preliminaries about the representation theory of the symmetric- group. In Chapter 2 the generic ring corresponding to a Coxeter system is introduced. The irreducible representations of the generic ring fJtCB^) of a Coxeter system of type B^, defined over the polynomial ring Q[x, y] are constructed i n Section 2.2. They are parametrized by pairs of partitions (a), (g) with |a| + |31 = n and are rational representations, i.e. they are defined over K = Q(x, y). The representations of the generic rings 0\ (A^) and 0\(Dn), defined over Q[x], corresponding to Coxeter systems of type A and D are obtained in Section 2.3 as coro-. J r n n l l a r i e s by considering appropriate specialized algebras of . For the specialization x —> 1, the representation of 0} (A^) obtained specialize to give the semi-normal matrix representations of the symmetric group. Chapter 3 i s concerned with the degrees of the irreducible consti-G tuents of Lg . It is f i r s t shown in Section 3.1 that the representations obtained are of parabolic type, i.e. they appear with multiplicity one i n G some permutation representation lp , where P i s a parabolic subgroup of G. The generic degree d of an irreducible character x °f t n e generic X algebra i s introduced, such that the degree of the corresponding irreducible character of G is obtained by specializing d . In Section 3.2 an inte-X resting induction formula i s derived for d , where x i - s a n irreducible X character of 0[(B^). This formula, and lengthy computations, enables us to prove in Section 3.4 an explicit formula for d as a rational function i n X the indeterminates x and y, in terms of the hook lengths of pairs of Young diagrams. The generic degrees of a l l the irreducible characters of 0\(An) and almost a l l the irreducible characters of OK^) are obtained as corollary. CHAPTER 1 3 PRELIMINARIES 1 .1 . PARTITIONS AND TABLEAUX A pcuvtction o f n i s an ordered se t of p o s i t i v e numbers (a) = (a^, a^) such tha t n = + . . . + , a r a 2 — * " — a k ' k a r b i t r a r y . Such a p a r t i t i o n (a) i s s a i d to have k pcwtb and i t s Z.2JflQtky | a | , i s n. Thus (4, 3, 3, 1) i s a p a r t i t i o n o f 1 1 . I t may "2" • a l s o be w r i t t e n (4, 3 , 1) and a s i m i l a r n o t a t i o n w i l l be used e lsewhere . We represen t (a) by a Young d<LagfLam, D (a ) , hav ing squares i n the f i r s t row, o.^ squares i n the second row and so on , the j * * * 1 squares o f the rows making a column. D(a) i s s a i d to have A>h.apz ( a ) . Thus 2 th i s the Young diagram of shape (4 , 3 , 1 ) . The square i n the i row and j * " * 1 column o f D(ct) i s s a i d to have CO0Kdlncut2A ( i , j ) and i s c a l l e d the ( i , j ) - 4 q u a A e . L e t denote the number o f squares appear ing i n the i*"*1 column o f D(a)• The Young diagram D ( a ' ) , obta ined by i n t e r c h a n g i n g the rows and 4 columns of D (a ) , i s c a l l e d the conjugate, o f D(a) and the p a r t i t i o n (a ' ) = (a^, . . . » a^) o f n i s c a l l e d the coyijugcutd o f (a) . The n ZeXtZAA 1, .'. . , . n _ may be arranged i n the squares of D(a) i n n! ways. Each such arrangement i s c a l l e d a tohLzanx oft ikapz (a). A tab leaux i s c a l l e d a btandaJid tdotzcuix i f the l e t t e r s i n every row i n c r ease from l e f t to r i g h t and i n every column from top to bot tom. Thus the tab leaux ( i i ) and ( i i i ) are s tandard tab leaux w h i l e ( i ) i s n o t . The tab leaux of (1) (11) F igu re 1. ( H i ) shape (a) w i t h the n l e t t e r s arranged i n consecu t i ve o rder i n the rows, s t a r t i n g w i t h the f i r s t square i n the f i r s t row i s c a l l e d the canonlcaZ tabttaux o f shape (a) . The tab leaux ( i i i ) above i s the c a n o n i c a l tab leaux of shape ( 3 , 2 ) . The number, f a , o f s tandard tab leaux of shape (a) i s determined as f o l l ows (see [12 ] , p . 44 ) . The ( i , j ) - s q u a r e of D(a) determines the (i,j)-fo.Oofe c o n s i s t i n g of the ( i , j ) - s q u a r e a long w i t h the c u - i squares to tlx tlx the r i g h t i n the i row and the a j~J squares below i n the j column. Thus the l e n g t h of the ( i , j ) - h o o k i s (1 .1 .1 ) = ( a ± - i ) + (a'. - j ) + 1. Then 5 ( 1 . 1 . 2 ) f° = n! / J[h • / i , j J A douhtu p<Wtl£L0Yi of n i s an ordered p a i r of p a r t i t i o n s (y) = (a, 3) with |a| + | B | = n.. I f (a) = (a^, ..., a g) and (8) = (8^ , . . . , 8 t ) , we write the double p a r t i t i o n (y) as (y) = (y-j,, u s + t ) where y ± = ou, 1 <_ i <_ s and y.. = $^ where j > s, j = s+i . We allow e i t h e r (a) or (8) to be a p a r t i t i o n of n i n the above, i . e . l e t ( 0 ) denote the empty partition. For (a) a p a r t i t i o n of n, ( ( a ) , ( 0 ) ) and ( ( 0 ) , (a)) are d i s t i n c t double p a r t i t i o n s of n . We represent (y) by an ordered p a i r of Young diagrams D(y) = (D(a), D ( 8 ) ) , c a l l e d the Voting diagram o& &ka.p<L (y) . D(y) i s considered to have s+t rows, where the i ^ row of D(a) i s the i ^ row of D(y) and the j*"* 1 row th of D(B) i s the -s+j 1 row of D(y). The Young diagrams of shape ( ( a ) , ( 0 ) ) and ( ( 0 ) , (B)) are taken to be (-, D(a)) and (D(a), -) . The squares of D(y) are i d e n t i f i e d by t h e i r coordinates i n the diagrams D(a) and D(B). Thus the square of D(y) which i s i n the t^1 row and tti j column of D(a) (resp. D(B)) i s c a l l e d the (i,j ) -4qaaAe o{. D(a) (resp. D(B)) and has coolttlnaXeA ( i , j ) . Hence d i s t i n c t squares of D(y) can have the same coordinates (for instance, the f i r s t square i n the f i r s t s t row of D(y) and the f i r s t square i n the s + 1 row of D(y) both have coordinates ( 1 , 1 ) , where s i s as above). A tahtzaux o{, ihape. (y) = (a, B) i s any arrangement of the l e t t e r s • _ 1 , ..., ii i n D(y) . Thus a tableaux T^ i s an ordered p a i r ct B (T , T ) where, for complementary subsets K and L of { 1 , n} with |K| = -ja| , |L| = |B|-, T a denotes any arrangement of the l e t t e r s of K i n 6 D(a) and T denotes any arrangement of the l e t t e r s o f L i n D(B). The tab leaux T y i s a AtandaAd tableaux i f the arrangement of the l e t t e r s i s i n i n c r e a s i n g order i n the rows and columns of both T a and T^. Thus 3 5 1 3 4 1 2 7 1 6 1 0 1 1 2 8 9 F igu re 2 . i s a s tandard tab leaux of shape ( ( 3 , 2-, 1 ) , ( 4 , 2 , 1 ) ) . The canonical tableaux o f shape (y) i s the tab leaux where the l e t t e r s are arranged c o n s e c u t i v e l y i n the rows o f D ( y ) , s t a r t i n g w i t h the f i r s t square i n the f i r s t row o f D ( y ) . The number, f^, o f s tandard tab leaux of shape (u) = ( a , 3 ) i s ( 1 . 1 . 3 ) , f y = ( ^ ) f a f 8 , k = |a | , where f ( 0 ) = 1 . We order the s tandard tab leaux of a g i ven shape as f o l l o w s : DEFINITION ( 1 . 1 . 4 ) Let T y , T!J- denote Atandand tableaux ofa &kape ( y ) . We. iay T y p r e c e d e * T y i£ the l e t t e u n , n - l , n-r+l appean In the 4>ame now in both tableaux but the letten. n-r appeau in a lowen. now in T: than in T y . The enumehation o£ the btandafid tableaux accofitting to thein. 7 otideAlng 4J> catted -pie. lcu>t leXteJi sequence. Thus i n the l a s t l e t t e r sequence a l l t ab leaux which have the l e t t e r i i i n the l a s t row precede those which have n i n the second to the l a s t row. These l a t t e r t ab leaux precede those which have n i n the t h i r d to the l a s t row and so on . Those tab leaux which have n i n the same row are arranged by the same scheme acco rd ing to the p o s i t i o n o f the l e t t e r n-1 and so on . I t i s e v i den t tha t the c a n o n i c a l tab leaux i s the f i r s t t ab leaux i n t h i s o r d e r i n g . We g i ve an example of t h i s o r d e r i n g . For the double p a r t i t i o n ( ( 2 , 1 ) , ( 2 ) ) , there are 20 standard t ab l eaux . Arranged acco rd ing to the , l a s t l e t t e r sequence they a r e , 12 13 12 13 23 14 14 24 12 13 3 2 4 4 4 2 3 3 5 5 45 45 35 25 15 35 25 15 34 24 23 14 24 34 15 15 25 15 25 35 5 5 5 5 2 3 3 4 4 4 14 23 13 12 34 24 14 23 13 12 F igu re 3. - F i n a l l y we de f i ne the n o t i o n o f a x i a l d i s t a n c e . DEFINITION (1 .1 .5 ) Von. &quan.eA A and B in a young diagram D(y) \&lik coordinates ( i , j ) and ( s , t ) pzctivetu, dzfttne. ike, OXAJOI distance., p, friom A to B to be. p -;" ( t-s) - ( j - i ) 8 A x i a l d i s t ance has a s imple g r a p h i c a l i n t e r p r e t a t i o n . Suppose the •squares A and B are i n the same diagram o f D(u) = (D(a ) , D(g)) . S t a r t i n g from A, proceed by any r e c t angu l a r rou te one square a t a t ime u n t i l B i s reached. Count ing +1 f o r each step made upwards o r to the r i g h t and -1 f o r each s tep made downwards o r to the l e f t , the r e s u l t a n t number o f s teps made i s the a x i a l d i s t ance from A to B . For squares b e l o n g i n g to d i s t i n c t d iagrams, a x i a l d i s t ance i s the d i s t ance of any r e c t a n g u l a r r o u t e , counted as above, i n the diagram obta ined by super impos ing D(B) upon D(a) . F i n a l l y DEFINITION (1 .1 .6 ) Thz axial distance, hfiom the Izttzn. _p_ to tkz I z t t z n ^ in a tableaux T y ij, thz axial didtancz {nom tkz Aquanz ofa T y in which, JD appzau to t/iz hquanz oi T y in which cj_ appzam. Thus i n F igu re 2 the a x i a l d i s t a n c e from 4^  to 13_ i s 3 , the a x i a l d i s t a n c e from 13 to j4 i s - 3 , the a x i a l d i s t ance from _6 . to 3_ . i s -1 and the a x i a l d i s t ance from 9 to 7 i s 0 . 9 1.2 . THE SEMI-NORMAL REPRESENTATIONS OF THE SYMMETRIC GROUP We b r i e f l y d e s c r i b e the i r r e d u c i b l e semi-normal r ep r e sen t a t i ons of the symmetric group on n l e t t e r s . The conjugacy c l a s s e s of S^ are paramet r i sed by the p a r t i t i o n s of n . In ( [18]) Young cons t ruc ted f o r each p a r t i t i o n (a) o f n an i r r e d u c i b l e r e p r e s e n t a t i o n [a] of S n of degree f 0 1 by c o n s t r u c t i n g p r i m i t i v e idempotents , the " n a t u r a l i dempoten ts " , i n the group a l g e b r a QS^ from the s tandard tab leaux o f shape (a) . The d i s t i n c t i v e f e a tu r e o f these r ep resen ta t i ons i s tha t they are i n t e g r a l , i . e . , m a t r i x r e p r e s e n t a t i o n s a f fo rded by the min imal l e f t i d e a l s generated by these idemptoents have e n t r i e s i n Z . In a subsequent paper ( [20] ) Young cons -t r u c t e d an equ i v a l en t form of these r ep re sen t a t i ons by means of the " s em i-normal i dempoten t s " . Whi le the cor respond ing m a t r i x r ep r e sen t a t i ons are not i n t e g r a l , Young showed an e legant c o n s t r u c t i o n to h o l d f o r the ma t r i c es of the t r a n s p o s i t i o n s ( i - 1 , i ) by means of the s tandard t a b l e a u x . For a t ab leaux T a of shape (a) , l e t ( i - 1 , i ) T a denote the tab leaux ob ta ined ct by i n t e r c h a n g i n g the l e t t e r s i-1 and i^  i n T . I f T i s a s tandard tab leaux and the l e t t e r s i-1 and jt do not occur e i t h e r i n the same row or column of T a , then ( i - 1 , i ) T a i s aga in a s tandard t ab l eaux . Young's fundamental theorem g i v i n g the semi-normal form of the r ep r e sen t a t i ons of of S can now s t a t e d as f o l l o w s , n THEOREM (1 .2 .1 ) Let T ° , T^ , f = f a , be the. anxaYigement oft the. Atandand tableaux oft thape (a) according to the. la&t IctteJi Ae.que.nce.. To conitsuict the. f x f matrix. h.epn.eA cnting ( i - 1 , i ) -cn the. voieducihle. lepneAzntation [a] oft S con/teAponding to (a), place. 10 ( i ) 1 in tke. p.p*" zntny wkviz kab i - l and i_ -in tkz ( i i ) -1 in tkz p ,p f c znPuj wkznz T° hoi i - l ana! ±_ in tkz now p now q th - PtI iamz now, th P.I 4ame column, ( i i i ) #ie matnix. -P P+1 ^ 1-p p T T ^ T '' column p column q .cn -the p , p t t l , p , q t h , q » p t h fl-^d q , q t h zntnizi wkznz p < q , = ( i - l , i ) Tp ana! 1/p -ci -the ax-ca£ dibtancz [i>zz ( 1 .1 .6 ) ) ifaom i to i - l -cn T a , — p ( i v ) ZQJWA zlAZWkzJlZ. The importance of the semi-normal form i s tha t i t p rov ides an i n d u c t i v e c o n s t r u c t i o n of the i r r e d u c i b l e r ep resen t a t i ons o f and moreover i s de f ined i n terms of the generators and r e l a t i o n s of S . Young 's fundamental theorem can be extended to y i e l d r e p r e s e n t a t i o n s o f S n co r respond ing to double p a r t i t i o n s (y) = ( a , 8) o f n . I f (a) i s a p a r t i t i o n o f k and (8) i s a p a r t i t i o n of £ w i t h k+£ = n, l e t [a]*[8] denote the r e p r e s e n t a t i o n of S n induced from the d i r e c t product r e p r e s e n t a t i o n [a] x [8] of the subgroup S, x S 0 o f S . The represen-t a t i o n [a]* [8] i s c a l l e d the outzn pnoduct nzpnzizntation o f [a] and • [ 6 ] . . For a standard, tab leaux T U = ( T a , T 8 ) of shape (y) = ( a , 8), i f the 11 Cfc l e t t e r s i-1 and jL do not occur e i t h e r i n the same row or column of T B y or T , ( i - 1 , i ) T i s aga in a s tandard t ab l eaux . A r rang ing the s tandard tab leaux of shape (y) a cco rd ing to the l a s t l e t t e r sequence, we have (see [12 ] , p.54) THEOREM (1 .2 .2 ) To com>tmict the matniceA ph.eAenti.ng ( i - 1 , i ) In the outen. product n.epn.ei>entation [ a ] - [ 6 ] oft S n , apply the conit/iuction given i.n Theorem (1 .2 .1 ) to the itandand tableaux ( T a , T 6 ) oft 6hape ( a , 6 ) , b i t t i n g p = 0. In ( i i i ) <lft the. letteu i-1 and 1 belong to dUtlnct tableaux T a , T S . The hypeAOCtahedAal gfioup of order 2 n n ! i s the group of s igned permutat ions on n l e t t e r s . I t ,can be regarded as a c t i n g on an or thonormal b a s i s e^, . . . , of R n by means of permutat ions and s i g n th changes. Denote the k s i g n change, e ^ — > -e^ by - ( k ) . The se t of t r a n s p o s i t i o n s ( i - 1 , i ) , i = 2, n , and the f i r s t s i g n change, - ( 1 ) , generate . In ( [19] ) Young showed the conjugacy c l a s ses o f H to be paramet r i sed by double p a r t i t i o n s (y) = ( a , B) of n and cons t ruc ted f o r each double p a r t i t i o n (y) an i r r e d u c i b l e r e p r e s e n t a t i o n [y] o f of degree f*\ by c o n s t r u c t i n g p r i m i t i v e idempotents i n QH^ analogous to the " n a t u r a l idempotent " of Young d i d not cons t ru c t the analogous of the " semi-norma l " idempotents f o r H . I t i s i m p l i c i t i n h i s work, however, t ha t a "semi-normal fo rm" can be cons t ruc ted f o r the r ep r e sen t a t i ons [y] u s i n g Theorem ( 1 . 2 . 2 ) . In p a r t i c u l a r , u s i ng the mat r i ces f o r the t r a n s p o s i -t i o n s ( i - 1 , i ) i n the outer product r e p r e s e n t a t i o n [ a ] * [3 ] , we need on l y cons t ruc t a m a t r i x f o r the f i r s t s i g n change -(1) which s a t i s f i e s the 12 r e l a t i o n s o f ff^ . I t can r e a d i l y be shown (see c o r o l l a r y ( 2 . 2 . 1 5 ) ) , u s i ng the i n d u c t i v e o r d e r i n g p rov ided by the l a s t l e t t e r sequence on the s tandard tab leaux of shape ( a , ( 3 ) , that . THEOREM (1 .2 .3 ) To conitnuct tkz matnizzi, nzpnziznting ( i - l , i ) in the. isiie.ducA.blz nzpnzizntotion6 [u] = [a , 6] 0& H , apply Theorem ( 1 . 2 . 2 ) . To com>tnucX, tkz matnix nzpnzt> znting - ( l ) plazz t h ( i ) 1 in tkz p,p zntny ifi thz Izttzn 1 appzam> in tkz tablzaux Ta o£ T y = ( T a , T S ) , p J P P P ' t"rl ( i i ) -1 in thz p,p zntny i i tkz Izttzn 1 appzam> in thz tablzaux T 6 ol T u = ( T a , T 6 ) , P 0 P P P ( i i i ) zznoi ztiewhznz . Thus, analogous to Theorem ( 1 . 2 . 1 ) , the i r r e d u c i b l e r e p r e s e n t a t i o n s o f ff" can be de f i ned i n d u c t i v e l y and i n terms of generators and r e l a t i o n s . 13 CHAPTER 2 REPRESENTATION OF THE GENERIC RING CORRESPONDING TO A COXETER SYSTEM OF CLASSICAL TYPE 2 . 1 . HECKE ALGEBRAS AND FINITE GROUPS WITH BN-PAIRS L e t B be a subgroup of a f i n i t e group G and l e t k be a f i e l d o f c h a r a c t e r i s t i c z e r o . Set e = | B | ^ £ b i n the group a l geb ra kG . beB Then e a f f o r d s the 1-representa t ion . 1^ of B and the l e f t kG-module G kGe a f f o r d s the induced r e p r e s e n t a t i o n 1 D . o DEFINITION (2 .1 .1 ) Tke. Hzcke. aJLazbna. \(G, B) -a tke. buhaJLaohna. oft kG g-ivzn by e (kG) e . TheHecke a l geb ra ac t s on kGe by r i g h t m u l t i p l i c a t i o n and the a c t i o n de f i ne s an isomorphism between ^(CI* B) and the endomorphism a l g e b r a END^ (,(kGe) . The double coset sums £ x , g e G, form a b a s i s f o r xeBgB H ^ G , B) (see [14 ] , Lemma 84) . I n t h i s t h e s i s we w i l l be concerned w i t h f i n i t e groups G w i t h 'M-pCLUU, o f subgroups (B, N) s a t i s f y i n g the axioms of [ 17 ] . Then H = B A N i s a normal subgroup of N and the WzyZ gfioup W = N/H. has a p r e s e n t a t i o n w i t h a s e t o f d i s t i n g u i s h e d i n v o l u t i o n a r y generators R and d e f i n i n g r e l a t i o n s r = 1 , r e R , (2 .1 .2 ) ( r s . . . ) n = ( s r . . . ) n , r , s E R, r ? s , r s r s 14 where n i s the order of rs i n W and (xy ...) denotes a product r s J m r of a l t e r n a t i n g x ' s and y ' s w i t h m f a c t o r s . The p a i r (W, R) i s c a l l e d a CoxeteA AyAtem. The group G i s s a i d to be o f type (W, R) . I f w ^ 1 e W, we denote by £(w) the l e a s t l e n g t h I of a l l exp ress ions (2 .1 .3 ) w = r ^ . . . r ^ , r^, . . . , r e R . (2 .1 .3 ) i s c a l l e d a reduced exp re s s i on f o r w i n R i f I = £(w) . There i s a b i j e c t i o n between the double cosets B \ G / B and the elements w e W r e s u l t i n g i n the Bfuxhat decompob-ition G = U BwB . weW The s t r u c t u r e of the Hecke a l geb ra H^(G, B) o f a f i n i t e group w i t h a BN-pair w i t h r espec t to a B o r e l subgroup B was shown i n [9] and [11] to be as f o l l o w s . THEOREM (2 .1 .4 ) H^(G, B) hoi, k-b<X6-6i {S w : w e W} wkzfld xeBwB with S 1 the Identity element. M u l t i p l i c a t i o n i& determined by the ftonmula S w S r = S w r ' r £ R ' £ ( w r ) > ^ w > > S w S r = q r S w r + ( q r " 1 ) S w ' r e R, Jl(wr) < £(w) iwhefie the { q r , r e R} one the Index pa/icmetefu (2 .1 .5 ) q r = |B : (B f l rBr) | . 15 Ton. any xndaced expftu&lon w = r.^ .-. . r Ion. w In R, w ^ 1 S S • • • S • r l vl ThuA H, (G, B) IA gmnfiato.d by {S , r e R} and Itai defining naloXiom, 2 S r = q r S l + ( ( l r " 1 ) r ' r e R ' (2 .1 .6 ) ( S S . . . ) = (S S . . . ) , r s n s r n r s r s wneAe n <c6 ah i.n ( 2 . 1 . 2 ) . r s Le t (W, R) be a Coxeter system and l e t {u^ , r .e R} be inde te rmina tes over k, chosen such tha t u = u i f and on ly i f r and r s 1 s are conjugate i n W . Le t D be the po l ynomia l r i n g D = k [ u r : r e R ] . Then there e x i s t s an a s s o c i a t i v e D-algebra Ol w i t h i d e n t i t y , f r e e b a s i s { a w , w e W} over D and m u l t i p l i c a t i o n determined by the formulas a a = a , r e R, w e W, £(wr) > £(w) , w r wr ' ( 2 .1 .7 ) a a = u a + (u - l ) a , r e R, w e W, £(wr) < £(w) , w r r w r r ' w ' ' ' . (see [ 2 ] , p.55) . The D-algebra CX i s c a l l e d the gznQJtLc. tvlng cor respond ing to the Coxeter system (W, R) . Analogous to Theorem ( 2 . 1 . 4 ) , the gene r i c r i n g CX has a p r e s e n t a t i o n w i t h generators {a ' , r e R} and r e l a t i o n s 2 a = u 1 + (u - l ) a , r e R , r r r r ( 2 .1 .8 ) ( a r a g . . . ) n = ( a g a . . . ) , r , s e R, r ^ s r s r s 16 w i t h n as i n (2.1.2). r s The Hecke a lgeb ra IL^(G, B) can be compared w i t h the group a l g e b r a kW as f o l l o w s . Let L be any f i e l d of c h a r a c t e r i s t i c zero and <J> : D — > L a homomorphism. Cons ider L as a D-module by s e t t i n g d-X = o>(d)X , d e D, X e L . Then the s p e c i a l i z e d a l geb ra (2.1.9) c^ L = L 8 Da i s an a lgeb ra over L w i t h b a s i s {a = 1 ® a } , generators {a , , r e R} w<j> w r<p and d e f i n i n g r e l a t i o n s obta ined from (2.1.8) by app l y i ng <f>. Thus, i f <J) : D —> k i s de f ined by <|>(ur) = 9 r » r e R, q r the index parameters (2.1.5), then (2.1.10) ^ V G , B ) w h i l e i f (j) : D —-> k i s de f i ned by <j> (y ) = 1, f o r a l l r e R, then o o r (2.1.11) 01 - kW . <j,o,k We say the Coxeter system (W, R) i s of claM&-ic.al ttjpz i f W i s of type A , B , n>2 or D , n > 4 . In t h i s chap te r , we w i l l determine • ' r n n — n ' — the i r r e d u c i b l e r ep resen t a t i ons o f the gener i c r i n g cor respond ing to a Coxe te r system of c l a s s i c a l type and by means of the appropr i a t e s p e c i a l i z e d a lgeb ras the i r r e d u c i b l e r ep resen t a t i ons of the Hecke a lgebras ^ ( G s B ) ° f groups w i t h BN-pair o f c l a s s i c a l t ype . 17 2 .2 . THE REPRESENTATIONS OF 0\ K ( B n ) I f a Coxeter system (W, R) i s of type B^, n ^ 2 , W(BQ) i s i somorph ic to the Hyperoc tohedra l group, the group of s ighed permutat ions on n - l e t t e r s (see 1.2 ) . Thus w ( B n ) has a p r e s e n t a t i o n w i t h generators R = {w.^ , W r } where w^ = ( i - l , i ) , i = 2, n , and = - ( 1 ) , the f i r s t s i g n change and r e l a t i o n s W 1 W 2 W 1 W 2 = W 2 W 1 W 2 W 1 ' w .w_ ,w. = w. , ,w.w, , . , i = 2, . . . , n-1 : l l + l l l + l l l + l ' ' ' ' • -. w.w. = w.w. , | i - i > 1 i J J i (see [ 4 ] ) . Furthermore the s e t of generators R i s p a r t i t i o n e d i n t o 2 se t s under c o n j u g a t i o n ; namely, w^ i s conjugate to w^ . f o r i , j >_ 2 w h i l e the nega t i ve one-cyc le w^ i s not conjugate to any w.., j >_2 . For the Coxeter system (W(B n ) , R) taken as above, we take the gener i c r i n g Q (B n ) to be de f i ned over the po l ynomia l r i n g D = Q [ x , y ] , x , y indete rminates over Q . I t has a p r e s e n t a t i o n w i t h generators a = a. , w. e R, and r e l a t i o n s w. l ' 1 ' i BI a* = y l + ( y - l ) a i , 2 B2 a^ = x l + (x- l ) a^ , i = 2, n ; B3 a i a 2 a l a 2 = a 2 a l a 2 a l ' 18 B4 a i a i + l a i = a i + l a i a i + l ' i = 2, . . . . n-1 ; B5 i . a . = a . a . , I i - j I > 1 . We depart from the no t a t i ons o f . ( 2 . 1 ) s t r i c t l y f o r n o t a t i o n a l conven ience , i . e . , we s w i t c h from Q ( u ^ , U 2 ) to Q ( X , y) to avo id c a r r y i n g around s u b s c r i p t s . We now cons t ruc t f o r each double p a r t i t i o n (y) = ( a , g) o f n , K n >_ 2, a K- represen ta t ion of (B n ) = K § Q (B^), K = Q ( x , y) . The method i n v o l v e s the c o n s t r u c t i o n of f^ x mat r i ces over K f o r each of K the generators a^ of CX (B n ) i n a manner analogous to the c o n s t r u c t i o n of the mat r i ces of the t r a n s p o s i t i o n s ( i - 1 , i ) f o r the ou te r product r e p r e s e n t a t i o n [ a ] *[6] o f ' S . For any i n t e g e r k, l e t k A (k , y ) = x y + l . Denote by M(k, y) the 2 x 2 m a t r i x (2 .2 .1 ) M(k, y) = A (k , y) (x-1) A(k+1, y) k { x A ( k - l , y) x y ( x - l ) J Then t r a c e M(k, y) = (x-1) , det M(k, y) = -x , so the c h a r a c t e r i s t i c po l ynomia l of M(k, y) g i ves (2 .2 .2 ) M(k, y) = x i + (x- l )M(k , y ) , I the 2 x 2 i d e n t i t y m a t r i x . 19 For k >_ 1, l e t k-1 . " A ( k , -1) = I x 1 . i=0 Denote by M(k, - 1 ) , k _> 2, the 2 x 2 m a t r i x (2 .2 .3 ) M(k, -1) = A (k , -1) -1 x A ( k - l , 1) A(k+1, -1) } As M(k, -1) i s obta ined from M(k, y) by s e t t i n g y = -1 , (2 .2 .1 ) shows (2 .2 .4 ) M(k, - l ) 2 = x l + (x- l )M(k , -1) Denote by D (z , w) the 2 x 2 d i agona l m a t r i x D (z , w) = z 0 0 w Then (2 .2 .5 ) D ( z , -1) = z l + ( z - l ) D ( z , -1) In what f o l l o w s , we employ the d e f i n i t i o n s and n o t a t i o n s of (1.1) i n regards to double p a r t i t i o n s , s tandard t ab l eaux , and a x i a l d i s t a n c e . DEFINITION (2 .2 .6 ) Let (u) = (a , B) be a double, partition o f n and let T y , . . . , T y , f = f y be. the. ondenlng ofa the. &tandand tableaux o£ &hape (y) according to the l a i t letteA sequence. Comt/iuet f x f matfu.c.et> M y ( i ) , i = 1, n , oven. K = Q(x, y) Oi, {.ollotti, : . • -(1) Con&tAuct M y ( l ) by placing 20 ( i ) y in thz p , p t h zntny ±i tkz Izttzn 1 appzau In ojj T y = ( T a , T B ) , P P P t i l ft ~ ( i i ) -1 In thz p ,p zntny ii tkz Izttzn 1 appzau In I T oi T y = ( T a , T e ) , P P P ' ( i i i ) zznoA zJUzwhznz . Conitnuct M ^ i ) , i = 2, n , by placing (1) x in tkz p ,p zntny ii tkz Zzttzu i - l and 1 appzan in tkz 6amz now oi T 0 1 on T 8 oi T y = Vf1, T 6 ) , -u P p 0 p p ' p ' ( i i ) -1 in tkz p , p t h zntny ii tkz Izttzm, i - l and 1 appzan in tkz Aamz column oi T 0 1 on T 8 oi T y ; 0 P P 0 p ( i i i ) The matnix M ( k , -1) -cn t h e p , p t h , p , q t h , q , p t n and q , q t h zntnizi zonnzhponding to thz tablzaux T y and Tq1 wheAe (a ) p < q , ( i - l , i ) T y = T y and thz Izttzm, i - l and i . p q — - — appzan zitkzn both in T 0 1 on T 6 oi T y , p P 0 p (b) k ^ tkz axial dUtanzz inom i to i - l -cn T y ; - P t i l t t l t i l ( i v ) Thz matnix M ( k , y ) in tkz p , p , p , q , q ,p aiid t i l q , q zntnizi,- connzi>ponding to tkz tablzaux and T y •dxznz (a) p < q , ( i - l , i ) T y = T y and thz Izttzm i - l and i. a p p e a r -cn diUznznt tablzaux oi T y , (b) k -ed t h e axta£ dUtancz inom 1 to i - l -cn T y ; ( v ) zeAo^ zlszwkznz. 21 Le t denote the f r e e Q-module generated by t^, . . . , t^ , f = f y cor respond ing to the standard tab leaux T^, o f shape (y) ordered acco rd ing to the l a s t l e t t e r sequence. For any f i e l d L of c h a r a c t e r i s t i c zero se t V L = V 8 L . The corresponding b a s i s elements y y . . t . 8 1 o f V L w i l l be denoted s imply by t , . Set K = Q(x, y ) . Def ine i . y i l i n e a r opera tors Z£, i = 1, . . . , n , on such that the m a t r i x o f ZY w i t h r espec t to the b a s i s { t 1 > t f } o f i s g i ven by M y ( i ) . THEOREM (2.2.7) Let K = Q(x, y) and l e t < 3 l K ( B n ) denote the gmehJie King oft the. CoxeteA &y&tem (w(Bn), R) a6 beftone. Let (y) be a double pa/Ltitcon oft n, n >_ 2. Then the K-Lineat map 7 r y : G K ( B n ) > END (V^) defttned by ^ ( a ^ = lt> a fiepneAentatLon oft OA K(B n). PROOF : We need to show the r e l a t i o n s (B1-B5) are s a t i s f i e d w i t h Z^ i n p l a c e of a^. We argue by i n d u c t i o n on n . For n = 2 i t i s a case by-case v e r i f i c a t i o n . The double p a r t i t i o n s ((2), ( 0 ) ) , ((0), (2)), 2 2 ( (1) , (0)) and ( ( 0 ) , (1) ) are c l e a r l y seen to y i e l d the w e l l known one- . d imens iona l r ep r e sen t a t i ons o f Q\ ( B 2 ) ( [ 6 ] , 1 0 ) . For the double p a r t i t i o n LU LH ( ( 1 ) , (1)) there are two standard t ab l eaux , and . From (2.2.6) (Tf m M ( U ) , ( ! ) ) ( ! ) = D ( y > _ 1 } a n d M ( d > , ( 1 ^ ( 2 ) = M(0, y ) . D i r e c t computat ion v e r i f i e s the r e l a t i o n M(0, y ) D(y, -1)M(0, y ) D(y, -1) = D(y, -1)M(0, y ) D(y, -1)M(0, y ) . 22 Thus the r e l a t i o n s (B1-B3) are s a t i s f i e d w i t h and Z^ i n p l a c e o f a^ and a 2 by the above computat ion , (2 .2 .2 ) and ( 2 . 2 . 5 ) . Now l e t (y) = (y^» •••> U g ) be a double p a r t i t i o n of n . D e l e t i o n of the l e t t e r n_ from a s tandard tab leaux au toma t i c l y y i e l d s a s tandard tab leaux i n v o l v i n g n-1 l e t t e r s . In f a c t d e l e t i o n of n_ from a l l s tandard tab leaux hav ing n_ at the end of the i f c ^ row w i l l y i e l d a l l s tandard t ab leaux of shape (y^, y^-1 , u ) . Denot ing t h i s p a r t i t i o n by (y^~) and us ing the f a c t tha t a l l s tandard tab leaux w i t h n i n th. tlx the i row precede a l l tab leaux w i t h i i i n the j row f o r i > j when ordered acco rd ing to the l a s t l e t t e r sequence, we have (2 .2 .8 ) V*" = . . . . . . . N • y ( y g - ) CVj-) and the corresponding ma t r i x b l o c k form M M ( i ) = M ( i ) + . . . . + M ( i ) , i < n as by ( 2 . 2 . 6 ) , M P ( i ) depends on ly on the l e t t e r s i-1 and jL . I t i s understood tha t (y^-) i s taken to equa l zero i f n cannot appear i n the th i row and the above summation, here and e lsewhere , w i l l be taken over those (y.j-) which are non-zero . By the i n d u c t i o n hypothes i s i t t h e r e f o r e s u f f i c e s to check the r e l a t i o n s (B1-B5) as they p e r t a i n to Z W . n The ma t r i x M y (n ) from (2 .2 .6 ) i s composed of the mat r i ces M(k, y) and M(k, -1) cen t red about the d i agona l a long w i t h d i agona l e n t r i e s x and - 1 . Thus the r e l a t i o n ( Z y ) 2 = x i + ( x - l ) Z U n n 23 f o l l o w s from ( 2 . 2 . 2 ) , (2 .2 .4 ) and (2 .2 .5 ) . Let V. . denote the subspace of w i t h b a s i s t ^ ' ^ , . . . . tg"'-1 , cor responding to the s tandard tab leaux o f shape (u) w i t h the l e t t e r _n appear ing i n the i row and n-1 appear ing i n the j row, the o rde r i ng of the b a s i s taken acco rd ing to the l a s t l e t t e r sequence. Then = © V. . , 1.3 the summation taken over a l l a l l o w a b l e i , j such tha t i i appears i n row i and n-1 appears i n row j , and t h i s decomposi t ion i s c o n s i s t e n t w i t h the l a s t l e t t e r sequence arrangement of the b a s i s of . Thus, whenever n and n-1 are i n d i s t i n c t rows and columns, we have V. . - V . . as - i » J J , i K 0\ (B^^)-modules f o r ri appear ing i n row i , n-1 appear ing i n row j , where W(B _) = <w n, . . . , w „> . n—L l n—z Suppose f i r s t tha t n and n-1 appear i n d i s t i n c t rows and columns, i n the tab leaux cor respond ing to V. . ; n i n row i , n-1 i n row j . Then i i appears i n row j and n-1 appears i n row i i n the tab leaux cor respond ing to V. . and the map t 1 ' " ' > t ^ ' 1 , p = 1, . . . . s . .=s. . , J » i P P V ' i , j 3,1 ' ~ K g i ves an isomorphism V. . - V. . as (B „)-modules, as the c o n f i g u r a t i o n v i , j j , i n-2' of the f i r s t n-2 l e t t e r s i n the tab leaux cor responding to t^'^ *- s t n e same as the c o n f i g u r a t i o n of the f i r s t n-2 l e t t e r s i n the tab leaux c o r r e s -ponding to t- '* 1 . In p a r t i c u l a r the ma t r i x of Z? , k = 1, . . n - 2 , P K on V . , f V. , i s 24 where i s the m a t r i x of ^ ( a^ ) on V i . On the o ther hand, " the m a t r i x . of z£ on K t p , J © K t j j ' 1 i s , by ( 2 . 2 . 6 ) , M(£, y) o r M(£, - 1 ) , £ the a x i a l d i s t ance from n to n-1 . Thus the m a t r i x of Z y on — . ... n V. . © V. . i s ^ l 1 a 2 2 I where (a . . ) = M(£, y) or M(£, - 1 ) , I the s . . * s. . i d e n t i t y m a t r i x . Then S S. = S, S f o r k = 1, n-2 , and (B5) h o l d s . The on l y n k k n o ther p o s s i b i l i t y i s when n and n-1 appear i n the same row o r column o f the tab leaux cor responding to ^. But i n t h i s case the m a t r i x of Z^ on V . . i s the s c a l a r ma t r i x x i or -I by (2 .2 .6 ) and thus commutes w i t h Z n ° n V k - 1 . , n-2*. This proves (B5) f o r a l l c a se s . To check the r e l a t i o n (B4) , we cons ider the r e s t r i c t i o n o f Z y , n—1 and Z y to subspaces w i t h b a s i s {t . } cor responding to a l l t ab leaux TV n l x hav ing a f i x e d arrangement of the f i r s t n-3 l e t t e r s and a l l p o s s i b l e rearrangements of the l e t t e r s .. n-2, n-1 and n_ . Let V = « V P P P denote the cor respond ing decomposi t ion of , the o r d e r i n g of the b a s i s 25 o f each V p taken w i t h respec t to the l a s t l e t t e r sequence. Then each V i s i n v a r i a n t under Z y , and Z y and i t s u f f i c e s to check (B4) f o r P n-1 n the va r i ous p o s s i b l e arrangements of the l a s t 3 l e t t e r s i n a case by case b a s i s . In what f o l l o w s , M p ( i ) , i = n o r n-1, w i l l denote the m a t r i x of Z P on V . 1 P Case 1 - the l e t t e r s n-2, 'n-1 and n i n the same row o r column. Then V p i s one d imens iona l and M p ( n ) = M p ( n - 1 ) = x o r -1 by ( 2 . 2 . 6 ) . Thus M p ( n ) M p ( n - l ) M p ( n ) M p ( n - l ) M p ( n ) M p ( n-1) and (B4) i s s a t i s f i e d . Case 2 - the l e t t e r s n-2, n-1 and n i n two ad jacent rows and two ad jacent columns o f the same d iagram. Then V p i s two-dimensional w i t h b a s i s elements corresponding to tab leaux where the c o n f i g u r a t i o n of the l a s t 3 l e t t e r s i s (a) n-2 n-1 > n-1 n n-2 n or (b) n-2 n-1 n n-2 n n-1 •ordered accord ing to the l a s t l e t t e r sequence. Then by ( 2 . 2 . 6 ) , M p ( n - 1 ) = M(2, -1) and M p ( n ) = D(x, -1) i n (a) and M p ( n - l ) = D(x , -1) and M p ( n ) = M(2, -1) i n (b) . Thus (B4) i s s a t i s f i e d i n b o t h cases by d i r e c t v e r i f i c a t i o n of the r e l a t i o n 26 M(2, - l ) D ( x , -1)M(2, -1) = D(x , -1)M(2, - l ) D ( x , -1) . Case 3 - the l e t t e r s n-2, n-1, and n. i n two rows and three columns or three rows and two columns. Then V i s three d imens iona l w i t h P b a s i s elements cor responding to tab leaux where the c o n f i g u r a t i o n of the l a s t 3 l e t t e r s i s one of n-2 n-1 2 i n-2 n 3 (a) 1 r r n-1 r n-2 n-1 n (b) 1 n-2 2 n-1 3 n n-1 n —I n-2 n —J ' n-2 n-l-J (c) 1 n-2 2 n-2 3 n-1 I I I n-1 n r n r n-1 | n n-2 (d) 1 n-2 2 n-1 n-1 J , n-2 -J n 3 n J I n-2 ordered acco rd ing to the l a s t l e t t e r sequence. I f we se t a l l a 1 2 a 2 1 a 2 2 b l l b 1 2 b 2 1 b 2 2 we have, by case ( 2 . 2 . 6 ) , i n case ( a ) , M ( n - 1 ) = S.. and M ( n ) = S„ where p i p 2 ( a _ ) = M ( d 1 , e ) , (b ) = M ( d 2 , e ) , E = y or - 1 , and c = x . Here d x i s the a x i a l d i s t ance from n-1 to n-2 i n 2 and d 2 i s t h e , a x i a l 27 d i s t a n c e from ri to n-1 i n 1 so tha t - d^ + 1. In case. ( b ) , M ( n-1 ) = S 0 and M ( n ) = S, w i t h the same e n t r i e s i n S. as i n case P 1 P 1 i (a) ., The a n a l y s i s of (c) and (d) i s s i m i l a r except tha t now c = -1 i n and S,,. Thus i n a l l cases (B4) i s s a t i s f i e d by LEMMA (2 .2 .9 ) Lzt S 2 b<L OA abovd and l e t (a....) = M(a , y ) , ( b ± j ) = M(b, y) . Then iofi \ ( i ) c = x and b-1 = j ( i i ) c = -1 and b+1 = we kavz S;js2s1 = S ^ s , ^ PROOF : Observe tha t S ^ S j . . = i f f . ' 1 1 ^ " l l ' " a l l u 1 2 " 2 1 (2) c a 2 2 ( c - a 2 2 ) = b ^ ^ , (3) V 3 l l b 22 + C ( b l l - a l l ) ] " 0 , • J , (*) a i j t a l l b 2 2 + c ( a 2 2 ~ b 2 2 ) ] = 0 '  1 * i > (5) a l l b 2 2 ( b 2 2 ' a l l } = C ( a 1 2 a 2 1 ~ b 1 2 b 2 1 ) ' For ( 1 ) , c b n ( c - b n ) - a u b 1 2 b 2 1 (2 .2 .10) = ^—^ A(b , y ) 2 A ( a , y) 2 A (.a, y ) [ c A (b , y) - c ( x - l ) ] - x A ( b - l , y ) A ( b + l , y) 28 Now f o r c = x , and f o r c = - 1 , c. M b , y ) - c ( x - l ) = x M b + i , y ) c A ( b , y) - c ( x - l ) = x M b - 1 , y) Hence (2 .2 .10) equals zero f o r c = x , b+1 = a and f o r c = - 1 , b-1 = a . The r e l a t i o n (2) i s e n t i r e l y s i m i l a r . For (3) and (4) we have a n b 2 2 + c ( b n - a n ) (x-1) A ( a , y )A (b , y) x y ( x - l ) + c [ A ( a , y) - A ( b , y) ] and a n b 2 2 + c ( a 2 2 - b 2 2 ) (x-1) A ( a , y )A (b , y) xb y ( x - l ) + c [ x a y A ( b , y) - x b y A ( a , y ) ] But f o r c = x , b-1 = a and c = - 1 , b+1 = a , a b A ( a , y) - M b , y) = x y A ( b , y) - x yA ( a , y) = x y ( l - x ) . For ( 5 ) , f i r s t note the u s e f u l f a c t o r i z a t i o n x A ( a - l , y ) A ( a + l , y) A ( a , y ) 2 f x A ( b - l , z ) A ( b + l , z) } A (b , z ) 2 29 (x-1) ' A ( a , y ) 2 A ( b , z ) 2 a / b .' i \ 2 b , a , % 2 ] x yCx z + 1) - x z ( x y + 1) (2 .2 .11) = Now (2 .2 .12) But , T> 2 . a+b . . b a . (x-1) (x yz - l ) ( x z - x y) A ( a , y ) 2 A ( b , z ) 2 . 1 a l l b 2 2 ( b 2 2 ~ a l l > ( x - 1 ) 2 A(b , y ) 2 A ( a , y ) 2 b+1 b w x Y ~ x y 1 a+b 2 x y — 1 1 r b+1 b . - [x y - x y] b a x y - x y f o r c = x , b-1 = a and f o r c = - 1 , b+1 = a . So f o r bo th cases , (2 .2 .12 ) equa ls a 1 2 a 2 1 " b 1 2 b 2 1 us ing (2 .2 .11) w i t h z = y . Case 4 - the l e t t e r s n-2, n-1 and n i n three d i s t i n c t rows and three d i s t i n c t columns. Then V i s 6-dimensional w i t h b a s i s elements P cor respond ing to tab leaux where the c o n f i g u r a t i o n of the l a s t 3 l e t t e r s i s n-2 n-1 n n-1 — 1 , n-2 — I , n n-2 n-1 n n-2 n - 2 - I -n - 1 — I n-2 J 6 30 ordered accord ing to the l a s t l e t t e r sequence. Let S l " r a u a 1 2 a 2 1 a 2 2 b l l b 1 2 b 2 1 b 2 2 C l l C 1 2 C 21 C 2 2 r C l l * C 1 2 ' • b. S 2 = '11 ' / "12 • C 21 * C 22 * * . . . a n . a 1 2 . b 2 i ' ' b 2 2 * * 3 21 * a 2 2 Then i f a l l rows are i n the same diagram we have by ( 2 . 2 . 6 ) , M ( n-1) = S 1 P X and M ( n ) = S 2 where (a . . ) = M(d , - 1 ) , (b. .) = M ( d 2 , - 1 ) , and ( c ^ ) = M(d.j, - 1 ) . Here d^ i s the a x i a l d i s t a n c e from n-1 to n-2 i n 1 , d 2 i s the a x i a l * d i s t ance from n-1 to n-2 i n 3 and d^ i s the a x i a l d i s t ance from n-1 to n-2 i n 5 so tha t d^ + d^ = d 2 and a l l d^ >_ 2 . I f two rows are i n one diagram and the t h i r d i n the second d iagram, we assume, w i thout l o s s of g e n e r a l i t y , the lowest box to be i n the second d iagram. Superimposing the second.diagram upon the f i r s t aga in does not a l t e r the r e l a t i o n d^ + d^ = d 2 except t ha t now on ly d^ >^  2 . I n t h i s case M ( n - 1 ) = S. and M ( n ) = S„ , where now (a . . ) = M ( d , , -1) , p 1 P 2 1 J 1 ' ' ( b _ ) = M ( d 2 , y) and (<-..) = M ( d 3 , y) . Thus f o r both cases (B4) i s s a t i s f i e d by LEMMA (2 .2 .13) Let S 1 and S 2 be at, above, and l e t ( a . . ) = M(s , w ) , (b/,) = M(p, y ) , ( c _ ) = M ( t , z) with' s+t '= p . Then ftoh. ( i ) w = - 1 , y = z , s _> 2 , ( i i ) z = - 1 , y = w, t _> 2 , ( i i i ) w = y = z = - 1 , s , t , p _> 2 , 31 we have. s 1 s 2 s i = S 2 S i S 2 * PROOF : F i r s t , e i t h e r ( i ) o r ( i i ) c l e a r l y imply ( i i i ) . Now S 1 S 2 S 1 " S 2 S 1 S 2 i f f (1) (2) a, 0 [ c . . a . . +• b . . ( a . . - c . . ) ] = 0 , k ^ JI, i ^ i ; K . ,£ i i i i i i J J r i ' . ' c, „ [c. . a . . + b. . ( c . . - a . .)] = 0 , k f Si, i ± j ; K . , £ 1 1 11 1 1 J J i i ' • * J ' (3) b, ' [ c . a . . - c . . . b . . - a . . b . . ] = 0 , k 4 A, ± 4 j ; (4) a. . c . . (a . . - c. .) = b . . (c . . c . . - a . . a . . ) , i ^ j ; i i i i i i i i i i i j j i 13 j i ' (5) a . . b . . ( a . . - b . . ) - c . . ( b . . b . . - a . . a . . ) , i 4 j ; i i J] i i JJ - J i i J J i i j J i J (6) c. .b . . ( c . . - b . .) = a . . (b. .b .. - c . . c . . ) , i f j ; i i JJ i i JJ JJ i J J i i J J i and as these r e l a t i o n s are symmetric i n the ( a . . ) and ( c . . ) , i t s u f f i c e s i J i J to prove the lemma f o r ( i ) . For ( 1 ) , ( 2 ) , and (3 ) , we observe tha t E1 = A ( s , -1) + x S A ( t , y) - A ( p , y) = 0 , E 2 = x t y A ( s , -1) - A ( t , y) + A (p , y) = 0 . A l s o s e t D = ^ A (P , y ) A ( t , y)A.(s, -1) We then o b t a i n 32 C l l a l l + b l l ( a 2 2 - ' c l l > ' . . = " D E 1 = 0 » c 2 2 a 2 2 + b 2 2 ( a n - c 2 2 ) = - x P y D E 2 = 0 , C l l a l l + b l l ( c 2 2 " a l l ) . - DE 2 = 0 , C 2 2 a 2 2 + b 2 2 ( c l l " a 2 2 ) = x P y D E i = 0 » C l l a 2 2 " c l l b 2 2 " a 2 2 b l l = ~ x S d E 2 '* ' 0 ' c 2 2 a n - c 2 2 b u - a n b 2 2 = - x t y D E 1 = 0 For (4) we have ^ i - a n c 1 1 ( a 1 1 - c u ) = b ~ a 2 2 c 2 2 ( a 2 2 " C 2 2 } _ ( x - l ) 2 ( x S + t y + lXxV + xS) • • A ( s , - l ) 2 A ( t , y ) 2 ~ 1 2 C 2 1 - a 1 2 a 2 1 from ( 2 . 2 . 1 1 ) , s e t t i n g a = t , b = s , z = -1 . The r e l a t i o n s (5) and (6) are handled i n an e n t i r e l y s i m i l a r manner. Th is completes the proof of the lemma and the proof o f the theorem. THEOREM (2 .2 .14) Let K be as beftore. The representations T r y oft C l K ( B n ) are i r r e d u c i b l e , pdirwise inequivalent and are, ap to. isomorp'riism, a complete bet oft' I r r e d u c i b l e , inequivaZent representations oft 01 ( B R ) . In p a r t i c u l a r K is a & flitting ftleld ftor Q K ( B ^ ) . 33 K PROOF : By i n d u c t i o n on n . For the r ep resen t a t i ons o f 01 (S^) i t i s a mat ter o f d i r e c t computat ion to check i r r e d u c i b i l i t y and i n e q u i v a l e n c e . C o n s i d e r a t i o n of degrees shows a complete s e t of i n e q u i v a l e n t r ep r e sen t a t i ons i s o b t a i n e d . For Q (B n ) we employ the decompos i t ion (2 .2 .8 ) a f f o rded by the l a s t l e t t e r sequence and the p o s i t i o n o f the l e t t e r n i n a s tandard t ab l eaux . Le t (y) be a double p a r t i t i o n o f n . The f3 K (B Q )-module i s e i t h e r i r r e d u c i b l e o r (2 .2 .8 ) i s the decomposi t ion o f i n t o i r r e d u -c i b l e i n e q u i v a l e n t CI (B R ) components, i n e q u i v a l e n t because each of the double p a r t i t i o n s (y^-) of n-1 i s d i s t i n c t . But f o r each p a i r (y^-), ( y . - ) , i 5* j , there e x i s t s a tab leaux T y w i t h n i n row i , n-1 i n row j and (n-1, n ) T y = T y a tab leaux w i t h n i n row j and n-1 i n row i . Thus the a c t i o n of i r y ( a ) does not decompose w i t h respec t to the ( y r ) n K V , i = 1, s+r . Hence V i s i r r e d u c i b l e . Furthermore the y double p a r t i t i o n (y) i s complete ly determined by the se t o f double p a r t i -t i o n s (y^-), i = 1, s+r , o f n-1 . Thus by the i n d u c t i o n hypo thes i s as (B )-modules i m p l i e s (y) = ( y ' ) . From ([19]). , I (fV) = 2 n n ! , (y) a double p a r t i t i o n of n . Thus Cl (B n ) i s 11 y semis imple and as the i r y are de f ined over K, K i s a s p l i t t i n g f i e l d f o r K Q[ ( B n )• This completes the p r o o f . I t i s c l e a r tha t the above r ep resen t a t i ons o f the gene r i c r i n g K y i e l d r ep resen t a t i ons of a wide v a r i e t y of s p e c i a l i z e d a lgebras o f Q ( B n ) . S p e c i f i c l y , se t n-1 . P (B n ) = x TT ( x 1 + y ) ( x X y + 1) (1 + . . . + x 1 ) e D = Q[x, y] . i=0 34 COROLLARY (2 .2 .15 ) Let L be, any ftleld oft characteristic zero, <j> : D — > L a komomoA-plvUm 6uck that <|>(P(Bn)) ? 0 . let (y) be a double panjtition oft n and l e t z y denote Pie llnean. opejuxtoh, on obtained by the 6ubititwtlon x —>. <))(x), y —> <j>(y) In the entity oft M y ( i ) . Then Zi<j) ^ we££ defttned and the L-lineaA map ^ T : ° L T ( B ) —> E N D (vL) <j>,L <j),L n y deftlned by i r y T ( a . ) = Z Y ^ a - tep^eientatlon oft 0{ (B ) . Trie 9 , L i i<j) u <p»L n representations { i T y } a t e . a complete set oft tn/ieductble, inequtvalent <P»L ^ e p ^ e n A i t c o R i Ojj CTf (B ) . PROOF : I f <j>(P(Bn)) ^ 0 , (2 .2 .1 ) and (2 .2 .3 ) show the mat r i ces M(k, y) and M(k, -1) are w e l l de f i ned under the s u b s t i t u t i o n x —> <KX)> y — > 4>(y) f o r -n+1 <_ k <_ n-1 . I t i s c l e a r from the d e f i n i t i o n tha t a x i a l d i s t ance i n a Young diagram corresponding to a double p a r t i t i o n o f n cannot exceed n-1 i n abso lu te v a l u e . Thus by ( 2 . 2 . 6 ) , Z Y i s w e l l de f i ned f o r a l l i . As 0 l . _ has a p r e s e n t a t i o n w i t h generators {a. ,} and r e l a t i o n s obta ined from <p,L 19 (B1-B5) by app l y i ng ij>, the p roo fs of Theorem (2 .2 .4 ) and (2 .2 .14) c a r r y over to t h i s case . Le t A be a separab le a lgeb ra over a f i e l d L and l e t L be an a l g e b r a i c c l o s u r e of L. Def ine the numer i ca l i n v a r i a n t s of A to be the se t of i n t e g e r s {n^} such that A^ i s i somorph ic to a d i r e c t sum of t o t a l m a t r i x a lgebras A L - <& M ^ d ) . 35 Thus for <J> defined as i n Corollary (2.2.15) the algebras Ol K(B n) and 01 have the same numerical invariants. In particular Corollary (2.2.15) <p,L, gives the well known result (see [1]) that for G a f i n i t e group with BN-pair with Coxeter system (W, R) of type ^ n » HQ(G, B) - QW . Indeed i n ([1]) this i s shown to be the case for a l l Coxeter system with the possible exception of (W, R) of type . Finally, we remark that, for the specialization <f>o : D —> Q defined by <f> (x) = d> (y) = 1, the representations { i r y n} are the irreducible representations of W(Bm) given by Theorem.,(1.2.3) . 36 2.3. THE REPRESENTATIONS OF 0\K(A ) AND OtK(D ) n n We now obtain the representations of the generic ring of a Coxeter system of type A and D . J J v n n If (W, R) i s a Coxeter system of type A ,,• W(A ,) i s n—l n—1 isomorphic to the symmetric group S^  and we take the set R to be {w2> ...» w^ } where wi = ( i - l , i ) , i = 2, n. We take the generic ring 0 (^ n_^ ) t o be defined over the polynomial ring D = Q[x]. I t has a presentation with generators a = a., i = 2, n, and relations i (B2, B4, B5). Set K = Q(x). The. representations, of OL 0 ^ ^ ) = Cl ( A ^ ) § D K are readily obtained from the results of the previous section. As the matrices M(k, -1) are defined i n Q(x), (2.2.6) shows the matrices M ( a , (°))(i)j i = 2 n, are defined i n Q(x) and Z^ *' ( 0 ) ) can be regarded as a linear operator on ,„>.•. Thus .(a, (0)) THEOREM (2.3.1) let a be. a. pantition oi n, n _> 2 and K = Q(x) The. Y.-linean. map * ° ! ^ K ( A n - l > ~> END<V(a, (0))) defined by 7 r a(a i) = z£ a' ^ \ i = 2, n, it> a nepnebentation of K a CI ( A n _ i ) •• The ne.ph.eAentatiom, {TT } , one. a complete&et ol ijoteducihle, K inequivalent nepneA entatio ni ofi 01 ( A n - 1 ) • 37 PROOF : Theorem (2 .2 .7 ) shows the {IT } are r e p r e s e n t a t i o n s of 0 ( A Q ^) . I r r e d u c i b i l i t y and i nequ i va l ence f o l l ows from Theorem (2 .2 .14) as the m a t r i x of Z , a ' on V , i s the s c a l a r m a t r i x y l . As (see [19]) 1 (a , (1)) £ ( f a ) 2 = n ! , a a p a r t i t i o n of n , a the {TT } are a complete se t o f i n e q u i v a l e n t r ep r e sen t a t i ons and are a b s o l u t e l y i r r e d u c i b l e . The r ep re sen t a t i ons of the s p e c i a l i z e d a lgebras are handled e n t i r e l y analogous to C o r o l l a r y ( 2 . 2 . 1 5 ) . Set n . P(A ) = x I (1 + . . . + x 1 ) . n i = l Then from the above and C o r o l l a r y (2 .2 .15) we have COROLLARY (2 .3 .2 ) Lzt L be. any iletd oi ckanacXeJuAtia zzno, <f> : D = Q[x] — > L a homomonphtAm 6uch that d>(P(A )) ^ 0 . Thzn ion. (a) n a partition oi n >_ 2, thz tinzan opznatonA z ^ ' i = 2, n , an.z iA)ztt dzi-inzd and thz L-linzah. mapi, % , L : ^ , L ( A n - l > ~ > E N D < , (0))> dzilnzd by Aa. J = z f " ' u> a KzpKZAzntatlon oi Ot T (A .) . • ^ ( p j L i f p n}> u 9>L n—1 Thz {TT° L> anz a aomplztz 6Zt oi ihJizduziblz, inzqvuivatznt n.zpn.z6zntatlom> 38 K Thus f o r <Ji as above, the a lgebras Cl (A ) and OL . (A ) n <p, L n have the same numer i ca l i n v a r i a n t s . We remark tha t f o r the s p e c i a l i z a t i o n x —> 1 the d e f i n i t i o n s o f the ma t r i c es M(k, -1) shows the semi-normal m a t r i x r e p r e s e n t a t i o n of i s ob ta ined (see Theorem ( 1 . 2 . 1 ) ) . I f (W, R) i s a Coxeter system of type D^, n _> 4 , w ( D n ) c a n be regarded as a subgroup of index 2 i n W(B ) ; W(D ) a c t i n g on an n or thonormal b a s i s o f R n by means o f permutat ions and even s i g n changes. A s e t o f d i s t i n g u i s h e d generators f o r W(D n) can be ob ta ined from the s e t {w^ W r } of W(B n) g i ven i n s e c t i o n (2.1) by s e t t i n g w^ = W j W 2 w ^ and t a k i n g . t h e s e t R to be (w. , w 2 , •• •, w } (see [4]) . L e t <j> : Q[x, y ] — > Q[x] be d e f i n e d by <{>(y) = 1. Then the s p e c i a l i z e d r i n g Gf, _ r ,(B ) has b a s i s {a w c W(B )} w i t h r e l a t i o n s 2 obta ined by app l y ing <f> to (B1-B5). In p a r t i c u l a r ( ^ ^ . ~ Set l(j) W^W^ ^ l } ) As w-w.w, i s reduced i n (W(B ) , R) we have a , = a - , a O J a . . , by 1 2 1 n ' w^w2w^4> l(j) 2(j> lej) (2 .1 .8 ) • App l y i ng <f> to (B1-B5) i t i s r e a d i l y seen that — 2 — B ' l = x l + (x- l ) a 1 ( ^ , B'2 al<j>a3<i>alcj>. a3<j)al^a3<!) . ' 39 As any reduced exp ress ion of w ^ 1 e WCD )^ i n the generators {w 1 9 . . . » w }^ i s a reduced exp ress ion f o r w i n the generators {wp W r ) of W(B n ) , the r e l a t i o n s ( B ' l -B ' 3 ) show the s u b r i n g of . 0\ «r i ( B ) generated by {a, •, a 0 j , . . . . a , } has f r e e b a s i s <p,Q[x] n ' 6 ^ 1<J)' 2<j>' ' ncj> {a , , w e W(D )} . As a l l the generators {w.. , w 0 , w } are conjugate w<p n .L z n i n W(D ) , the subr ing generated by { a 1 i 5 a 0 , , a ,} i s i somorph ic to n l ? c^p nip the gene r i c r i n g of a Coxeter system of type n ^ 4. Denote t h i s sub r i ng by O(D n >. Thus the r ep resen t a t i ons of CX (B ) g iven by C o r o l l a r y (2 .2 .15 ) K prov ide us w i t h r ep resen t a t i ons o f C\ ( D n ) . Young [19] showed the r e s t r i c t i o n s o f the r ep resen t a t i ons o f W(B f l) to w ( ° n ) co r respond ing to a double p a r t i t i o n (a , g) o f n remain i r r e d u c i b l e i f (a) ^ (g) and decomposes i n t o two i r r e d u c i b l e components when (a) = ( g ) . Ue show tha t t h i s ho lds t rue i n a " g e n e r i c " sense. R e c a l l that a s tandard tab leaux T f o r the double p a r t i t i o n a. 8 ( a , g) of n i s an ordered p a i r T = (T , T ) . Then the tab leaux g ct T* = (T , T ) i s a s tandard tab leaux of shape (g , a ) , c a l l e d the conjugate tableaux o f T. Moreover the map T —> T* i s a b i j e c t i o n from the s tandard tab leaux of shape (a , g) to the standard tab leaux of shape ( g , a ) . Take (a) ^ (g ) . I f 1^, T , . . . , T , . . . , T f , f = f a ' ^ , i s the arrangement of the s tandard tab leaux of shape ( a , g) acco rd ing to the l a s t l e t t e r sequence, order the tab leaux of shape (g , a) acco rd ing to the scheme; T* 40 precedes I* i f T p precedes i n the l a s t l e t t e r sequence. C a l l t h i s the conjugate, ordering of the tab leaux of shape (3 , a ) . Let I* denote the n * n m a t r i x n I* = n 1 . . . 0 LEMMA (2 .3 .3 ) let M " ' e ( a ) denote, the matrix oft V * ' B ( a ) with respect to the basis {t±} oft ^ ordered according to the last letter sequence, a e 01K(D )• Then i * M ° ' ' B ( a ) l * , f = f a , B , Jis tlxe matrix oft T r ? , a ( a ) with n r t p i <p respect to the conjugate ordering oft ike basis {t±} oft v g a • Thus the restrictions oft the representations ^ ound ^ 0 1 ( ° n ) equivalent. PROOF : Le t T 1 } T^, . . . , T f be the arrangement o f the s tandard tab leaux of shape (a , g) a cco rd ing to the l a s t l e t t e r sequence. For f i x e d i , i = 2 n , w r i t e „ as the d i r e c t sum • „ = & V o f Z ? ' g a,g a ,6 p,q x$ i n v a r i a n t subspaces where V- „ i s taken to have b a s i s { t } i f p s p p i-1 and i appear i n the same row or column of T and V . has b a s i s ~ P P ' q {t , t } where ( i - 1 , i ) T = T , p < q i n the l a s t l e t t e r sequence. L e t P q P q = ® V* denote the cor respond ing decomposi t ion of , where 8,a p,q ^ & ^ 8 ,a V* has b a s i s {t* t*} cor responding to T* , T* , and where the o r d e r i n g p.q p q . ° p q of t * , t * i s taken w i t h respec t to the conjugate o r d e r i n g . We need to show tha t i f the mat r i x of Z ? ' B on V i s M, the m a t r i x o f Z ? ' B on V* x<p p,q 1 9 p,q i s I*MI*. This i s a s imple case by case v e r i f i c a t i o n . 41 1. I f i - l and i are i n the same row or column of T , they are - P likewise i n T* and the lemma i s shown f o r t h i s case. P 2. I f i - l and i _ are i n d i s t i n c t rows and columns of the same tableaux T a or T 6 of T = ( T a , T S ) , set T = ( i - l . i ) T and take P P P P P q P p < q. Then T* < T* i n the arrangement according to the l a s t l e t t e r sequence while T* < T* i n the conjugate ordering. The a x i a l distance, k, from i to i - l i s the same i n both T and T*. Thus from (2.2.6) - P P the matrix of Z ? J B on V i s M(k, -1) while the matrix of Z ? » a on i<j> p,q xi> V* i s I*M(k, -1)1* as i s required. ct 8 3. I f i - l and i are i n d i s t i n c t tableaux of T = (T , T ) set - P P P T = ( i - l , i ) T and take p < q . Then T* < T* i n both the ordering P q P q according to the l a s t l e t t e r sequence and the conjugate ordering. I f k i s the a x i a l distance from i to i - l i n T , -k i s the a x i a l distance from - — - p» i to i - l i n T* . Let M(k, 1) denote the 2 x 2 matrix obtained from - P M(k, y) under the s u b s t i t u t i o n y = 1. Then (2.2.6) shows the matrix of Z ? ' 6 on V i s M(k, 1) while the matrix of Z ? ! 0 1 on V * i s M(-k, 1) . 19 p,q 16 p.q D i r e c t computation v e r i f i e s the r e l a t i o n (2.3.4) I*M(k, 1)1* = M(-k, 1) as i s required. I t remains to show the lemma for. M°'' 8(a. J) . As Z ™ ' 8 acts on the basis { t ^ of ^ by s c a l a r m u l t i p l i c a t i o n , the decomposition of Q into Z ^ ' 8 i n v a r i a n t subspaces as above i s v a l i d f o r Z ^ ' ^ Z ^ ' ^ Z ? ' 8 as a,p zq) lip z<p 1(J) w e l l . I t i s furthermore clear from (2.2.6) that the action of Z ° ' 0 z " ' 8 Z ? ' 6 1<J> z<j> lq> 42 ot 8 d i f f e r s from that of Z ' only on the spaces V where the l e t t e r s 1 2<P 3 • / p,q -ct 8 and 2 appear i n d i s t i n c t tableaux of T = (T , T ). In t h i s case the - P P P matrix of Z?» e on V and on V* i s D ( l , -1) . Using i<p p,q p,q (2.3.4) , a simple matrix c a l c u l a t i o n completes the proof f o r t h i s case. This completes the proof of the lemma. We define the conjugate. ordering of the standard tableaux T = ( T 1 f T 2) of shape (a, a) as follows. Set T± = {I = (T,, T°) : n appears i n T°} , i = 1, 2. A l l standard tableaux belonging to 7"2 precede those belonging to i n the arrangement according to the l a s t l e t t e r sequence. Rearrange the l a s t i f a , a tableaux i n the l a s t l e t t e r sequence, i . e . those i n T l S as follows; f o r T^, T 2 i n T,, precedes T 2 i f . T* precedes T* i n the l a s t l e t t e r sequence arrangement of the tableaux i n T, 2 L E M M A (2.3.5) Let M°' , a(a) denote, the. matrix oft na'a(a) on with 9 9 a,a respect to the conjugate ordering oft the basis {t.} oft , a e tfK(D ) *^ ot y ot n Set r - I I f 2 _ »a f = f Then 43 (2.3.6) _ . . a . a . v -1 f <j> < a > R f M 1 (a ) M 2 (a ) PROOF : I f ( a , (ou-)) i s a double p a r t i t i o n o f n-1 conta ined i n ( a , a) then so i s ( ( a ± - ) , a ) , (a ±-) as i n the proof of Theorem (2.2.7). Thus we have the decompos i t i on , i n the conjugate o r d e r i n g , (2.3.7) V* = V . , a , a ( a , ( a - ) ) V ( a , ( a - j - ) ) ® V ( ( a 1 ~ ) , a) 0 " " ( ( a g - ) , a) v K K of V as 0\ (D ..)-modules, Q\ (D ) generated by ot y CL n x n—i. ^al<j>' a2<f)' * " * a(n-l)(J)^ * T h u s L e m m a (2.3.3) shows that f o r a e G (Dn_j_) ct ct M, ' (a) i s of the form 9 (2.3.8) A 0 % 0 I*AI* J which i s e a s i l y seen to commute w i t h R Hence we need to show (2.3.6) on l y f o r M™' a (a ) . I f the l e t t e r s <p nq> n-1 and n appear i n the tab leaux T^ of T p = ( T ° , T^) e T~2, the p roo f o f lemma (2.3.3) shows the m a t r i x of on the subspaces w i t h c o r r e s p o n -d ing b a s i s { t p , t*} o r { t p , t q , t * . t*} i f (n-1, n ) T p = T q , p 4 q i s of the form (2.3.8) and the above reason ing a p p l i e s . I f the l e t t e r s 44 n-1 and n appear i n d i s t i n c t tab leaux of T = (T?, T " ) , w i t h T e T.» p 1 z p 2 then (n-1, n)T = T e T. . Thus T* e T. and we can choose T such p q 1 q 2 p t ha t T p < T* i n the l a s t l e t t e r sequence arrangement o f the t ab l eaux be long ing to . Then T p < T* < T q < T* i s the arrangement of the tab leaux acco rd ing to the conjugate o r d e r i n g . Tak ing the same o rde r i ng of the cor respond ing b a s i s , the m a t r i x of Z ' on the subspace w i t h b a s i s n<p { t , t * , t , t*} i s of the form p q q p f  a 11 x21 22 l12 x12 22 J21 l l l where {a^} = M(k, 1 ) , M(k, 1) de f ined as i n lemma (2 .3 .3 ) and k the a x i a l d i s t ance from jL to i - l i n T p . A s imple m a t r i x c a l c u l a t i o n shows R f AR~ 1 ' A l 0 •» _ 0 A 2 J This completes the p roo f . Let V X a, a 1 a,a 2 a,a where, f o r b a s i s elements t corresponding to tab leaux T e T„, .v^ P p 2. 1 a , a 45 has basis {t p + t*} and ^ q has basis {t p - t*} . By Lemma (2.3.5) the K-linear maps .TT"'° : (31K(D ) —> E N D ( ) i <p n a,a ot ct where the matrix of .ir ' (a ) with respect to the above basis i s l <p w M (a ), w z W(D ), are representations of fl (D ) . i w n n THEOREM (2.3.9) Von double. pantltioni> (a, g), |a| < |g|, (a) 4 (g), and (a, a) oi n > 4, ike. nzpneAzntations T T " ' 0 1 and .-n~,a , i = 1, 2, ant — 9 1 9 a complete. &eX oi innzducible., inejqui.vale.nt nepnuentationi oi Ot ( ° n ) • PROOF : By induction on n . For n = 4, i t i s a matter cf direct verification. For n > 4, the induction assumption and the proof of Lemma (2.3.5) shows dim Hom^V^ o) = 2, Of = G|K(Dn), Homa(V^ ) generated by I, and I* f = f"'" . Thus and „ 7 T ' are irreducible t t 1 9 Z <p and inequivalent. The argument employed in Theorem (2.2.14) suffices for Ot OS the i r r e d u c i b i l i t y and inequivalence of the {TT ' } . By (2.3.7) 9 (2.3.10) .V* - V^, . •« ... 9 V^f v . , i = 1, 2, 1 a,a ( ( c t g - ) , a) ( ( c x j - ) , a) K ct 8 ot ct as 01 (D ..)-modules. Thus none of the ir ' are equivalent to . I T . ' , n—1 9 1 9 i =, 1, 2. Finally, consideration of degrees using the formula given i n Theorem (2.2.14) shows a complete set of inequivalent representations i s obtained, and the representations are absolutely irreducible. Thus K = Q(x) is a sp l i t t i n g f i e l d for (®^) . This completes the proof. 46 CHAPTER 3 DEGREES OF THE IRREDUCIBLE CONSTITUENTS OF 1^ 3.1. DEFINITIONS AND CHARACTERS OF PARABOLIC TYPE In t h i s chapter we g i v e some r e s u l t s on the i r r e d u c i b l e c o n s t i t u e n t s G of the induced r e p r e s e n t a t i o n 1^  o f B o r e l subgroup B o f a f i n i t e group G w i t h BN-pai r of c l a s s i c a l t ype . The f o l l o w i n g theorem i s b a s i c to the s tudy of these r e p r e s e n t a t i o n s . THEOREM (3.1.1) .([5]) Let 0 denote the algebnaic clobune oft Q. Each inxeduciJole Q-chanacten. x oft H— (G, B) lb the nestniction to HQ(G, B) G oft a unique inxeducible Q-ckanacten. x, oft G, buck t i a t ( ? , Jl_) > 0. X X B G Mo leaven, eveny inxeducihle constituent oft l g i s obtained in this way. . Tke degn.ee oft z. ii> given by X (3.1.2) deg r = |G : B| deg x I ( i nd w) x ( S „ ) x ( S „ ) A. . weW w w vAene §w is the basis element oft HQ(G, B) conn.esponding to w 1 and i n d w = |B : BC\B W | , w e W, B W = w - 1 Bw . Le t C\ be the gene r i c r i n g o f a Coxeter system (W, R) d e f i n e d over D = Q [ u r , r e R] as i n (2.1) and l e t K be the q u o t i e n t f i e l d o f D, K the a l g e b r a i c c l o s u r e of K. I t i s c l e a r from the r e l a t i o n s (2.1.8) (see e . g . [6], lemma 2.7 ) tha t there e x i s t s a unique homomorphism v : C\ — > D such t ha t v ( a ) = u , r e R. r r 47 DEFINITION (3.1.3) Ltt x bt an inAtduciblt K-chahactth. oi C l R . Set: f r 1 I v(a ) L w "•weW J I v(a w)- 1x(a w) x(a w) weW -1 whtnt w a w - l * W e d ^ e 9 e n e A ^ c dtQfizt aiAozlattd with x X Let M be an i r r e d u c i b l e K-matr ix r e p r e s e n t a t i o n o f (Ji . th K Le t M^ (a ) denote the i , j en t ry of M(a) , a e . Thus 1^. i s a K K f u n c t i o n from fl to K . The r i n g Cl i s a symmetric a l geb ra w i t h dua l b a s i s {a } and {v(a ) "*"a } (see e . g . [8], Lemma 5.1.). Then from W W W (.[7], Lemma 62.8) and Schur ' s lemma, we have (3.1.4) ( M . . , M ) i j r s ' I v(a ) _ 1 M. . . ( a )M (a ) T T w i i w r s w' weW J 1 i s j r where 6 i s the Kronecker d e l t a and i f x i s the cha rac t e r o f M, (3 .1 .5 ) C M = (deg x)" 1 I v ( a w ) ~ 1 X ( a w)x ( a w ) . weW Now l e t (X be the gene r i c r i n g o f a Coxeter system (W, R) o f c l a s s i c a l type and l e t G be a f i n i t e group w i t h BN-pair o f type (W, R) . Le t 9 : D — > Q be the homomorphism de f ined by <j>(yr) = Qr> r e R , q r the index parameters (see (2.1.5)). Le t P = ke r f and l e t 9* : D p — > Q be the ex tens ion of 9 to the r i n g o f f r a c t i o n s D p , regarded as a sub r ing of K . Le t x be an i r r e d u c i b l e cha rac te r of C f K . The r e s u l t s of chapter 2 (see a l so [6], P r o p o s i t i o n 7.1) show x(&w) e D p 48 for a l l w e W and the Q-linear map x± : Cl. —> Q defined by 9 9>Q (3.1-6) X ^ ) = 9*(x(a w)) i s an i r r e d u c i b l e cha rac t e r of C l . The map x — > X . i s a b i i e c t i o n 9.Q _ 9 between the i r r e d u c i b l e cha rac te rs of OT^  and those of As Q - H7r( G» b )> w e regard the s p e c i a l i z e d cha rac t e r x± a s »X H 9 an i r r e d u c i b l e cha ra c t e r of H—(G, B) aiid denote the cor respond ing i r r e d u -G c i b l e c o n s t i t u e n t of L, i n the sense of Theorem (3 .1 .1 ) by r, , . a X>9 PROPOSITION (3 .1 .7 ) With the notation* a& above, w ehave , 9 * ( d x ) = d e g ( ? X ) ( J ) ) . I v ( a w ) l = |G : B| and •WfW J PROOF : From ( [ 6 ] , lemma 5.9 ) , <|>(v(aw)) = i n d w . The statement now f o l l o w s from (3 .1 .2 ) and (3 .1 .6 ) and the d e f i n i t i o n of d X In p a r t i c u l a r i f 9 Q : D —> Q i s de f i ned by 9 Q ( u r ) = 1 f o r a l l . TJ ~ QW and (3 .1 .7 ) becomes deg ( x ) = deg(r, ^ . • y o ' x X>9, o We w i l l e va lua te d f o r the i r r e d u c i b l e cha rac te r x o f the X gener i c r i n g cor responding to a Coxeter system of c l a s s i c a l type i n the next s e c t i o n . We conclude t h i s s e c t i o n w i t h the f o l l o w i n g . r Let J c R and l e t W = <J> . J determines "a pahjabotie subgroup G = BW 3 . 49 D E F I N I T I O N (3 .1 .8 ) Let c be an iM.zda.dbl2. chan.acX.2Ji oft G Audi that G G. G r (C, l g ) > 0. t, is sold to be oft parabolic, type ift ( c , i £ ). = l fton. J some. J R . From the above there i s a n a t u r a l b i j e c t i v e correspondence C . — > C , between the i r r e d u c i b l e Q-characters £ , o f G and the X.9 X,<P0 X»9 i r r e d u c i b l e Q-characters of W. In ( [ 6 ] , T h e o r e m 7.2) i t i s shown tha t f o r a l l J C R. Thus to show the i r r e d u c i b l e c o n s t i t u e n t s o f Lg are of p a r a b o l i c type i t i s enough to show i t f o r the i r r e d u c i b l e cha rac te r s o f the Weyl groups . PROPOSITION (3 .1 .10) Evzty Wizdaciblz chanactzn. x oft W (A n ) , W(B n ) , n _> 2, and W(D n), n >_ 4 , i s o| paAabo£ic type. PROOF : Let (a) be a p a r t i t i o n ' o f n. Le t R(a) denote the group of row permutat ions of the c a n o n i c a l tab leaux of shape ( a ) . Then R(a) co i n c i de s w i t h W f o r some J C R, R the se t of d i s t i n g u i s h e d genera tors J f o r W(A ,) - S g i ven i n ( 2 . 3 ) . Order the p a r t i t i o n s o f n l e x i c o g r a p h i c l y n-1 n and l e t x° denote the cha rac t e r o f the i r r e d u c i b l e r e p r e s e n t a t i o n o f W(A n) cor respond ing to ( a ) . From ( [12 ] , pp.40-41) <3-1-1> 4(a) -  X a \ l a m * , / • raa,e±° \ S a , n Thus (x , 1 ^ ^ ) = 1, which i s w e l l known. 50 For a double p a r t i t i o n (ex, 6 ) o f n , (a) = (a^> a r ) , 3 = ( 3 . , • • • » 8 ) , l e t (a+8) denote the p a r t i t i o n of n de f ined by (a+8) = ( c ^ + 3 ^ a t + 8 t ) , t = max{r,s} . Let x ^ ' t ^ denote the cha rac te r of the outer product r e p r e s e n t a t i o n [ a ] - [ 8 ] of (see (1 .2 .2 ) ) From ( [ 12 ] , Theorem 3 .31 ) , [ a ] - [ 8 ] m- ot+3 + y y > Q Then by (3 ,1 .11 ) (3 .1 .12) U , 1R(a+8) ; U , X -> " But by Theorem (2 .2 .15) the r e s t r i c t i o n o f ' t h e i r r e d u c i b l e r e p r e s e n t a t i o n ct 8 ir ' of W(B ) to S i s the outer product r e p r e s e n t a t i o n [ a ] ' [ 8 ] • ° ct 8 ct 8 Thus, l e t t i n g x ' denote the cha rac te r o f it 1 , we have o U ' R (a + 8 ) ; K X ' R (a + 3 ) ; by (3 .1 .12) and Frobenius r e c i p r o c i t y . As R(cx+8) i s a p a r a b o l i c subgroup ct 8 of W(B n ) , the x ' a r e ° f p a r a b o l i c t ype . The r ep re sen t a t i ons of W(D n ) , n >^  4, are handled s i m i l a r l y . I f ct 8 (a) 4 ( 8 ) , X ' remains i r r e d u c i b l e when r e s t r i c t e d to ^(^n) by As S r i s a subgroup of W(D n ) , ( y a ' B l W ( D n ) )' = ( Y [ a ] ' t B ] l n ) = 1 K X ' R(a+3) ; U ' R(a+8) ; L ' 5 1 I f (a) = (8) the s i t u a t i o n i s on l y s l i g h t l y more c o m p l i c a t e d . Set J ' = {w 2 , w n _ T ^ ^ R» d i s t i n g u i s h e d generators o f W(D n ) . Then ct ct ot ct W_, - S , . Le t . Y ' denote the cha rac te r o f . I T ' , i = l , 2 . J n—1 l i <p T o From (2 .3 .7 ) (3 .1 .13) V ^ L •- x t a ] , [ a r - ] + . . . + x l 0 ? , [ ° l " 1 . i - l . 2. 1 Wj , . where (a.-) are the p a r t i t i o n s of n-1 conta ined i n 3 (a) = ( a 1 a ) . Now (a) + (o r -) > (a) + ( a ^ - ) >. . . . > (a) + (a^-) i n the l e x i c o g r a p h i c o rder so W(D ) S , / os,a 1 n » / a ,o i , n-1 . 4 X ' ± R ( ( a ) + ( a r - ) ) ; \L X \sn_±' 1 R ( ( a ) + ( a r - ) ) ; - ( x ( a ) + ( a r - ) } x («>+(«r-) . ) , i = i , 2 by ( 3 . 1 . 1 2 ) , (3 .1 .13) and Frobenius r e c i p r o c i t y . Hence i x a , ° ' , i = 1 > . 2 » . are o f p a r a b o l i c t ype . This completes the p r o o f . 52 3 .2 . AN INDUCTION FORMULA Let y = ( y 1 9 y^) be a double p a r t i t i o n of n and l e t x y denote the cha rac te r o f the r e p r e s e n t a t i o n i r y of Ol k (B ) . Set n (3 .2 .1 ) C y = ( f V 1 I v ( a w ) - 1 x y ( a w ) X y ( a ) W E W ( B ) W n We show that the i n d u c t i v e c o n s t r u c t i o n of the r e p r e s e n t a t i o n s i r y y i e l d s an i n d u c t i v e formula f o r C y , which w i l l p rov ide the means to determine y the gene r i c degree a s soc i a t ed w i t h x • Le t (W(B ) , R) be as i n Se c t i on 2.2 . Take J , <z R to be n ' ' n-1 the subset J = { w . . w .. } and l e t WT = <J , > . Then n—i i n—1 J , n—1 . n-1 WT - W(B .,). Le t M y ( a ) denote the m a t r i x o f T r y ( a ) . From the p roo f J - n-1 w w' • * n-1 of Theorem (2 .2 .7 ) we have the decomposi t ion y <y8-) • •• ( v r> (3 .2 .2 ) M (a ) = M (a.) + . . . + M (a ) , w £ W T W W W J , n-1 where the sum i s taken over those (y -) v/hich are non-zero . Le t g^ and f denote the p o s i t i o n , i n the arrangement acco rd ing to the l a s t l e t t e r sequence, o f the f i r s t and l a s t tab leaux of shape (y) r e s p e c t i v e l y , which upon d e l e t i o n of n y i e l d s tandard tab leaux of shape (y ~) . Then f o r a t , b r , g t £ a t £ f t , g r £ b r £ f r , (3 .2 .2 ) i m p l i e s (3 .2 .3 ) M y • (a ) = 0 f o r t ^ r , w e WT a .^ ,b w J , t r n-1 and 53 y < V > ( 3 .2 .4 ) M a t , b t C a w ) " \ , b t ( V • W e V l ' S ince | w ( B ) | = 2 % ! , | w ( B ) : W | = 2n . From ( [ 2 ] , p . 3 7 ) , n-1 there e x i s t s a se t { x^ | k = 1, . . . , 2n } o f coset r e p r e s e n t a t i o n o f the l e f t Wj -cosets of W ( B n ) such that £ ( x ^ ) = £(x^) + £(w) , n-1 w e Wj , k = 1, 2n. The a r e the unique elements o f min ima l n-1 l e n g t h i n the l e f t cosets • We w i l l determine these elements n-1 e x p l i c i t l y f o r our cho ice of R and J , i n the next s e c t i o n . n-1 DEFINITION (3 .2 .5 ) with the notation at, above, let (H -) 2n f t _ E = I I v(a )-\ (a )M y (a ) k= l j=g^ x k 1 » J k ^ ' * k We now prove THEOREM (3 .2 .6 ) fj = C E PROOF : Choose p such tha t g f c <_ p <_ f . Le t { x^ : k = 1, . . . , 2n } be the se t o f l e f t W -coset r e p r e s e n t a t i v e s of m in ima l l e n g t h as above. n-1 B y ( 3 .1 .4 ) and ( 3 . 1 . 5 ) , C y = (M y M y ) = I v ( a ) " V (a )M y . ( a ) l j P P ) 1 weW(B ) W X ' P W P ' 1 W 2 n _ i - 1 I T V ( P ) v ( a ) X (a a ) M y ( a a ) k = l weV V w l . P x k W P ' 1 W X k n-1 54 2n I I k= l weW, / X - 1 / N-1 v (a ) v (a ) t f Jn-1 I }£ . (a ) M y (a ) ' I M y . (a )MU. . ( a ) 2n f (3 .2 .7 ) I . I v ( a ) " 1 M y ( a , )M!j ( a ) * c=l i , j = l x k 1 » 1 *k 3 , 1 * k I v (a ) ~ V ( a ) M y . (a ) ] n-1 The second step f o l l o w s from the f a c t tha t by the cho i ce o f the { x ^ , £ ( x k W ) = £ (x k ) + £(w) . By (3 .2 .3 ) v (a ) M y (a ) M y . (a ) = 0 w £ W w i . p w p , ^ w ' J n-1 f o r e i t h e r i or j not l y i n g between g f c and f , w h i l e by (3 .2 .4 ) and (3 .1 .4 ) r - L u u ( y i - - ) ( y i - _ ) ( y * - _ ) I v (a ) X J aJ^ = (M. , M t ) = 6..C fc c-w W 1>P w P»3 w i»P P,J i j W E W n-1 (y -) t f o r 8 t £ i> j 1 f f c i as IT i s an a b s o l u t e l y i r r e d u c i b l e r e p r e s e n t a t i o n K of Ot ( Bn-±) b y t h e o r e m ( 2 . 2 . 1 4 ) . Combining the above formulae w i t h (3 .2 .7 ) g i ves C M = c - l.ti u ( y t _ ) ( y t - - ) X . (a ) M y Aa ) = C E ' =1 j = g t *k X » J *k J j l * k which i s the r e q u i r e d r e s u l t . 55 Whi le the above theorem prov ides the induc t ion , s tep f o r a v a r i e t y of f a c t o r i z a t i o n s o f C y , no such f a c t o r i z a t i o n lends i t s e l f to an e x p l i c i t fo rmula f o r C y w i t hou t ted ious c a l c u l a t i o n . To o b t a i n a formula f o r C y by i n d u c t i o n u s i ng Theorem (3.2.6) r e q u i r e s the e v a l u a t i o n of ( V ) E f o r some row t . The most convenient cho ice i s the f i r s t a l l o w a b l e row from which the l a s t square can be de t e t ed . Le t c denote the index of t h i s row. Then i n the Young diagram of shape (y) = ( a , g ) , the row c i s a row of the Young diagram of shape (ot) i f (a) * ( 0 ) . Moreover , the s tandard tab leaux of shape (u) wh ich upon d e l e t i o n of n y i e l d s tandard t ab leaux o f shape (y -) occur l a s t i n the o r d e r i n g a cco rd ing to the l a s t l e t t e r sequence. We de fe r the e v a l u a t i o n of E u n t i l the next s e c t i o n . We c l o s e t h i s s e c t i o n w i t h a deduct ion which w i l l prove v a l u a b l e i n the c a l c u l a t i o n s to come. PROPOSITION (3.2.8) Let (W(B n ) , R) be as beftore and l e t w be any element oft w(Bn) which can be expressed as a product oft d i s t i n c t generators chosen ftrom R in increasing order, i.e. w = w. . . .w^ , w i # e R , j = 1 , k ^ * 1 k 3 and < < ••• < l k • For the matrix representation M y oft 0 1 K ( B n ) , y = ( a , 8) a double p a r t i t i o n oft n , (3 2 9) M y Aa ) = T M y (a . ) M y (a. ) . . . M y . ( a . ) ^'•t-'?} c d v wv _ L c ,s x . s , t x ' u,d x, "' s , t , . . . ,u ' 1 2 k and there is at most one non-zero term in tlie above summation. In p a r t i c u l a r M y , (a ) 4 0 ift and only ift tlxere exists a w e w ( B ) , w = w. . . . w . , c , d w ' u J u v n ' J i l ' — ' 1 J s vhere j n < . . . < j and { j . , j } a { i . i , } , or w = 1 , J. S X • S X K. suclt that w _ 1 T y = T y . c d 56 PROOF : As the {w.} i = 1, . . . . k a re d i s t i n c t , a = a . . . . a . and (3 .2 .9 ) i s j u s t the d e f i n i t i o n of m a t r i x m u l t i p l i c a t i o n . Furthermore the second statement i n the p r o p o s i t i o n f o l l o w s immediate ly from the f i r s t s i n c e ' by d e f i n i t i o n (2 .2 .6 ) and theorem ( 2 . 2 . 7 ) , a m a t r i x en t r y M V . (a . ) ^ 0 , f o r k = 2 n , i f and on l y i f i = j or w f cT^ . = T y w h i l e M Y . ( a 1 ) ^ 0 i f and on l y i f i = j . Thus a product of m a t r i x e n t r i e s of the form i s non-zero i f and on l y i f there e x i s t s a w E W, w = w. . . . w . w i t h H 3 s j , < . . . < j g , { j , , j s > cr { i , , i } or w = 1 such tha t -1 w ~ x y = T y . Hence to complete the proof of* the p r o p o s i t i o n we must show there i s at most one non-zero term i n the summation ( 3 . 2 . 9 ) . We do t h i s by i n d u c t i o n on k . I t i s c e r t a i n l y t rue f o r k = 1. Assume t rue f o r k - 1. By the r u l e s of m a t r i x m u l t i p l i c a t i o n , (3 .2 .10) M y (a w ) - I M y ( a w , ) M y (a ) , w' = w . . . w ' s k 1 k-1 By the i n d u c t i o n hypo thes i s i t i s enough to show a t most one term i n (3 .2 .10) i s non-zero . Cons ider the p o s i t i o n of the l e t t e r s and i ^ - l m i , . a ( i ) I f they be long e i t h e r to the same row or column o f the tab leaux of shape (a) or the tab leaux of shape (3) of T^ , M y ^(a^ ) 0 ' k i f and on l y i f s = d . Thus 57 My , (a ) = My , (a ,)My , ( a . ) c ,d w' c ,d w d ,d x A and the p r o p o s i t i o n i s proved f o r t h i s case . ( i i ) I f the l e t t e r s i ^ and do n o t be long to the same row or column of e i t h e r the tab leaux T a or the t ab leaux T S of T{j = ( T a , T 6 ) , s e t = w. TV,' . Then e 4 d , My J ( a . ) 4 0 i f and on ly i f s = d or e and e i k d s , d v i ^ ' My , ( a ) = My , (a ,)My , ( a . ) + My (a , ) M P , ( a . ) . c ,d w c ,d w' d,d i ^ c,e w f e , d v x A Suppose My ^ ( a w i ) 4 0. By the i n d u c t i v e hypothes i s and the f i r s t p a r t o f the p r o p o s i t i o n , there e x i s t s a w c W e x p r e s s i b l e as a product of d i s t i n c t generators chosen from the se t {w. , w. } such tha t wT y = TV, xr V l c d-As i ^ < . . . < i ^ , the l e t t e r i ^ i s l e f t f i x e d under the a c t i o n of w on TV , i . e . i , occupies the same p o s i t i o n i n the s tandard tab leaux T^ c _ k r r c and T y . Therefore by the cho ice of d , the l e t t e r i ^ occupies d i f f e r e n t p o s i t i o n s i n the s tandard tab leaux T v and T y . But then there cannot r • c e e x i s t a w e W e x p r e s s i b l e as a product of generators taken from {w. , w. } such tha t wT u = T y , i . e . My (a ,) = 0 . Thus x i V i c e  MP . ( a . . . . a . ) = My (a . . . . a . )My , ( a . ) c ,d x± xk c , s x1 xk_± s , d x k where s = e or d , depending on the p o s i t i o n of the l e t t e r i ^ i n T^ . This completes the p r o o f . 58 The above p r o p o s i t i o n i s a d i s t i n c t i v e p rope r t y o f the shape o f the mat r i ces M y(a^) i n tha t i t c l e a r l y depends on l y on the p o s i t i o n of the zero e n t r i e s . Furthermore i f w e W(B ) i s as i n the statement o f the n p r o p o s i t i o n , the p r o p o s i t i o n i s c l e a r l y v a l i d f o r w 1 as w e l l . F i n a l l y , f o r the d i agona l e n t r i e s M y c ( a w ) > t n e p r o p o s i t i o n i s v a l i d f o r any w e x p r e s s i b l e as a product o f d i s t i n c t gene ra to rs , not n e c e s s a r i l y on i n c r e a s i n g (or decreas ing) product (see [13], pp 43-44). 3 .3 . THE EVALUATION O F E ° 59 (y -) Our aim i n t h i s s e c t i o n i s an exp re s s i on f o r E i n terms o f the po lynomia ls A(m, y) and A(m, - 1 ) . Throughout, y = ( a , 8 ) w i l l denote a double p a r t i t i o n o f n, w i t h the p a r t i t i o n (a) hav i ng s^ p a r t s (a) = ( a . , . . . . , a ) and the p a r t i t i o n (8) hav ing s p a r t s (8) = (8, , 8 ) . As be fo re (y -) denotes the non-zero double p a r t i t i o n 1 Sg c o f n-1 obta ined from (y) by d e l e t i o n of a square from the end of row c of the Young diagram D(y ) , i . e . the f i r s t a l l o w a b l e row. We f i r s t determine e x p l i c i t l y the elements t'KyJ ° f min imal l e n g t h i n the l e f t WT -cosets of (W(B ) , R ) , R and J .. d e f i ned as i n the J , n n—i n-1 prev ious s e c t i o n . We i n t r o d u c e some more n o t a t i o n . For any se t of consecu -t i v e i n t ege r s 1 _< k, k+1, £, <_ n , se t w(k, I) = W j ^ k + l * * , W £ s o t n a t w(k, k) = w^ . For ease of n o t a t i o n we a l so d e f i n e w (k+l , k) = 1 . Furthermore, se t w(k) = w(2, k) x W j W ( 2 , k ) , k = 1, n . Thus th w ( l ) = w, . w(k) i s the k s i g n ' change -(k) o f W(B ) . x n LEMMA (3 .3 .1 ) The. &<U S = {1} {w(k, n) : k = 2, . . . , n} {w(k)w(k+l, n) : k = 1, . . . , n} ii, thz unique. heX oft eJLemenXs oft minimal Izngtit in the. leftt WT -co&e£& n-1 Oft W(B n) . PROOF : W(B ) ac ts on a f i x e d orthonormal b a s i s { e n , c } of £ n n 1 n J as a l l permutat ions and s i g n changes. For the g iven cho i ce of R, the se t o f fundamental roo ts of W(B ) i s n 6 0 { £ 1 ' £ 2 e 1 , e 3 - e 2 , e n -and the se t of p o s i t i v e r oo t s i s { E . , £. - e . , £. + £. I 1 < i , j < n , j > i } see ( [ 2 ] ) . For an element x = w(k, n ) , x 1 i s the n - k + 2 c y c l e ' (n n-1 . . . k-1) , work ing from r i g h t to l e f t . Hence x sends the p o s i -t i v e r oo t s e. - £. - , j > k - 1, to the nega t i ve roo ts £. , - e , J k - 1 ' J 6 j - 1 n ' j = k, . . . , n . By the cho ice of ^ n i» these roo t s remain nega t i ve under .the a c t i o n of W - W(B ) . Thus £ (w 'x " 1 ) >_ I(x-1) f o r a l l J n-1 n - ± w' e Wj , as £(w) equa ls the number of p o s i t i v e roo t s sent to n e g a t i v e n-1 roo ts under w (see [2] o r [14 ] , append ix ) . There fore x i s of m in ima l l eng th i n the l e f t W -coset xW . A s i m i l a r argument shows n-1 J n-1 £(w'x 1 ) >_ £(x "*") f o r x = w(k )w(k+l , n ) , k = 1, . . . , n and f o r a l l w' e W . A s £(x) 4 £ (x ' ) f o r any x , " x ' £ S, the elements of S n-1 must be long to d i s t i n c t c o s e t s . As |w(Bn) : Wj | = 2n and |s| = 2n , n-1 S must be a se t of coset r e p r e s e n t a t i v e s f o r the l e f t W -cosets o f n-1 W(B n ) . This completes the p r o o f . Let a ,~ . . = a ( 2 , k ) and a n . = a(k) i n 01 (B ) . As the w(.z, W ( . K ; n express ions f o r w(2, k) and w(k) as a product o f generators from R i s reduced a ( 2 , k) = a 2 > . . a ^ (3 .3 .2 ) a (k) = (a^...a2)a1(a2...a^) 61 In order to s t a t e the next p r o p o s i t i o n concern ing the ma t r i c es M y ( a ( k ) ) cor respond ing to the k t b " s i g n change of W(B ) we need some n o t a t i o n s . Le t n p. , i = 2, . . . , n denote the a x i a l d i s t ance from i-1 to i i n the i , P — standard tab leaux . Le t P i p ( i ) = I (p + 1 ) , i « 2, n P j=2 J , P w \ and d e f i n e P p ( l ) = 0 f o r a l l i n d i c e s p = 1, . . . , f y . PROPOSITION (3 .3 .3 ) The matrix M y ( a ( k ) ) , k = l , " v . . , n is a diagonal th p o ( k ) matrix uiitk ike p,p entry equal z x p , where z = y Ift ike letter k appears in ike tableaux T a oft T £ = ( T A , T B ) and z = -1 ift k appears in the tableaux l e od . PROOF : The proof i s by i n d u c t i o n on k . For k = 1 the statement o f the p r o p o s i t i o n i s j u s t the d e f i n i t i o n o f M ^ a ^ ) . Now assume M y ( a (k-1) ) i s d i a g o n a l . By (3 .3 .2 ) M y ( a (k ) ) = M y ( a k ) M U ( a ( k - l ) ) M y ( a k ) . Wr i t e as the d i r e c t sum = & V of a, i n v a r i a n t subspaces , V y p,q k f » where V has b a s i s {t , t } i f ( k-1 , k )T = T , p < q , and V p,q p q p q p,p has b a s i s { t } i f the l e t t e r s k-1 and k appear e i t h e r i n the same row P — o r column of the same t ab l eaux . With t h i s o r d e r i n g of the b a s i s , M y ( a ( k ) ) has the cor respond ing b l o c k form M y ( a (k ) ) = 4- M y ^(a(k) ) Thus the usua l case by case argument on the c o n f i g u r a t i o n of the l e t t e r k-1 and k w i l l s u f f i c e . 62 !• k-1 and k i n the same row or column of the same of T P 0 0 = P. I f k-1 and k are i n the same row, p. = 1 . There fore k,p 5 ( k- l ) + 2 . As M p ) p ( a k ) = x by theorem (2 .2 .7 ) M y (a (k) ) = x P»P f P p (k-D^i z x F P (k-l )+2 p (k) X = z x = z x ! f k-1 and k are i n the same column, p, = -1 . There fore k,p P p ( k ) - P p t k - 1 ) . As M y p ( a k ) = -1 by theorem (2.2 7) M y n ( a ( k ) ) = (-1) P »P r p „ ( k - D zx (-1) = zx P_(k) 2. k-1 and k i n d i s t i n c t rows and columns of T y . - p Set Tj| = ( k - 1 , k ) T y and take p < q . The d e l e t i o n of a l l l e t t e r s > k - 1 i n T y and T y y i e l d the same tab leaux of k-2 — P q l e t t e r s . Therefore p (k-2) = p (k-2 ) . Let p q k-2 k-1 denote the c o n f i g u r a t i o n of the l e t t e r s k-2, k-1 and k. i n T^ , w i t h e^, z,y the r e s p e c t i v e a x i a l d i s t a n c e s . With t h i s n o t a t i o n we have , p (k-1) = p (k-2) + c,..+ 1 P P 1 P „ (k ) = P (k-2) + e + ' e 9 + 2 P P 1 2 p (k-1) = p (k-2) + e , + e 0 + 1 q q 1 2 P q ( k ) = .p (k-2) + e 2 + 2 •. 6 3 Therefore i f k-1 and k be long to the same tableaux o f T y , p (k-2)+e.+l E M p , q ( a k } = M ( £ 2 ' a n d ^ q ^ ^ - D ) = z x P D ( l , x Z ) by theorem (2 .2 .7 ) and the i n d u c t i o n h y p o t h e s i s . D i r e c t computat ion v e r i f i e s the r e l a t i o n e 2 e 2 M ( e 2 , -1 )D(1 , x " ) M ( e 2 , -1) = xD(x 1) Therefore p (k-2)+e +2 e M y (a (k ) ) = zx P A D(x , 1 ) p»q P (k) P (k) = D(zx p , zx q ) I f k-1 and k be long to d i s t i n c t tab leaux of T y , • M y (a,) = M(-e_, y) — p p,q k I by theorem (2 .2 .7 ) . As we have taken p < q w i t h r espec t to the l a s t l e t t e r sequence, k-1 appears i n the tab leaux cor respond ing to (a) of T^ . Thus by the i n d u c t i o n hypo thes i s p (k-2)+e,+l e , M y ( a (k- l ) ) = x P 1 D(y , -x . p»q D i r e c t computat ion v e r i f i e s the r e l a t i o n £ £ M(-e 2 , y )D(y , -x 2 ) M ( - £ 2 , y) = xD(-x 2 , y) . There fore p ( k - 2 ) + E +2 £ ? p (k) p (k) M y (a(k) ) = x p X D(-x , y) = D(-x P , yx q ) p > q Th is completes the p roof of the p r o p o s i t i o n . 64 We use t h i s p r o p o s i t i o n to a f f e c t a r e d u c t i o n of E P a i r i n g the W -coset r ep resen t a t i ons a ( k ) a ( k + l , n) and a ( k + l , n) n-1 of W(B n) (lemma (3 .3 .1 ) ) we have M l , i ( a ( k + 1 ' n ) ) + ^ ± ( a (10a (k .+ l , n)) 1 (3 .3 .4 ) 1 + M j ^ C a O O ) M y . ( a ( k + l , n)) , i = 1, f , as M y ( a (k ) ) i s d i agona l by p r o p o s i t i o n (3 .3 .3 ) Furthermore f o r l e t t e r s k-1 and k bo th appear ing i n the same row of the canon i c a l tab leaux of shape (u) (see s e c t i o n 1.1), w^ = ( k-1 , k ) , k = 1, n , i s a row permutat ion of the c a n o n i c a l t ab l eaux . As any row permutat ion w can be w r i t t e n as a product of such t r a n s p o s i t i o n s (3 .3 .5 ) M j f l ( a w ) = M y ) l ( a w ) = 0 , 1 * 1 , and (3 .3 .6 ) M y ) l ( a w ) = M ^ C ? ) = x by theorem (2 .2 .7 ) , w a row permutat ion of . Let r^, i = 1, tlx s = s + s D , denote the l a s t l e t t e r i n the i row of the c a n o n i c a l Ot p tab leaux T y , i . e . r. = J a . i f i < s 1 j = l 3 S a k r. T a. + T 8. i f i = s + k . 65 Set, r. x (3 .3 .7 ) R = I l + v(a(k)) V .(a(k)) k=r. .+1 v v ( a ( k + l , r ^ ) " 1 x H l , l < a ( k + 1 ' r i ) ) 2 * ( y c - ) Combining (3 .3 .5 ) and (3 .3 .6 ) w i t h the d e f i n i t i o n s of R ± and E and us ing the e x p l i c i t coset r ep r e sen t a t i ons g i ven by lemma (3 .4 .1 ) we have (3 .3 .8 ) (yr-) I R v ( a ( r +1, n ) ) " 1 £ M y . ( a ( r . + l , n)) x i = l 1 j=g^ X ' J x ( a ( r .+ l , n)) Set A(m, y - 1 ) = l+x^V 1 . Then P R O P O S I T I O N ( 3 .3 .9 ) , Ton 1 < i < s denote. R . by R . FoA. — — ex l a . l s „ + 1 £ i £ s + s , denote R , by R . wheAe i = s + i . Then U CX p 1 p • CX J R „ = A ( a , - 2 i + l , y ) A ( a , , -1) , R = A(B - 2 i + l , y X )A (B -1) i PROOF : F i r s t l e t 1 < i < s . The a x i a l d i s t a n c e from r. to r . + l i s — — a _ i I -ou w h i l e f o r k and k+1 i n the same row o f T y , the a x i a l d i s t ance from k to k+1 i s 1 . Therefore r i - l + 1 p l ( r i - l + 1 ) = i - l j i - l I (p + 1) = I (2(a -1) + (1-a ) ) = I ( a . =2 K , J - j = l J J j = l J 66 A l s o v ( a ( r . _ .+ l+k ) ) = yx where m = £ a . and k = 0, a . . j = l 2 X There fore v (a ( r ^ ^ l + k ) ) _ 1 M ^ 1 (a ( r ^ + l + k ) ) 2 = [ y x 2 ( ^ k ) ] - l y 2 x 2 [ m - ( i - l ) + 2 k ] = . ^ - U l - V + T k f o r k = 0 , . . . , . a.-k Furthermore v ( a ( r . .,+1+k, r . ) ) = x and r - l i " ' 2 ( r . - r .-k) 2(a -k) M l 1 ( a ( r 1 _ 1 + l 4 f c t r±)) = x - x f o r k = 1, . . . , by ( 3 . 3 . 6 ) , as we have de f ined a(m+l, m) = 1 . Thus (3 .3 .7 ) i m p l i e s R - l' (1 + y x ' 2 ( i " 1 ) + 2 ( k " 1 ) ) x a i _ k  a i k-1 a i a .-k a .+k-2i = I (x 1 + yx 1 ) k= l a . - 2 i + l a . - l = (1 + yx 1 ) (1 + . . . + x 1 ) = A ( a i - 2 i + l , . y ) A ( o i , -1) . We now tu rn to the second pa r t o f the p r o p o s i t i o n . Observe f i r s t tha t the a x i a l d i s t ance from r to r +1 i n the c a n o n i c a l t ab leaux T y s s 1 a .a i s by d e f i n i t i o n (1 .1 .6 ) the a x i a l d i s t ance from the l a s t square i n the diagram (a) to the f i r s t square i n the f i r s t row of ( a ) . There fore the 6 7 path t r a ve r sed from the l e t t e r . 1^  to the l e t t e r r +1 i s a c l osed pa th ct r +1 s a i n terms of a x i a l d i s t a n c e . As a r e s u l t Y o . , = 0 and J - 2 ^ r +1 s a p. ( r +1) = T p. . + 1 = l a ! . For i > s s e t % = i - s . Then 1 S a j=2 J ' 1 a a r i - l + 1 P l ( r i - 1 + 1 ) " W + J R + 2 P k , l + 1 s a £-1 = l«l + I 6. - 1 . j = l 2 A l s o v (a (r ._ . .+l+k) ) = - x 2 ( m + k ) where m = f ot J + £ B - and k = 0 , ...... p . 1 j = l J * Therefore / / w v l J ft ,,,, X N r 2 ( m + k ) , - l 2[m-(£-l)+2k] v ( a ( r i _ 1 + l + k ) ) M ^ 1 ( a ( r i _ 1 + l + k ) ) = [yx ] x -1 -2(£-l)+2k • « y x . The argument used i n the f i r s t p a r t o f the p r o p o s i t i o n can now be used to g ive R 3 > = A(B.-2i+ l , y " 1 ) A ( 8 i , -1) . This completes the p r o o f . We now s t a r t e v a l u a t i o n of the sum f y I M y (a ( r +1, n ) )M y ( a ( r . + l , n)) j = g c ' 2 2 ' 1 68 i n the exp re s s i on f o r E g iven by (3 .3 .8 ) PROPOSITION (3 .3 .10) Let c be as above.. Then ( i ) . (a (k , n) ) = 0 fton, k > r +1 and ftoh. a l l j _> g i » J c c ( i i ) M^ . . ( a ( r +1, n)) = 0 ftoh, a l l j > g while i1J c c n-r c • itf ( a ( r +1, n)) = T T M y (a _,_.) l , g c . ' ' e. , , e . • r + i & c 1=1 1-1' 1 c where T y = T y , the canonical tableaux oft shape ( y ) , and o T y = w ^ . T y : , i > 1 . e. r + i e. ' — i c 1-1 ( i i i ) Mj , ( a ( k , n)) = 1 > f . and k < r J — 1 — c f • 1 ) H. . ( a (k , r )) l » i c My ( a ( r +1 , n)) tfo* f c-1 f ( i v ) I Mj ( a ( r + l , r )) = f f f I M y (a ( r +1, r )) j = l X » J K c . i=k 4=1 X > : 1 1 1 + 1 J •^ CM k < r PROOF : ( i ) and ( i i ) . For j -fl g , . the l e t t e r n occup ies the l a s t square i n row c by d e f i n i t i o n of (y - ) . Suppose M y . (a(k, . n) ) / 0 , c _L, j j >. g c - By p r o p o s i t i o n (3 .2 .8 ) there e x i s t s a w e W such tha t wT y = T y , w e x p r e s s i b l e as a product of d i s t i n c t generators chosen from {w^, wn> . I t f o l l o w s that w f i x e s a l l l e t t e r s <_k-1 i n T y . Hence i f k > r c + l , the l e t t e r r £ occupies the same p o s i t i o n i n the t ab leaux T y and T y , i . e . the l a s t square i n row c . Th is i s i m p o s s i b l e . Therefore M y . ( a (k , n)) = 0 f o r k > r +1 which proves ( i ) . »3 c 69 On the other hand, i f k = r + 1 , wT y = T y i m p l i e s w = w'w c 1 j v r c + l ' s i n c e the l e t t e r r must be moved under the a c t i o n of w. Therefore c r +1 i s i n the l a s t square of row c i n w ,. ,T y = T y . S i m i l a r l y , as c ^ r +1 1 e, J ' c 1 w + ^ does not occur i n the exp re s s i on of w' as a product o f d i s t i n g u i s h e d c gene ra to r s , w' = w"w , „ because the l e t t e r r +1 must be moved under the r +2 c a c t i o n o f w' . Therefore r c + 2 i s i n the l a s t square of row c i n w ,oT y = T y . C o n t i n u a t i o n of t h i s argument a l l ows us to conclude c e l e 2 j = g c and M y (a + 1 . . . a ) = M y (a , 1 ) M y (a . . ) . : . M W - (a ) l , g c r c + l n l , e i r c + l e ^ r+2 e ^ . g ^ n ' where w r + ± T y = T y . Th is proves ( i i ) c i-1 i ( i i i ) and ( i v ) . Le t T M denote a s tandard tab leaux obta ined from e r y  T y by any rearrangement of the l e t t e r s _1, . . . , r^ , f o r j <_ c . Th i s amounts to a rearrangement of the above l e t t e r s among the f i r s t j rows of (u) • Because the f i r s t j rows, ( j <^  c) are of equal l e n g t h , any such arrangment of _1, r^ i n the f i r s t j rows must have i n the l a s t box of row j . Thus the p o s i t i o n of the l e t t e r s r n i s the same i n wT y and wT y , f o r any w whose reduced exp re s s i on as a product o f elements from R i s made up e n t i r e l y o f the generators w^, i = r j + i » •••> n -I t f o l l o w s that M y , ( a ( r , + l , k ) ) = M y , ( a ( r , + l , k ) ) e , i j x , i j f o r j < c and k > r . + l . - - J Therefore 70 S , t , . . . , U M l t ( a ( r j + 1 + 1 ' r j+2 ) )--- Mi , i ( a < r c - l + 1 » r c » c-1 f y TT I ^ k ( a ( r i + 1 » r i + 1 » i= j k= l 1 , k 1 1 + 1 by the above argument and the o rde r i ng o f the l a s t l e t t e r sequence, the same reason For M y . ( a ( k , n)) = I M y , ( a ( k , r ) )M y , ( a ( r +1, n ) ) I ^ (a (k , r )) j = l 1 ' J J MT (a ( r +1, n) ) Th is proves ( i i i ) and ( i v ) Us ing t h i s p r o p o s i t i o n and p r o p o s i t i o n (3.2.8) i t i s s t r a i g h t (^") can be w r i t t e n as forward from (3.3.8) that E c _(PC-) = I R v (a ( r +1, n))" 1 f M y (a(r.+l, n))M y . (aTr^lTn")) i=l j = g J-.J i j , l r - D l D 2 where 71 I R ± v ( a ( r +1, r )) 1 TT I < , ( a ( r +1, r . ) ) i = l C j = l t k = 1 .-»•»* 3 J+ l M j f l ( a ( r j + 1 , r . ^ ) ) D, = v ( a ( r +1, n ) ) " 1 ! ^ (a ( r +1, n ) ) M y (a ( r +1 , n) ) . Cons ider a pa r t of the Young diagram of shape ( a , B) cons i s t i ng-of the l a s t box i n the p t h row and the e n t i r e q t h ~ row, p < q . Le t T y o be a s tandard tab leaux w i t h t+1 l e t t e r s _£, £+1 £+t d i s t r i b u t e d i n £+1 £+t row q row q t boxes i n c r e a s i n g order i n t h i s p a r t . Set T = w w„ . ,T , i = 1, . . . . t e. £+i £+1 e ' ' 1 o Then w ^ + ^ i s a row permutat ion of T y f o r j 4 i - l or i , as the j l e t t e r s £+i-l and £+i have e i t h e r not been moved from row q or have been re turned to row q . Hence f o r k 4 e^ and j 4 i - l or i , by theorem (2 .2 .4 ) . There fore I M y , ( a ( £+ l , £+t))M y (a (£+l , £+t)) J ' e o j = l o ' J " ."IT t«e •.. < » m > i : 1=1 o o 72 t (3 .3 .12) + JJ'KV (a„^-)M V ( a 0 J . . ) , . 1 ' e. .. , e. £+i e. , e. , Z+x i = l i - l i i ' i - l by p r o p o s i t i o n ( 3 . 2 . 8 ) . L abe l these three terms A , B, and C r e s p e c -t i v e l y . As the e n t r i e s o f M y(a^) depend on l y on the p o s i t i o n o f the-l e t t e r s i-1 and i i n a s tandard t ab l eaux , the above computat ion i s independent of the l e t t e r £_ and depends on l y on the rows p and q . Hence s e t F = v (a (£+ l , £+t) )~ 1 (A + B) = x _ t ( A + B) , (3 .3 .13 ) f = v (a (£+ l , £+t))~ 1C = x _ t C . p»q We now r e w r i t e (3 .3 .11) as f o l l o w s . Le t d . be such that w(r ' , r.y^T^ = T, . Then d . = e 3 c+1 j 1 d . 3 r . - r c where the e^'s are de f i ned as i n ( i i ) of p r o p o s i t i o n ( 3 . 3 . 1 0 ) . The p roo f of ( 3 . 3 . 1 0 ( i i ) ) shows the l e t t e r s r,., r^.+l, r j + j _ occur i n the l a s t square o f row c and i n the j + l s t row o f T^ i n i n c r e a s i n g o r d e r , i . e . i n a c o n f i g u r a t i o n as desc r i bed above.. Hence ( 3 . 3 . 1 0 ( i i ) ) and p r o p o s i t i o n (3 .2 .8 ) show v ( a ( r +1, n ) ) " V ( a ( r + 1 , n ) )M y n (a1>"+l7~n)) • JTv(.<r +1, V i " " ' I, H e . „ . . ( S <*r « > j=c . J i = r . - r +1 i - l i c i ' i-1 c 3 c s (3 .3 .14 ) = TT f * • j=c+l C ' J 73 S i m i l a r l y , f o r j < c the l e t t e r r . , r . + l , r - appear i n the l a s t square o f row j and i n the j + l s t row of T y . Not ing that f ^ = 0 f o r rows p and q of equa l l e n g t h and bo th be long ing to the same tab leaux of T y the computat ion (3 .3 .12) shows 1 f W  v ( a ( r . + l , r . + 1 ) ) ^ M ^ . ( a ( r . + 1 , r j + 1 ) ) M . } 1 ( a ( r .+1, r j + 1 ) ) (3 .3 .15) = F j > j + 1 f o r j < c . S u b s t i t i o n of (3 .3 .14) and (3.3.15) i n t o (3 .3 .11) now g ives (3 .3 .16) E (u c~) r c - l c-1 y R. TTF. ... + R u f . . k i = l j=x J , J •'i=c+l PROPOSITION (3 .3 .17) Lzt m bz thz coUal dUtanzz ifiom tkz i i u t hquafiz in tkz q t h now to tkz lat>t iquanz in tkz p t h fiow and OAhurnz fww p -c6 a now o^ thz tablzaux T a and now q <a> a now of* thz tablzaux T s oi T e = ( T a , T 6 ) . Lzt t bz thz Iznqtk oi tkz q t h now. Tkzn ( 4 \ f = A(m+1, y)A(m-t, y) W P,q A(m, y )A (m-t+l , y) » w _ ( x - l ) 2 A ( t , -1) p,q xA(m, y )A(m-t+l , y) PROOF : We use the n o t a t i o n s of ( 3 . 3 . 1 3 ) . ( i ) Because the l e t t e r _£ appears i n the l a s t box i n row p and £+1 i n the f i r s t box of row q , the a x i a l d i s t a n c e from £+1 to _£ i n T y i s m and the a x i a l d i s t a n c e from £+i to £+i-l i n T y i s e e. . o i - l m-i+1, i = 1, t . Therefore 74 M V ( a n j _ . )M U (anj_.) e. , , e . £+3.' e . , e . , £+i' i - l i i ' i - l . xA (m- i , y )A(m-i+2, y ) [A (m-i+l , y ) ] 2 by theorem (2 .2 .4 ) . Thus (3 .3 .11) imp l i e s C = JJ *A(m~±> y)A(m-i+2, y) i = l [A(m-i+l , y ) ] 2 x A(m+1, y)A(m-t, y ) A(m, y )A (m-t+l , y) As f = x tc , we have the r e q u i r e d r e s u l t , ( i i ) We have M ( a „ , , ) = x-1 ' i - l ' i - l A (m- i+ l , y) ' i 1, . . . , t , by theorem (2 .2 .7 ) and as i s a row permutat ion of T W f o r i > j + l , M e e ^ a£+i^ = x f o r 1 •  > b y t h e o r e m (2 .2 .4 ) . Hence (3 .3 .11 ) i m p l i e s y 3 A = x 2 ( t - 1 > ( x - l ) 2 [A(m, y ) ] 2 Fur thermore , u s i ng ( i ) and (3 .3.11) ^ [ JJ xA (m- i , y)A(m-i+2, y) | ( x - p V ^ " 1 k ) B = k= l M=l [A(m-i+l , y ) ] [A(m-k, y ) ] ' C 1 x k A(m+l , y)A(m-k, y) . ( x - l ) 2 x 2 ( t 1 k ) V x A^m+i, y;At,m-k,. ;  k ^ A(m, y)A(m-k+l, y) [A(m-k, y ) ] x t ~ 1 ( x - l ) 2 A ( m + l , y ) ^ 1 1 A(m, y) t - l - k A(m-k+l, y)A(m-k, y) 75 An easy i n d u c t i o n argument, us ing the f a c t tha t xA (k , - l ) A ( m - k - l , y) + A(m, y) = A(k+1, - l )A (m-k, y) shows t-1 t - l - k . , . y x = A ( t - 1 , -1) k £ 1 A(m-k+l, y)A(m-k, y) A(m, y )A (m-t+l , y) *• There fore TI _ x t " 1 ( x - l ) 2 A ( t - l , - l )A (m+l , y) - — —2 •• [A(m, y ) ] A (m-t+l , y) F i n a l l y , the above computations g i v e A + B „ x t " 1 ( x - l ) 2 ^t-1 , A ( t - 1 , - l )A (m+l , y) r w s •, 2 A (m-t+l , y) [A(m, y ) ] = x f c X ( x - 1 ) 2 A(m, y ) A ( t , -1) r. , <. n 2 A(m-t+l , y) [A(m, y ) ] v J J = x f c X ( x - l ) 2 A ( t , -1) A(m, y )A(m-t+l , y) As F = x t [ A + B ] , we have the r equ i r ed r e s u l t . P »H COROLLARY (3 .3 .18) l e t p and q be. rows belonging to tlie same tableaux oft . With the same notations as In proposition (3 .3 .17) we have e o (±\ f = A(m+1, - l ) A ( a - t , -1) K J "p,.q A(m, - l )A (m- t+ l , -1) (11) F = A ( t , -1) v 1 p,q xA(m, - l )A (m-t+ l , -1) 76 lvJitkoJvmon.z, p = q-1 and the, nam q and q-1 have: -tfie -same Zzngtk;-then F . = x 1 . q - l , q PROOF : I f the rows p and q be long to the same tab leaux o f T y , t h e o mat r i ces M(k, y) are r ep l aced by the mat r i ces M(k, -1) i n p r o p o s i t i o n -( 3 . 3 . 1 7 ) by theorem (2 .2 .7 ) . The mat r i ces M(k, -1) are ob ta ined from. M(k, y) by s e t t i n g y = - 1 , whence the f i r s t s ta tement . Fo r the second statement we have m = t and A ( l , -1) = 1 . 77 3.4 . GENERIC DEGREES R e c a l l from (1.1) tha t ( a ) ' = (a, ' , a 1 , ) denotes the p a r t i t i o n X s ct conjugate to the p a r t i t i o n (a) = ( a , , a ) a DEFINITION (3 .4 .1 ) Let ( a , 8 ) be a double p a r t i t i o n with corresponding ordered pair oft Young diagram (D(a ) , D ( 8 ) ) . For Pit ( i , ^ - s q u a r e oft D(a ) , bet h i , j = ( a i " j ) + ( a j " i ) + 1 ' •8? A - («± - j ) + (Bt - i ) + 1 , X » J J where 8 ! = 0 fton, j > s o l . Jon, the ( i , j ) - s q u a r e oft D ( 8 ) . set 3 P h B = ( 6 . - j ) + ( 8 ' - i ) + 1 , J-» 2 •>• J g i , = ( 8 , - j ) + ( a ! - i ) + 1 , wfieAe a j = 0 ^ J > s a i • W e- h ? j U e a p . h B ) the hook length oft the ( i , j ) - s q u a r e oft D(a) (/ie6p. D ( B ) ) . We c a l l g? . [resp. g? .) l » j l » j the s p l i t hook length oft the ( i , j )-square oft D(a) (^ie6p. D ( 8 ) ) • From ( 1 . 1 . 1 ) , h? . (hf .) •is the l e n g t h of the ( i , j ) - h o o k of i , 3 i , 3 t i l D(a) ( D ( 8 ) ) . As the l a s t square i n the i row of D(ct) has coord ina tes ( i , cu) w h i l e the l a s t square i n the j t b column o f D(ct) has coord ina tes ( a ! , j ) , (1 .1 .5 ) shows h? . equals the a x i a l d i s t ance from the ( a ! , j ) -3 i ,3 3 8 square to the ( i , a . ) - s q u a r e p lus one i n D (a ) . h. . has the same i i , 3 i n t e r p r e t a t i o n f o r the diagram D ( 8 )• The s p l i t hook l engths have a 78 cor respond ing i n t e r p r e t a t i o n . Namely, g? . i s the a x i a l d i s t ance from the square i n the j * " * 1 column of D(B) , the (8^, j ) - s q u a r e , to the l a s t square i n th ft the i row of D (a ) , the ( i , a . ) - s q u a r e , p l u s one. S i m i l a r l y g. . equa ls the a x i a l d i s t a n c e from the end o f the column o f D(a) tp the end o f the i f c k row of D(8 ) , p l u s one. PROPOSITION (3 .4 .2 ) ( i ) Ton. the double. paJvUtiotx ( a , (0)) ( (a - ) , (0)) E = A ( a c - c , y ) H a , whene . A (h . , -1) a -1 . ,, ct ... c-1 J ,ot c A(h . , -1) „•. - TT c- TT —§^  TunXkenmon.e j = l xA(h - 1 , - l ) j = l A ( h " - 1 , - l ) J ' a c ' J ot -1 . , a v c A(g . , y) A(a -c, y) = A ( g " , y) TT c j = l A(g - 1 , y) *-» J ( i i ) Ton. the. double, p a r t i t i o n ( ( 0 ) , 8) ( ( 0 ) , (B -) E = A ( 8 c - c , y )H , p wkene H u> defined cain ( i ) tkz pa/itiXion (g) replacing the panXLtion ( a ) . Tuntkenmon.e B c _ 1 A ( - g B y) A ( 8 c - c , y _ 1) = ( y x 1 _ C ) ~ 1 A ( - g ^ B , y) TT ^ c j = l x A ( - g ; +1, y) ( i i i ) Ton. the double paAtitlon ( a , B ) , (a) 4 ( 0 ) , (8) 4 ( 0 ) , 79 Izt now d bz thz now ol D(B) -6ucA that thz a c column ofi D(8) e.ndi -aa /tow d .^jj a c — 6 1 * ^ 8 1 < a c ' d = 0 • T^** ( (a - ) , (8)) ((a - ) , (0)) E = E G whznz -A ( g a , y) a -1. . a \ , A(-g 6 . , y) A<VC»-*> j = l A(g« . - 1 , y) j i l A(-g^ + 1 . y ) I £ -ci undzutood that thz la&t pnoduct In thz dz{Lnitlon ofi G - a takzn to bz zqual to 1 ifa d = 0 . PROOF : We w i l l show the exp ress ion f o r E g iven by (3 .3 .16 ) has the d e s i r e d form f o r each of the cases ment ioned. D i r e c t computations are a l l tha t i s r e q u i r e d . ( i ) For the double p a r t i t i o n ( a , ( 0 ) ) , the row c i s a row o f D(a) and = a £ f o r i = 1, . ' . . , c . Then by P r o p o s i t i o n (3 .3 .9 ) and C o r o l l a r y (3 .3 .18) c-1 c-1 1=1 1=1 R c c A(a . -1) I A(a - 2 i + l , y ) x i - l l-C ( 3 . 4 . ) = x 1 - C A ( a c , - l ) A ( c , - l ) A ( a c - c , y) . Let m. denote the a x i a l d i s t ance from the f i r s t sauare i n row i to the l * l a s t square i n row c, i > c. Then m^+j_ = and us ing C o r o l l a r y (3 .3 .18 ) 80 s s a a A(m +1, -l )A(m -a , -1) TT f = T T — — i=c+l C ' 1 i=c+l A ° V - D A C ^ - a . + l , -1) A(mc+rac+r - 1 } ^ A ( m . + 1 - a . + 1 > -1)  A ( mc + r -x> i=c+l A ^ i + l - a i ' Set a . = 0 f o r i , > s and r e w r i t e the above as J a i i i f i c « i " A ( m c + i > - 1) i i l i kIL1+i A < m i+ r k ' ~» where the second product i s taken equa l to 1 f o r a l l i such that o u + ^ = ou. I f  ai+±  K a i - a ' i + i e 1 u a l s t n e number o f columns o f D(ct) which end i n row i w h i l e i f a , . , = a . , no column ends i n row i . Then i + l l f o r i >_ c such that <* i + 1 < we have by (3 .4 .1 ) and (1 .1 .5 ) m. . , - k + 1 = a + ( i-c ) - k + l = h a , , a . , ,+1 < k < a . . i + l c ' c ,k ' i + l — — i Thus ' a i A ( m , . , - k + l , -1) a i A ( h " -1) U 1 + 1 : = TT HA , k = a . x 1 +l A ( m . + 1 - k , -1) k = B . ± 1 + l A ( h B , - l , -1) i + l i + l c ,k Fur thermore , by the cho ice of the row c , m , n = a > a ,, and h a = 1 , c+1 c c+1 c , a c so tha t A ( m c + r a c+r ~» _ i Y A ( h<U> - x ) A ( m c + 1 , -1) A < V 1 } k=ac+1+l A ( h^ k -1, -1) The above computations show 81 (3,4 a c A(h . , -1) 4) , : L - • S C T - . I T ±=c+i — . - ^ c ' i = i A ( h a : - i , - l ) c ,x F i n a l l y , c = h ° , so tha t x A(h ° , -1) 1-c, , , x l - c . „ _ o t " C — J ' a -(3 .4 .5 ) x x L A ( c , -1) = x x c A ( h " , -1) = T T 1 » ° c j = l - " a c xA (h , „ - 1 , -1) ( (a - ) , (0)) -" c (3 .4 .3 ) - (3 .4 .5 ) combine to g i v e the d e s i r e d exp re s s i on f o r E As the second p a r t i t i o n i s the empty p a r t i t i o n ( 0 ) , a -c = g . and C C y X • Ot ot g . = g . + 1, j = 1, c-1 , so tha t the exp ress i on f o r A(a - c , y ) C » J c » J ' i c g iven i n the statement of ( i ) i s j u s t a t e l e s c o p i c p roduc t . ( i i ) The proof of ( i i ) i s e n t i r e l y s i m i l a r to ( i ) and w i l l be o m i t t e d . S imply r ep l ace (a) w i t h (B) and note tha t by P r o p o s i t i o n (3 .3 .9 ) —1 3 y must be r ep l a ced w i t h y i n ( 3 . 4 . 3 ) . A l s o $ c ~ c = 8 C ^ s o tha t A(8 - c , y x ) = (yx m ) x A(m, y) , m = -g^ - . ( i i i ) In t h i s case the row c i s a row of D ( o t ) . Thus employing (3 .3 .16) ( (a - ) , ( 3 ) ) c-1 i _ ^ % S a ^ 8 E C I R i TT F i 1 + 1 + R c TT f c i TT f c i i = l 1 j = l 3 ' J + X C Ji=c+1 C ' x i=s + i 0 , 1  J a S 8 -TT f c l • ( (a - ) , (0)) s 3 E c i=s +1 a I t s u f f i c e s to show (3 .4 .6 ) TT f . = G . i=s +1 a 82 Let n u , i = 1, s , denote the a x i a l d i s t ance from the f i r s t box i n th the i row of D(3) to the l a s t box i n row c o f D (o ) . Then m. . , = m. + 1 and as i n ( i ) i + l l s +s. a 3 . T T + 1 £ c i i=s +1 a A(m +1, y) s Q , . s g ^ . A U . - g . , y) A(m, y) i s N A ( m ± + 1 - e i , y) * Let 8. = 0 f o r j > s„ and l e t d be as i n the statement o f the theorem. J 8 Rewr i te the above as (3 .4 .7 ) r A ( m d + 1 - d +1, y) d Ado^-B^- y) A(m>y) " ill A ( m i+ r B i » -y> 1 f A ( m d + r B d + r y ) - r f A ( m i + r 6 i + r y ) ' I A ( m d + 1 - a c + l , y) J ' + 1 A(m.+1-8i, y) J Labe l the two b racke ted express ions i n (3 .4 .7 ) A and B r e s p e c t i v e l y . . By (3 .4 .1 ) and ( 1 . 1 . 5 ) , m, - - a + 1 d+1 c (a -a ) + (d-c) + 1 c c = g c , a and m. - 8. = a + ( i-c ) - 1 - 8. 1 1 c i = "[(8.-a ) + (c- i ) + 1] = -g; Therefore A(C • y ) d AKa ' y)  A = L_9 T T S A ( ( V C > y> i = l A ( - g f +1, y) 83 As g,,, < ct , and as f o r B. . •, < k < 6.-1 , d+1 c l + l — — l m i + l - k = « c + ( i " c ) " K = A U + 1 » computations i d e n t i c a l to those employed i n ( i ) show a -1 . , a v c A(g , y) B = TT 1=1 A(g ° - 1 , y) The above express ions f o r A and B and (3 .4 .7 ) show (3 .4 .6 ) as r e q u i r e d . Th is completes the proof of ( i i i ) and the p r o p o s i t i o n . We can now o b t a i n an e legant fo rmula f o r C y = ( d egxV 1 I v ( a )-V(a )xy(a) weW(B ) W w w For the double p a r t i t i o n (u) = ( a , B) se t 1-a. (3 .4 .8 ) H" . = x J A ( h " . , -1) , J-tj -L»J a 1-3, o H. . = x J A (h . . , -1) , • = (yx m ) X A(m, y) , m = -84 u K THEOREM (3 .4 .9 ) Ton. the. irreducible, character x oft C\ ( B N ) » (y) = ( a , 8) a double, p a r t i t i o n oft n >_ 2 , ,P = TT u a r-a TT u8 ^3 TT H a . G a . JT H? .G. . ( i , j ) e ( a ) 1 ' 3 ^ ( i , j ) £ ( 8 ) '"•J PROOF : By i n d u c t i o n on n >. 2- The statement o f the theorem r e a d i l y be K checked f o r the r e p r e s e n t a t i o n of 01 (B 2 ) g i ven by Theorem ( 2 . 2 . 7 ) . Le t n > 2. The Young diagram D(y c ~) i s ob ta ined from the Young diagram D(u) by d e l e t i n g the l a s t square from row c . Thus the hook l eng ths of the squares of D(y c ~) d i f f e r from the hook lengths o f the squares of D(y) on ly f o r squares i n the c row and or 8 c column depending on whether the row c i s a row of the diagram D(a) o r the diagram D(8)' of D(y) = (D(ct), D ( 8 ) ) . Indeed, the hook lengths of these squares of D(y c ~) are one l e s s than the cor respond ing squares o f D ( y ) . S i m i l a r l y the s p l i t hook l eng ths of the squares of D(y c ~) d i f f e r from the s p l i t hook l eng ths o f th the squares o f D(y) on ly i n the " c row and, i f c i s a row of D(ct) and B c * 0 , i n the column of D ( B ) . Aga in the s p l i t hook lengths o f these squares o f D(y c ~) are one l e s s than the cor responding squares of D(y) . Thus, from the computations o f P r o p o s i t i o n (3 .4 .2 ) we have tha t E ^ c \ i s of the form ( V ) G l • H s t 1 » j s > t where X = a or 8, depending on whether the row c i s a row of D(oc) o r D (8 ) , and where ( i , j ) and ( s , t ) run over the app rop r i a t e squares o f 8 5 D ( u c ~ ) , mentioned above, f o r which, the hook l eng ths and s p l i t hook l eng ths d i f f e r from those of D(u) . As C y = c ^ c ^ C ^ c - ) b y Theorem (3.2.6) , t h i s completes the i n d u c t i o n and the p roo f o f the theorem. We can now g ive an e x p l i c i t exp re s s i on f o r the gene r i c degree K d ( 3 . 1 . 3 ) , o f the i r r e d u c i b l e cha rac te rs Y of (X (B ) . L e t P_ (x , y ) X n B n K be the PoinaaAt polyn.omi.aJi of (X (B n ) , (3 .4 .10 ) P R (x , y) = I v ( a ) . n weW(B ) n From ( [ 1 0 ] ) , n-1 . . P_ (x , y) = I (1 + x y ) . ( l . + . . . + x 1 ) . B n i=0 COROLLARY (3 .4 .11) To A ike. iAhtduciblt choAacttA x P oi 0 l K ( B n ) » (u) = ( a , B) , d y = P B ( x , y ) / " TT H« Gj TT X P n / ( i , j ) e ( a ) X » J ^ ( i , j ) e ( B ) ± t 2 X » J PROOF : Th is f o l l o w s from the d e f i n i t i o n o f d ( 3 . 1 . 3 ) , (3 .4 .10) and •X Theorem ( 3 . 4 . 9 ) . K The gene r i c degrees of the r e p r e s e n t a t i o n of 0\ (A n ) and K C\ (D ) are r e a d i l y ob ta ined from Theorem (3 .4 .9 ) as w e l l . I n p a r t i c u l a r COROLLARY (3 .4 .12) Let cf> : D = Q[x, y] —> Q(x) bt Hit homomoApki&m dtiintd by cp(y) = 0 . Let (a) bt a poAtition oi n and Ztt x ° and 86 x ( o , ( 0 ) ) b e ^ £ vvteducible characters oft C J t K ( A n _ 1 ) and C ( K ( B n ) corresponding to (a) and (a , ( 0 ) ) . Let <j>* : D p — > Q(x) be the extension oft $ to the King oft ftractions, P = k e r <p . Then d ( a , (0)) £ ° P a n d * H d ( « , <0))> " d a * A. A. A. PROOF : For' w e W(B ) , de f i ne JZ,n (w) to be the number o f times w. n 1 1 occurs i n a reduced exp re s s i on f o r w i n R , and se t J2.^ Cw) = &(w) -Then v ( a ) = y x , a E 0((B ) • w • . ' w n We f i r s t show (3 .4 .13 ) x < a ' < 0 » (O - y V"V"-' < 6 ) )«t). V W W Jr n Le t M(a) denote the m a t r i x of i r ^ a ' (a) w i t h respec t to the b a s i s { t x , t f } of ( Q ) ) , a e 01 K ( B n ) . By ( 2 . 2 , 6 ) , M(a^) = y l and thus commutes w i t h M(a^), i = 2, n . Hence f o r w e W(B n) A, (w) _ M(a w ) = (M(a x ) ) A M(a w ) where - r< W W • . r , a = > c a , c e Q[x] W geW(A n _ 1 ) ^ 8 From ( 2 . 2 . 6 ) , x^"' ( a w ) e Q(x) C Dp, D p cons idered as a sub r i ng of K = Q(x, y ) . Thus we have shown ( 3 . 4 . 1 3 ) . , The r e s t of the proof i s now c l e a r . As £^(w) = £^(w x ) , we have 87 9*(C ) = 1 v C O X (a ) x (a ) = C . weW(A ,)' • n-1 S i m i l a r l y weW(B ) weW(A .) n n-1 and the statement o f the c o r o l l a r y f o l l o w s . Thus by the above c o r o l l a r y and Theorem (3 .4 .9 ) we have , f o r (a) a p a r t i t i o n o f n , (3 .4 .14) d a - P A W/.ITHJ.J x n-1 / x , j where n P. (x) = T]" (1 + . . . + x 1 ) . n-1 1=0 COROLLARY (3 .4 .15) Lzt 9 : D = Q[x, y] —> Q(x) be thz homomonphiAm dziinzd by <j>(y) = 1. Lzt (y) = ( a , B) be a double, partition oi n witk (a) ^ (6) ami l z t x y &ttd be ike. iAAe.ducA.blz chanactzm, oi C \ K ( B n ) K and C\ (D ) connz6ponding to (y) . Then <j>*(d ) = d , whznz N X P TJ»V • 9* : Dp — > Q(x) ij> tkz zxtzmion oi $ to tkz Aing oi inaction* D p , P = k e r 9 u K PROOF : From Theorem (2 .3 .9 ) , ijr i s the r e s t r i c t i o n to Q[ (D R ) o f the i r r e d u c i b l e cha rac te r X i ° f ® ± T^(B ) , K = Q(x) , where 9 9 ,K n X^ ( a w ) = cb*(x y ( a w ) ) . Thus by the d e f i n i t i o n of gener i c degree , to prove the c o r o l l a r y i t i s s u f f i c i e n t to prove 8 8 (3 .4 .16 ) I v(a .)'V(a )i'j(a .) = 2 £ v(a. . )"V(a , )X»(a) weW(B ) w * 9 w 9 <p w<p w£w(D') w* * w* * * n n f o r (y) as i n the statement of the c o r o l l a r y . For any w e w ( B n ) and cor respond ing b a s i s element a . e <JT n, . (B ) , we have a ,a ' = a w<p <p, Q (x) n ' w<p w^ <p ww^<p' 2 as (a ) = 1. Thus, u s ing the o r t h o g o n a l i t y r e l a t i o n s (3 .1 .4 ) and the w-j_<P coset decomposi t ion W ( B n ) = W.(Dn) U W(D n )w 1 , we have <p*(C ) - I «<^-)rf(aw.)xJ(a ) weW(B ) W9 9 w<p 9 w<p n -I I v ( a ) V . ( a A ) M u . ( a A ) w£w(B ) i-1 W * 1 1 W * 1 1 W * n' f P (3 .4 .17) = 1 I v ( a J " 1 weW(D ) i = l W * n M V . ( a . )M? . (a .) + X X W(p I X w<p M?. (a x a j M y ( a A a ,) . xx w<p w^9 w^ <p w<j> J where M (a ,) i s the m a t r i x of *Try(a ) w i t h r espec t to the l a s t l e t t e r W 9 9 w r sequence arrangement of the b a s i s { t . , t,.} of . Then M V ( a ,) 1 f y w^9 i s a d i a g o n a l m a t r i x w i t h e n t r i e s + 1 b y (2 .3 .9 ) so M V . ( a A a . ) M V . ( a . a .) = M? . (a j M V . ( a J . xx w<j> w^9 xx w^tp w<p xx W9 xx W9 Then (3 .4 .17) becomes f y 9*(c ) = 2 y f ,. ^ i i i v V ^ y f t c v " i 2 ' v ( a J ' V Y * N u,» 89 u K as the r e s t r i c t i o n of TT, to Q[ (D ) i s an a b s o l u t e l y i r r e d u c i b l e K r e p r e s e n t a t i o n of 0\ (D )• Th is proves (3 .4 .16) and completes the p roo f . We conclude w i t h an example. We c a l c u l a t e the gene r i c degree o f the milrlcZion nzph.eAZn£a£ion o f the gene r i c a l geb ra of c l a s s i c a l t ype , a computat ion a l s o g i ven i n ( [6] ) (as a po l ynomia l i n one v a r i a b l e ) . The double p a r t i t i o n (y) = ( (n-1) , (1)) y i e l d s the /t^lzctLon kZpAZAtnicuUon of Cl K ( B n ) . n-1 squares • (D (n- l ) , D ( l ) ) We have ir~~3 n. 2 c ( ( n - l ) , (1)) m y - l ( 1 + ^ - 1 ^ % ( 1 + x l y ) f f (1 + . . . + x 1 ) i = - l 1=1 Thus d = p R (x , y ) / c y = y ( 1 + • • • + * n ~ 1 ) ( 1 + * n 2 y } X y B n / d + x ^ y ) S e t t i n g y - 1 i n the above we have by (3 .4 .15) the gener i c degree of the r e f l e c t i o n r e p r e s e n t a t i o n o f Q (D R ) . The p a r t i t i o n (a) = (n-1, 1) o f n y i e l d s the r e f l e c t i o n r e p r e s e n t a t i o n of Cl (A ,) . 90 l _ J n~L squares D(n-1, 1) K The gener i c degree of the r e p r e s e n t a t i o n of 0{ (B n ) corresponding- to the double p a r t i t i o n (a , (0)) = ( (n-1, 1 ) , (0)) i s , = x ( l + x n -"-y) (1 + . . . + x n " 2 ) x c . <o» - ( 1 + ] f i S e t t i n g y = 0 i n the above we have by (3 .4 .12 ) the gene r i c degree of the K r e f l e c t i o n r e p r e s e n t a t i o n o f d (A R ± J ' F i n a l l y we remark tha t from (3 .1 .7 ) the above c o r o l l a r i e s g i v e Q the degrees of the i r r e d u c i b l e c o n s t i t u e n t s of 1 f o r G a f i n i t e group B w i t h BN-pair w i t h Coxeter system of c l a s s i c a l type by s u b s t i t u t i o n o f the index parameters i n the formula f o r the gene r i c degree. In p a r t i c u l a r , these computations app ly to the f a m i l i e s of Cheva l l ey groups A^(q ) , B^(q), 1 2 1 2 1 2 D (q) , A 2 £ ^ q ^' A 2 £ - l ^ q )» D £ (9 ) • '^ i e degrees of these cha rac te r s f o r the f a m i l i e s of type A (q) were a l ready known (see [16 ] ) . 91 BIBLIOGRAPHY Benson, C.T. and C u r t i s , C.W., On the Degrees and R a t i o n a l i t y o f C e r t a i n Charac ters of F i n i t e .Cheval ley Groups, T r ansac t i on o f the American Mathemat ica l S o c i e t y , 165 (1972), 251-273. B o u r b a k i , N . , Groupes e t A lgebres de L i e , Chap. 4, 5, 6, P a r i s , Hermann, 1968. , Alge"bre Commutative, Chap. 5, 6, P a r i s , Hermann, 1964. C a r t e r , R.W., Conjugacy C lasses i n the Weyl Group, Composi t io Mathemat ica, 25 (1972), 1-59. C u r t i s , C.W. and Fossum, T . V . , On C e n t r a l i z e r Rings and Charac ters of Representa t ions o f F i n i t e Groups, Mathematische Z e i t s c h r i f t , 104 (1968), 402-406. C u r t i s , C.W., Iwahor i , N. and K i l m o y e r , R., Hecke A lgebras and Characters of P a r a b o l i c Type of F i n i t e Groups w i t h ( B ,N )-pa i r s , I n s t i t u t des Hautes Etudes S c i e n t i f i q u e s , P u b l i c a t i o n s Mathematiques, 40 (1971), 81-116. C u r t i s , C.W. and Re i ne r , I., Represen ta t ion Theory of F i n i t e Groups and A s s o c i a t i v e A l g e b r a s , New Yo rk , John Wi ley and Sons ( I n t e r s c i e n c e ) , 1962. Green, J . A . , On the S t e inbe rg Charac ters of F i n i t e Cheva l l ey Groups, Mathematische Z e c t s c h r i f t , 117 (1970), 272-288. Iwahor i , N . , On the S t r u c t u r e of the Hecke R ing of a Cheva l l ey Groups over a F i n i t e F i e l d , J ou rna l of the F a c u l t y of S c i ences , Tokyo U n i v e r s i t y , 10(2) (1964), 215-236. MacDonald, I .G . , The Poincare" Se r i e s of a Coxeter Group, Mathematische Anna len , 199 (1972), 161-174. Matsumoto, H . , Generateurs e t Re l a t i ons des Groupes de Weyl Generalises, Comptes Rendus, L*Academic Des Sc iences P a r i s , 258 (1969), 3419-3422. Rob inson , G. de B., Representa t ion Theory of the Symmetric Group, Toronto^ U n i v e r s i t y of Toronto P r e s s , 1961. R u t h e r f o r d , D.E . , S u b s t i t u t i o n a l A n a l y s i s , Ed inburgh , Edinburgh U n i v e r s i t y P r e s s , 1948. 92 14 . S t e i n b e r g , R., Lec tu res on Cheva l ley Groups, Lec tu re No tes , Y a l e U n i v e r s i t y , New Haven, Conn; , 1967. 15 . •• _, V a r i a t i o n s on a Theorem of Cheva l l e y , P a c i f i c J o u r n a l o f Mathemat ics , 9 (1959) , 875-891. 16. , A Geometr ic Approach to the Representa t ions of the F u l l L i n e a r Group over a G a l o i s F i e l d , T ransac t i ons of the American -Mathemat ica l Soc i e t y 41 (1951) , 279-282. 17 . T i t s , J . , A l g e b r a i c and A b s t r a c t Simple Groups, Annals o f Mathemat ics , 80 (1969) , 313-329. 18 . Young, A . , On Q u a n t i t a t i v e S u b s t i t u t i o n a l A n a l y s i s IV , P roceedings of the London Mathemat ica l S o c i e t y , 31(2) (1930) , 253-272. 19 . , On Q u a n t i t a t i v e S u b s t i t u t i o n a l A n a l y s i s V, Proceedings of the London Mathemat ica l S o c i e t y , 31(2) (1930) , 273-288. 20. , On Q u a n t i t a t i v e S u b s t i t u t i o n a l A n a l y s i s V I , P roceedings of the London Mathemat i ca l S o c i e t y , 34(2) (1932) , 196-230. 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            data-media="{[{embed.selectedMedia}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
https://iiif.library.ubc.ca/presentation/dsp.831.1-0080112/manifest

Comment

Related Items