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UBC Theses and Dissertations

Characters of the special linear group Bates, Susan 1971

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THE CHARACTERS OF THE SPECIAL LINEAR.GROUP SL (2 ,q ) by Susan Ba te s B.A. ( H o n s . ) , U n i v e r s i t y o f B r i t i s h Co l umb i a , 1969 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS.FOR THE DEGREE OF MASTER OF ARTS i n the Department o f Ma themat i c s Ve a c c e p t t h i s t h e s i s as c o n f o r m i n g to t he r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA September, 1971 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced deg ree a t t he 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 a g r e e t h a t t he 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 t u d y . I f u r t h e r ag ree 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 rpo se s 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 . I t i s u n d e r s t o o d t h a 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 o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f The U n i v e r s i t y o f B r i t i s h Co l umb i a Vancouve r 8, Canada S u p e r v i s o r : Dr. B. Chang ABSTRACT The purpose of t h i s t h e s i s i s to determine the o r d i n -ary and p-modular i r r e d u c i b l e c h a r a c t e r s and the c h a r a c t e r s of the p r i n c i p a l indecomposable modules of the group SL ( 2,q), q=p n, f o r odd p. The decomposition matrix a.nd the Cartan matrix f o r SL(2,q) are a l s o given. ACKNOWLEDGMENTS I would l i k e to express my g r a t i t u d e to Dr. Chang f o r a l l the help and encouragement he so r e a d i l y o f f e r e d w h i l e t h i s t h e s i s was being prepared. I would a l s o l i k e to thank Dr. Ree f o r reading the t h e s i s ; and the N a t i o n a l Research C o u n c i l and the U n i v e r s i t y f o r f i n a n c i a l a s s i s t a n c e . TABLE OF CONTENTS S e c t i o n I I n t r o d u c t i o n 1 S e c t i o n I I The Conjugacy Classes of SL(2,q).... 2 S e c t i o n I I I The Ordinary I r r e d u c i b l e Characters of SL(2,q) 12 S e c t i o n IV The Modular I r r e d u c i b l e Characters of SL(2,q) 33 S e c t i o n V The Decomposition M a t r i x f o r SL(2,q) 40 S e c t i o n VI The Cartan M a t r i x f o r SL(2,q) 49 S e c t i o n V I I The Characters of the P r i n c i p a l Indecomposable Modules 56 B i b l i o g r a p h y 68 I I n t r o d u c t i o n L e t F be the f i n i t e f i e l d w i t h q = p n e l e m e n t s where p q i s an odd p r i m e . L e t G be the group SL(2,q) o f a l l 2 x 2 m a t r i c e s w i t h d e t e r m i n a n t 1 o v e r t h e f i e l d F . The aim o f t h i s t h e s i s i s t o q d e t e r m i n e the o r d i n a r y i r r e d u c i b l e and p-modular i r r e d u c i b l e c h a r a c t e r s o f G , and t h e c h a r a c t e r s of the p r i n c i p a l i n d e c o m p o s a b l e KG-modules, where K i s a f i e l d of c h a r a c t e r i s t i c p . I n S e c t i o n I I , we examine the S y l o w subgroups of G t o d e t e r m i n e the c o n j u g a c y c l a s s e s . We f i n d q c l a s s e s o f p - r e g u l a r e l e m e n t s , t h a t i s , , e l e m e n t s whose o r d e r i s p r i m e t o p , and 4 c l a s s e s o f p - i r r e g u l a r e l e m e n t s , t h a t i s , e l e m e n t s whose o r d e r i s d i v i s i b l e by p .. I n S e c t i o n I I I , we f i n d the o r d i n a r y i r r e d u c i b l e c h a r a c t e r s o f G , i . e . the c h a r a c t e r s a f f o r d e d by 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 v e r a f i e l d o f c h a r a c t e r i s t i c 0 . These c h a r a c t e r s were f i r s t d e t e r m i n e d by S c h u r [4] i n 1907 and i n d e p e n d e n t l y by J o r d a n [3] i n the same y e a r . I n S e c t i o n I V we d e s c r i b e t h e p-modular i r r e d u c i b l e r e p r e s e n a t i o n s o f G , t h a t i s , 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 v e r a f i e l d o f c h a r a c t e r i s t i c p . These r e p r e s e n t a t i o n s were d e t e r m i n e d by B r a u e r and N e s b i t t [1] i n 1941. We f u r n i s h t h e i r p r o o f s t h a t t h e s e r e p r e s e n t a t i o n s g i v e the q d i f f e r e n t i r r e d u c i b l e p-modular r e p r e s e n t a t i o n s of G . We a l s o d e s c r i b e t h e s y m m e t r i c power r e p r e s e n t a t i o n s o f G ( c a l l e d " i n d u c e d m a t r i x " r e p r e s e n t a t i o n s i n [ 5 ] ) . I n S e c t i o n V, we show t h a t t h e d e c o m p o s i t i o n m a t r i x ( [ 2 ] p. 591) f o r G has a l l e n t r i e s e q u a l t o 0 o r 1 , and i n d i c a t e where the non-z e r o e n t i r e s o c c u r . We use a s l i g h t l y d i f f e r e n t a p p r o a c h f r o m t h a t o f S r i n i v a s a n [ 5 ] , who f i r s t obtained these r e s u l t s i n 1962. I n S e c t i o n V I , we compute the diagonal e n t r i e s of the Cartan matrix using a w e l l known theorem of Brauer ( [ 2 ] , p . 593) on the r e l a t i o n between the decomposition m a t r i x and the Cartan matrix. We i n d i c a t e b r i e f l y where the non-zero o f f - d i a g o n a l e n t r i e s l i e . In S e c t i o n V I I , the characters of the p r i n c i p a l indecomposable modules.are computed. Again we use a s l i g h t l y d i f f e r e n t approach from t h a t of S r i n i v a s a n [ 5 ] . N o t a t i o n and Terminology We use the standard c h a r a c t e r theory terminology throughout. Any p e c u l i a r i t i e s of n o t a t i o n w i l l be e x p l a i n e d as they occur. Roman numerals r e f e r to s e c t i o n s ; a r a b i c numerals r e f e r to primary and secondary d i v i s i o n s of s e c t i o n s , that i s , I I . 3 . 1 f o r example, r e f e r s to s e c t i o n I I , p a r t 3, p a r t 1 of I I . 3 . The lemmas are numbered i n order of occurrence. I I The Conjugacy Classes of G = SL(2,q) . Let U H w B x e F h e F I q^ H U 3. Lemma 1. (Bruhat Decomposition) G = B I) B w B = HU J HU w U gives a double coset decomposition of SL(2,q) w i t h respect to B Furthermore, i f g = hu^ or hu^ w U2 h E H , , £ U then h , , and are uniquely determined by g e G Proof. L e t g = e SL(2,q) . We have two cases: ( i ) c = 0 Then g = /a b ( 0 a -1 fa 0 ^ fl a " 1 b 0 a 0 V e HU Uniqueness: I f g = / h O \ 1 x j e H U , then we have h = a 0 h 1/ \ 0 1 hx = b hence x = a ^ b On the other hand, i f a b 0 d = /h 0 0 \ / ' l 3 I l V 0 1 V 1 X 2 > l H V 0 I A - 1 0 A 0 1 j E HU w U , then we have h = 0 , which i s im p o s s i b l e . 4. ( i i ) c t 0 Then g = / - c " 1 0 I 0 -c 1 ac ; \(o 1 c d j e HU w U 0 1/ l - l 0 j \ 0 11 / \ Uniqueness: C l e a r l y g \. HU I f g = / h 0 - h " 1 - c - h _ 1 x 2 = d e HU w U , hence h = -c -1 x^ = ac - 1 , x 2 = c d As an immediate consequence of Lemma 1, we have |G| = ( q - l ) q + (q-1) q.q = q ( q - l ) ( q + D Lemma 2. SL(2,q) - U(2,q) . Proof. The u n i t a r y group U(2,q) i s defined by U(2,q) = f / a b \ a a q + b b q = 1 ; a , b f F 2 Let a be the Frobenius automorphism of SL(2,q 2) , i . e . , of a b \ = / a ^ b q \ c d c q d q 5. Then SL(2,q) and U(2,,q) may be c h a r a c t e r i z e d as subgroups of 2 SL(2,q ) as f o l l o w s : SL(2,q) = {x e SL(2,q 2) | x° = x} U(2,q) = {y e SL(2,q 2) | w y° w"1 = y} where w =/ 0 1\ as before. ^-1 0, where X^ , X 2 e F 2 , X^ * = -X^ ,' X 2 = q 2 We show that the conjugation of SL(2,q ) by z r e s t r i c t e d to SL(2,q) gives the d e s i r e d isomorphism. We observe that z ^ z - w , so that f o r x e SL(2,q) we have . -1 . a - l - a a a -1 w(z x z ) w = w z x z w -1 a = Z X z -1 = z X z . _] 2 Hence z " x z e U(2,q) . Therefore, conjugation of SL(2,q ) by z maps SL(2,q) on U(2,q) , i . e . , SL(2,q) - U(2,q) 6. The Sylow Subgroups of G = SL(2,q) . Since | G | = q(q-l)(q+l) and (q-1 , q+1) = 2 , for any prime d i v i s o r d of | G | , d 4 2 , only one of the following occurs: (i) d I q ( i i ) d I q - 1 ( i i i ) d I q + 1 ( i ) Sylow p-subgroups of SL(2,q) . x e F ") i s order q = p n , every p-element Since U = 1/1 x 1 O I of SL(2,q) i s conjugate to an element of U . The centre Z ( G ) of G i s a 2-group generated by f-1 0 \ . Now i f x e U , then C(x) = U Z , where \ 0 -1 ) C(x) i s the c e n t r a l i z e r of x . Hence i f the order of an element of g E G i s d i v i s i b l e by p then |g| = p or 2p and g i s conjugate to an element i n U Z . Next we f i n d how many conjugacy classes are determined by the elements of order p and 2p . Suppose g - 1 / l x \ g = / l y | 0 1 0 1 2 A simple computation shows that g = fa b \ and x = a y . Hence 0 a" 1) 1 1 \ i s conjugate to / 1 x \ whenever x i s a square i n F^ , i . e . 0 1 / \ 0 1 V 2 * whenever there ex i s t s y e F s.t. x = y . I f A e F i s a non-square, q q 7. then f l X \ i s conjugate to /1 x \ whenever x i s a non-square i n \0 1 / • I 0 1/ Furthermore, / 1 1 \ and f1 X \ are not conjugate. 0 1 J [oi] S i m i l a r l y , f-1 -1 \ i s conjugate to f-1 -x \ whenever x i s a square i n F , and / -1 -X \ i s conjugate to /-1 -x \ whenever x i s [ o - i j [o-i * a nonsquare i n F^ . ( i i ) Sylow r-subgroups of SL(2,q) , r [ q - l , r ^ 2 . * Since H = |/h 0 N h e F I i s c y c l i c of order q - 1 , q 0 h any r-element of SL(2,q) i s conjugate to an element of H . I f D = /rh o \ e H , h ^ + 1 , then C(D) = H . Hence i f the order of an -1 ^0 h element x £ SL(2,q) i s d i v i s i b l e by r , then x i s conjugate to an element of H . Next, suppose that g ^ / h o \ g = /h'0 \ , h ^ + 1 ( o h " 1 ) [o h ' " 1 I f g \. H , then a computation shows that g = j 0 'b \ f o r some b and [-b- 1 0 —1 * h' = h . Therefore i f h e F i s a generator of the m u l t i p l i c a t i v e q group F , then / h" 0 \ f o r a = 1' , 2 , ... , form a s e t of q 0 h " a r e p r e s e n t a t i v e s of the conjugacy c l a s s e s of elements x e G such that |x| * 2 , (|x| , q-1) + 1 . ( i i i ) Sylow s-subgroups of SL(2,q) , s[q + 1 , s / 2 . Let * q "^ ' i a e F ^  > aa = 1j q — 2 — Then K <_ U(2,q) <_ SL(2,q ) , K i s c y c l i c of order q + 1 . By Lemma 2, there e x i s t s a c y c l i c subgroup K of order q + 1 i n SL(2,q) such that z ^  Kz = K . Hence any Sylow s-subgroup (s|q + 1 , s £ 2) of SL(2,q) i s conjugate to a subgroup of K . Since conjugacy r e l a t i o n s i n U(2,q) determine conjugacy r e l a t i o n s i n SL(2,q) , we have f o r x e K , C(x) = K • So i f x e SL(2,q) , s||x| then x i s conjugate to an element of K . We need to f i n d how many conjugacy c l a s s e s of U(2,q) the elements of K determine. Suppose g 1 / a O \ g = / b O \ e K U a q 0 b q g e U(2,q) , a , b e F 2 , a ^ + l I f g | K then g = A ) d \ and b = /o „q Let / k G \ be a generator of K . 0 k q ) Then / k 0 \ , 8 = 1 , 2 , . . . , ^ r - , form a set of re p r e s e n t a t i v e s of conjugacy c l a s s e s of x e U(2,q) , such that | x | ^ 2 , (Jxj,q+1) ^ 1 9. Denote the class of SL(2,q) corresponding to kP 0 \ = /k P 0 o k e q / Vo k Sylow 2-subgroups of SL(2,q) . Case 1: 4 |q - 1 . We show that every 2-element is conjugate to an element in H . We have | G | =q(q-l)(q+l) . Suppose 2 A ||G| , 2* + 1 :'(:\G\ . If 4.|q - 1 , then 4 \ q + 1 so 2 is the highest power of 2 which divides q + 1 . 2 | q so we must have 2 3 ^ | q - 1 . Let W = <w> . Then |w| = 4 and |H W | = 2(q-l) . Hence 2 a | | H W| and HW contains a Sylow 2-subgroup of SL(2,q) . Therefore i f x e SL(2,q) is a 2-element, x is conjugate to an element in HW . It remains to show that every element in HW is conjugate to an element in H Then g / 0 h \ g = / / : i o \ e H . 0 - / - I 10. Case 2: 4 | q + 1 . We show that every 2-element i s conjugate to an element i n K . Note that w = /"0 l \ = / o l\ e U(2,q) . Then -1 o j 0 u s i n g Lemma 2 and an argument analogous to that i n Case 1 ? we have any 2-element x e U(2,q) i s conjugate to an element i n KW . Let W' = z W z ^ . We show that any element i n KW i s conjugate to an element i n K and conclude that any 2-element i n SL(2,q) i s conjugate to an element i n KW' and t h e r e f o r e to an element i n K . Let / 0 a \ e KW a a q = 1 - a q 0 Take g = / c A ac q \ e U(2,q) +/T a q c c q j q+1 * where c = 1/2 . ( I f e i s a generator of F ^ , q 2 i s an even power of e because |2| d i v i d e s or i s equal to p - 1 . 2 P = ( 1+1 ) P = 2 . Hence /2 e F* 2 so q could take c = 1//2 ) Then g 1 / 0 a \ g = / +/-1 0 \ e K - a q 0 1 [ o -VZi J This completes the determination of the conjugacy c l a s s e s of SL(2,q) . 11. Summary We summarize our r e s u l t s w i t h a t a b l e : Representative of the c l a s s The order of C e n t r a l i z e r No. of elements i n the c l a s s 1 0 0 1 '-1 0 0 h . 0 1 < a < 0 h q-3 2 -a q ( q + l ) ( q - l ) q ( q + l ) ( q - l ) q-1 q(q+D k 6 0 1 < 0 k~ q-1 ± 2 q+1 q ( q - D 0 1 1 0 X 1 -1 0 -1 0 -X -1 2q 2 2q 2 2q . 2q 2 We have 1 + 1 + + + 4 = q + 4 c l a s s e s i n a l l . 12. I l l The O r d i n a r y I r r e d u c i b l e C h a r a c t e r s o f SL(2,q) . T h e r e a r e q + 4 ( = number o f c o n j u g a c y c l a s s e s ) o r d i n a r y i r r e d u c i b l e c h a r a c t e r s of SL(2,q) . §111.1 A b r i e f d e s c r i p t i o n o f t h e methods u s e d to d e t e r m i n e the o r d i n a r y  i r r e d u c i b l e c h a r a c t e r s of SL(2,q) . The i n d u c e d c h a r a c t e r s o f the n o n - p r i n c i p a l l i n e a r c h a r a c t e r s o f B g i v e -^p- i r r e d u c i b l e c h a r a c t e r s o f G . ( S t e i n b e r g ( [ 6 ] , Theorem 47, p.246) has shown t h a t t h i s method i s a p p l i c a b l e t o a l l f i n i t e C h e v a l l e y g r o u p s ) . The i n d u c e d c h a r a c t e r o f the p r i n c i p a l c h a r a c t e r o f B i s the sum o f t h e p r i n c i p a l c h a r a c t e r o f G and an i r r e d u c i b l e c h a r a c t e r o f degree q , w h i c h i s known as a S t e i n b e r g c h a r a c t e r . 2 The c h a r a c t e r s o f S L ( 2 , q ) o b t a i n e d by the f i r s t method above r e s t r i c t e d t o SL(2,q) minus the i n d u c e d c h a r a c t e r s o f HW g i v e a n o t h e r •^2^ i r r e d u c i b l e c h a r a c t e r s o f G . ( U n l i k e t h e f i r s t method, t h i s one c a n n o t , i n g e n e r a l , be e x t e n d e d t o o t h e r C h e v a l l e y g r o u p s . ) Two o f t h e 4 c h a r a c t e r s i n d u c e d by t h e n o n l i n e a r ( d e g r e e 2) c h a r a c t e r s o f B minus the c h a r a c t e r i n d u c e d by a n o n - p r i n c i p a l , n o n - a l t e r n a t i n g l i n e a r c h a r a c t e r o f HW g i v e 2 i r r e d u c i b l e c h a r a c t e r s o f G ; and the o t h e r 2 c h a r a c t e r s i n d u c e d by t h e n o n - l i n e a r c h a r a c t e r s o f B minus th e c h a r a c t e r i n d u c e d by a n o n - p r i n c i p a l , n o n - a l t e r n a t i n g l i n e a r c h a r a c t e r o f KW' g i v e 2 i r r e d u c i b l e c h a r a c t e r s o f G . (Which i n d u c e d c h a r a c t e r o f B goes w i t h w h i c h d i f f e r e n c e depends on w h e t h e r 4 | q - 1 o r 4 | q + 1 .) We f i r s t d e t e r m i n e c h a r a c t e r t a b l e s f o r B , HW and KW' . 13. The f o l l o w i n g n o t a t i o n w i l l be used throughout t h i s s e c t i o n : ( i ) x(a) = / l a \ , a e F q ( i i ) X denotes a f i x e d nonsquare i n F q * ( i i i ) h i s a generator of F . I f 8 i s a Ct / ct character, we w r i t e 8(h ) f o r 8/h 0 o h " a ( i v ) k i s a generator of the c y c l i c subgroup or order q + 1 i n F 2 ' k B 0 \ = z / k P 0 \ z 1 e SL(2,q) O k " 3 O k I f 0 i s a cha r a c t e r , we w r i t e 8(k ) f o r e / k 3 o \ , e(k 3 ) f o r e/k 6 o \ V° k"e) 1° (v) I f a , b e SL(2,q) , we w r i t e a — b f o r " a i s conjugate to b " , and a f b f o r " a i s not conjugate to b " . §111.2. The o r d i n a r y i r r e d u c i b l e characters of B .. We note the f o l l o w i n g conjugacy r e l a t i o n s i n B : ( i ) f ha 0 \ f h~a 0 \ . Hence the U h " a 0 k a determine q - 3 conjugacy c l a s s e s i n B 14. ( i i ) I f 4 I q - 1 , then -1 i s a square i n * F^ . Hence the elements of the p - i r r e g u l a r c l a s s e s are conjugate to t h e i r i n v e r s e s . ( i i i ) I f 4 \ q - 1 , then -1 i s not a square i n F^ . The elements of the p - i r r e g u l a r c l a s s e s are not conjugate to t h e i r i n v e r s e s . We have [ x ( l ) ] " 1 — x ( X ) [ - x d ) ] " 1 ~ - x ( A ) The conjugacy c l a s s e s of B are as f o l l o w s : Class /1 r e p r e s e n t a t i v e 0 -1 0 -1 /I 0 h " AO 1 Jl 0 1 / \ 0 -1 C e n t r a l i z e r q ( q - l ) q ( q _ l ) q ~ l 2q 2q 2q Elements i n c l a s s Sol i l l a z l 2 2 2 Since H i s c y c l i c of order q - 1 , the ord i n a r y i r r e d u c i b l e characters of H are given by the l i n e a r r e p r e s e n t a t i o n s n i ( h ) = e 1 _< l £ q - 1 where e e C i s a p r i m i t i v e q - 1 s t root of u n i t y . 15. Since B/U ~ H , each of the q - 1 i r r e d u c i b l e characters of H gives an i r r e d u c i b l e character of B . Let 3^ be the chara c t e r of B given by , i = 1 , 2 , ... , q - 1 . For g = hu , h .e H , u e U we have 3^(g) = n^(h) . (By Lemma 1, h i s uniquely determined by g e B) . We have to determine 4 more i r r e d u c i b l e characters of B . Denote these characters by 3 ^ , 3 ^ , g ^ , . Let n. be the character of B induced by r\. . Then n. B (1) n . K (-1) = ( - D 1 q n i (h ) n. B (g) = o f o r g p - i r r e g u l a r . We have , B B. (n. , n. ) i ' " i 'B " q l F i T ( q q q ^ e e > a=l q-2 i a - i o u and f o r 1 < j ^ q - 1 , = 3 1 0 i f i = j otherwise We conclude that n ± B - 3 ± i s the sum of 2 of 6 ^ , 3 ^ , 3 ^ , 16. Since f o r i , j both even or both odd, we can s e t and S<3> + S W > - - 6 2 Thus f a r , we have determined the character t a b l e f o r B as f o l l o w s : Class r e p r e s e n t a t i v e t i o\f-i o\\fha o \ i / i ' i \ \ f i • x\\f-i -1V/-1 -A 0 h /; 10 1/ :\0 1/ I 0 -1/ A 0 -1 -a Elements i n c l a s s q - i 2 2 a i x„ x 2a -x 2a 3 1-x, (-D 1 (-1)1 -1-x, ! l - x r X6 1-x, 2a - y 2 a 1 - i - y : '4 -1-y, - l - y _ • -1-y, We use the o r t h o g o n a l i t y r e l a t i o n s between the rows and columns to solv e f o r the unknowns. ( i ) d e t e r m i n a t i o n of x^ , y^ . 17. We have I 3 ( 1 ) ( 1 ) 2 + T M l ) 2 = q(q-D i = l i = l Hence q - 1 + x 2 + ( q - l - X ; L ) 2 + y 2 + (q-l-y^2 = q ( q - l ) X l = " 2 (q-1) + / 4 [ ( q - l ) T 2 y i ] 1 Now x^ and y^ are the degrees of the re p r e s e n t a t i o n s a f f o r d i n g 6 ^ and 8 ^ , ther e f o r e are r e a l numbers. This i m p l i e s 2 y 1 = q ~ 1 y l 2 X l 2 ( i i ) determination of x 2 a , y 2 a • There are q elements i n the conjugacy c l a s s represented by V * 0 \ . Hence £ 8 ( ± ) (h 0 1) 8 ( i ) ( h ~ a ) + Y 8.(h a) 8. (h" a) -0 h" a> 1 = 1 1 = 1 q-1 I i = l We have £ e"1 ' ^ + 2 x 2 a x 2 a + 2 y a y 2 a = q " 1 * S ° t h a t 18. x ' = 0 2a 2a ( a denotes complex conjugate) . ( i i i ) determination of x„ , x. , x c , x, . 3 4 _> b Since f-1 0 \ i s not i n the k e r n e l of the r e p r e s e n t a t i o n 0 -1 a f f o r d i n g , we have x 5 = ~ X 3 x 6 = " X4 ' Using ( 8 ^ , B ^ ) - = 1 , we o b t a i n B 1 [ 2 ( ^ ) 2 + ^ ( 2 x 3 x 3 + 2 x 4 x 4 ) ] = 1 q(q-D - , - o+l X 3 X 3 X 4 X 4 2 I f 4 | q - 1 , then by ( i i ) on the conjugacy r e l a t i o n s the above equation becomes (*) x 2 + x 2 = ^ x 3 , x 4 e (R . I f 4 \ q - 1 , then by note ( i i i ) on the conjugacy r e l a t i o n s the above equation becomes 19. (**) V 3 = *V Now using ( B ^ , = 0 , we o b t a i n B 2(P-~-)2 + ^ [-2(x 3+x 3x 3) - 2 ( x 4 + x A x 4 ) ] = 0 x 3 + x 4 = -1 I f 4 | q - 1 , then w i t h (*) we o b t a i n = -1 ± ^  X 3 2 I f 4 | q - 1 , then w i t h (**) we o b t a i n 2 , _ o+l _ X 3 + X 3 4 " ° X 3 2 ( i v ) Determination of y 3 , y^ , y,_ , y^ Since / - l 0 \ i s i n the k e r n e l of the. r e p r e s e n t a t i o n 0 -1 (3) a f f o r d i n g 8 , we have y 3 - y 5 y 4 = y 6 . 20. Using ( 6 ( 3 ) , 6^ 3 )) = 1 , we o b t a i n B 2 y 3 y 3 + 2 y 4 y 4 - S t l I f 4 | q - 1 , t h i s becomes /*^  2 ^ 2 q+1 D (*) y 3 + y 4 2 y 3 ' y 4 E R I f 4 j q - 1 , t h i s becomes (**> y 3 y 3 = Using ( B ^ , B ^ ) _ = 0 , we o b t a i n B y 3 + y 4 = - i Hence, as befor e , i f 4 | q - 1 we have = -1 + ^ y 3 2 whereas i f 4 \ q - 1 we have = -1 ± /-q  y 3 2 21. This completes our determination of the or d i n a r y i r r e d u c i b l e c h a r a c t e r s of B . We have the character t a b l e : 3 i 1 0 1 \ o\ f-1 o\ fha o V / ' i l)fl 0 - 1 / \0 h ~ a l ( o l) \o 1 j\o -lj\o " I ) A \J-1 -1 j j-1 -A ^ (-D 1 a i (-D1 ("1) : B ( 1 ) q-1 2 _.<LT1 2 0 -l+/cq" 2 -l+/cq 2 l - / c q 2 l+/cq~ 2 e ( 2 ) q-1 2 2 0 - l - / c q 2 -l+/cq~ 2 l+/cq~ 2 l - / c q 2 g ( 3 ) q - i 2 q - i 2 0 -l+/cq 2 - l - / c q 2 -l+/cq 2 - l - / c q ~ q-1 2 q - i 2 0 - l - / c q ~ 2 -l+/cq* 2 - l - / c q 2 -l+/cq~ 2 Where 1 £ i 1 q - l , e e C i s a p r i m i t i v e q - 1 s t root of u n i t y and - ,t i C 1 i f 4 | q - 1 -1 i f 4 | q - 1 §111. 3. The o r d i n a r y i r r e d u c i b l e characters of HW . We note the f o l l o w i n g conjugacy r e l a t i o n s i n HW ( i ) h 0 1 0 h " a 0 V o h " a ( i i ) fo ha\ V -h" a 0 o h a t: i) fo h a \ _ / 0 - h ~ a 0 i f a i s even. i f a i s odd. V / -A"1 0 / 22. The conjugacy c l a s s e s are as f o l l o w s : Class jl 0 \ f-1 0 r e p r e s e n t a t i v e iQ ± , . Q _± / V o \ fo l\ f 0 {o h - J \ - i a q-3 v 1 <_ a <_ -^y-C e n t r a l i z e r 2(q-1) q ( q - l ) Elements i n 1 1 Class q-1 2 3=1 2 2 L e t n be the chara c t e r of H W induced by the c h a r a c t e r x of H . We have n. ( l ) n* (-1) = 2 (-D 12 * a n. (h a) • ax , -ax e + e ru (hw) = 0 f o r any h e H Then ( n i ' V H W B=l 2 1 /O 2J,O 2 _L V o / a i - - a i N 2 . - r y (2+2 + 2, 2 ( £ + e ) ) 2(q- l ) =1 1 i f i 4 or q - 1 2 i f i = or q - 1 * * a—3 Since n. = n , . , we o b t a i n ^r— d i s t i n c t i r r e d u c i b l e c h a r a c t e r s of x q - l - x 2 23. HW i n t h i s manner. Let Y . = n. i = 1 , 2 i i 7 ' 2 Denote the remaining i r r e d u c i b l e characters of HW by > Y ^ > Y ^ j Y ^ Since I Y ± CD2 + I Y ( 1 ) CD2 - 2 (q-1) , i = l i = l and we have ^3 2 1=1 = 2(q-3) , we conclude that A ^ , A ^ , A ^ and A ^ are l i n e a r . These characters are t h e r e f o r e determined completely by t h e i r values at fh 0 \ and at f 0 -1 0 ) I f 4 I q - 1 , then every element of HW i s conjugate to i t s i n v e r s e , hence the values of the 4 l i n e a r characters are r e a l roots o f u n i t y , i . e . 1 or -1 . This completely determines the l i n e a r c h a r a c t e r s as f o l l o w s : Y l (1) .(2) ,(3) 1 I 1 1 1 -1 0 0 -1 ( -DV ' 0 c a . - c a e 4e (-1)' (-D1 o - l 24. Where i = 1 , 2 , ... , S z l . I f 4 \ q - 1 , then whenever a l i n e a r character has the value -1 at / h 0 \ i t must have the value /-T or -/-l a t the elements 0 h" of order 4 . Hence we have the character t a b l e : 1 0 \ f-1 0 \ / h" 0 \ / 0 1\ f 0 X \ 0 1 ) [ 0 -1 ] I 0 h~ a y ^-1 Oj (v-^_1 0 ) Y i 2 (-l) i2 ea i - l e - a i 0 0 Y ( 1 ) 1 1 1 1 . 1 Y ( 2 ) 1 1 1 - 1 -1 Y Y( 3 ) i - l (-D a /=i -/=i ( 4 ) I - i < - i ) a -/=i / - i Where i = 1 , 2 , ' 2 §111.A. The ordi n a r y i r r e d u c i b l e characters of KW' . By Lemma 2, KW' = KW hence we determine the i r r e d u c i b l e characters of KW <_ U(2,q) . We note the f o l l o w i n g conjugacy r e l a t i o n s i n KW : ( i ) / k 3 0 \ 7 k" 8 0 0 k j \ / k 0 ! ( i i ) / 0 k 3 X j 0 1 \ i f 8 i s even u -•k 0 0 -1 o ^ 0 k B j / 0 k ^ i f 3 i s odd. 25. The c o n j u g a c y c l a s s e s a r e : f \ f ' \ ( R ^ f \ C l a s s 1 0 \ -1 0 / k p 0 j / 0 1 W 0 k r e p r e s e n t a t i v e \ ± j j Q -±j { o k~ B I (-1 0 / - k " 1 0 C e n t r a l i z e r q ( q + l ) q ( q + l ) q+1 No. o f e l e m e n t s 1 1 2 q+1 o + l 2 2 The i r r e d u c i b l e c h a r a c t e r s o f KW a r e d e t e r m i n e d i n e x a c t l y the same manner as t h o s e o f HW . That i s , we i n d u c t t h e q + 1 i r r e d u c i b l e c h a r a c t e r s o f t h e c y c l i c group K t o KW and o b t a i n d i s t i n c t i r r e d u c i b l e c h a r a c t e r s o f KW . Denote t h e s e c h a r a c t e r s by (f^ 5 <J>2 » ••• ' (i' q_ 1 • We have: <J>. (1) = 2 ^ (-1) = ( - 1 ) ^ 8 8 i — 8 i <j>, (k ) = co + i o where u e $ i s a p r i m i t i v e q + 1 s t r o o t o f u n i t y $ ± (g) = 0 f o r g = kw k e K . Denote the r e m a i n i n g 4 i r r e d u c i b l e c h a r a c t e r s by <j>^ , <f>^ > <j>^ » <J>^ As b e f o r e , we f i n d t h a t t h e s e c h a r a c t e r s must be l i n e a r and we o b t a i n the c h a r a c t e r t a b l e : 26. 1 o\ 0 1 r-i 0 \ / k B 0 \ / 0 0 -1 -8 1 / 0 k / / .-1 0 k K / 1 -1 0 / -k 0 • l 2 0 0 1 1 1 1 1 1 1 1 -1 -1 1 - c ( - D 8 -/=E •<«> 1 - c Where i = 1 , 2 , ... q-i » 2 c = 1 i f 4 | q - 1 -1 i f 4 I q - 1 §111.5. The ordinary irreducible characters of SL(2,q) §111.5.1 The characters of SL(2,q) induced b y - f i , l < _ i < . q - l . Let 3^ be the character of SL(2,q) induced by the character of B , l < i < q - 1 We have B. (1) q + 1 B. ( -D = c-ircq+D 3. (h a) c a , - c a e + e \ ( x ( D ) = I B* ( X ( A ) ) = 1 3* ( - x ( l ) ) = 3* (-x(X)) = 2 . 2 Then (3* , B*) p = 5— [2(q+1)2 + q ( q + l ) £ ( e a ± + e " a i ) 2 + ^ q ( q -1 a = l 1 i f i ^ o r q-1 2 i f i = o r q - 1 * q—1 Hence B^ » i ^ 2~ o r q - 1 » i s an i r r e d u c i b l e c h a r a c t e r o f SL(2,q) . S i n c e ft ft 3. = B. i f f j = i o r q - l - i , 1 J q_3 * ft * we o b t a i n d i s t i n c t i r r e d u c i b l e c h a r a c t e r s , namely 3^ » 3 2 > ••• > L e t x ^ be t h e p r i n c i p a l c h a r a c t e r o f SL(2,q) . Then ( 3 * , x ) = 1 2 [2(q+1) + q ( q + l ) ( q - 3 ) + ^ ~ -4] q ( q -1) = 1 , ft hence ~ x i ^ s a n i r r e d u c i b l e c h a r a c t e r o f SL(2,q) . Denote t h i s c h a r a c t e r by x 2 . ( T h i s i s the S t e i n b e r g c h a r a c t e r o f S L ( 2 , q ) . ) §111.5.2. The c h a r a c t e r s o f SL(2,q) i n d u c e d by Y.J_ , 1 _< i <_ . L e t be t h e c h a r a c t e r o f SL(2,q) i n d u c e d by t h e c h a r a c t e r Y . o f HW . We have 1 28. Y* (1) = q(q+l) Y* (-1) = (-D^Cq+l) * >, a* ai , - a i y± (h ) = e + e * ~fi" Y. O O = o Y^ (g) = 0 for g p-irregular. — 2 L e t 8^ be t h e c h a r a c t e r o f S L ( 2 , q ) o b t a i n e d i n t h e same manner ^ we o b t a i n e d t h e c h a r a c t e r 8.^  o f SL(2,q) . That i s , ( 8 ^ a r e i n d u c e d 2 2 2 by t h e l i n e a r c h a r a c t e r s o f the subgroup o f o r d e r q (q -1) i n S L ( 2 , q ) We r e s t r i c t 8^ t o SL(2,q) and o b t a i n the c h a r a c t e r g i v e n by 8. (1) = q 2 + 1 r, (-1) = ( - 1 ) 1 ( q 2 + l ) — ,, a> a i -ax 8 ± (h ) = e + z 77 8v _ Bx -Bx 3. (k ) = to + co 8. (x(l» = 1 8. ( x ( X ) ) = 1 f. ( - x ( l ) ) = ( - I ) 1 6 i (-x(A)) = ( - 1 ) 1 where e , w a r e ( q + l ) t h and ( q - l ) t h powers of a g e n e r a t o r o f the £ _ ^ m u l t i p l i c a t i v e group F „ . F o r 1 < i < , l e t 6. = Y. _ 8. . 2 — — 2 x x x q Then 29. <LTA 2 2 (6. , 8,)_ = \ [( q - 1 ) 2 - 2 + q ( q - l ) [ ( J * + c ^ V + ^  -4] 1 1 G q(q -1) 3=1 . = 1 , hence 6. i s i r r e d u c i b l e , l R e c a l l the character n , of HW . Let n' be the character of SL(2,q) q - i 2 * induced by nn_i * ^ e ^ ave n' (1) = q(q+D n' (-1) = (-1) 2 q(q+D £T_1 , 2 o - l n ' ( h a ) = (-1) 2 n ' (k 3 ) = 0 n 1 (g) = 0 f o r g p - i r r e g u l a r Then we can agian see that 6 . = n' - 3 , i s an i r r e d u c i b l e character 2 2 of SL(2,q) . §111.5.3. The characters of SL(2,q) induced by the l i n e a r i r r e d u c i b l e characters of HW and KW' , and by the n o n - l i n e a r i r r e d u c i b l e characters of B . There are 4 more i r r e d u c i b l e characters to determine. Denote these characters by x^ , x^ , x^ . , x^ . 30. ( i ) * Let 3 be the character of SL(2,q) induced by the character 6 ^ of B , i = 1 , 2 , 3 , 4. . We have B ( i ) ' " (-1) = \ - * ~ i = 1 , 2 i = 3 , 4 3 ( i ) * ( h a ) = 0 3 ( ± ) * ( ? ) - 0 3 ( 1 ) * (g) = 3 ( l )(g) for g p-irregular (4)* Let y be the character of SL(2,q) induced by the character (4) Y of HW . We have: ^ ( ^ - a i a t i l • « 4 I q - i -aSs^> i f 4 l q - i Y ( A )* (h°> - (-1)" Y ( 4 ) * ( k 3 ) = 0 (tW A Y (g) = 0 for g p-irregular. . i (41)* (3)* If 4 | q - 1 , let x 3 = Y W - &K } X4 Y (4)* Q(4) If 4 \ q - 1 , let x 3 = Y x 4 - Y (4)* _ B ( l ) * (4)* Q ( 2 ) * 31. 2 Then (x , x ) \ - [2(^±V + q C q + l ) ^ ) + ^ • (q+1) ] q ( q -1) = l , s i m i l a r l y (x^ , x^)^ = 1 , hence x_ and x. are i r r e d u c i b l e . 3 4 Let t ) ) ^ " be the character of SL(2,q) induced by the c h a r a c t e r <J>^ of KW' . We have 2 ( _ D - - i i a z l l i f 4 I q - 1 . i f 4 | q - 1 4<4>* ( h a ) - 0 (7) = ( - D * <j>^* (g) = 0 f o r g p - i r r e g u l a r . I f 4 I q - 1 , let x 5 = 8 ( 1 ) * - <j> D * , ( 4 ) * q) _ fl(2)* . ( 4 ) * Xg = 8 - <f> i (3)* ( 4 ) * I f 4 | q - 1 , l e t x 5 = - * Xg = 8 - <J> 2 Then (x , x ) = \— [ 2 ( ^ ) 2 + q ( q - l ) ^ + ^ (q+1)] q(q - l ) = i , s i m i l a r l y (x, , x,) = 1 , b o G hence x r and x, are i r r e d u c i b l e . 5 6 32. This completes our determination of the or d i n a r y i r r e d u c i b l e characters of SL(2,q) . §111.5 .4 . V The char a c t e r t a b l e f o r SL(2,q) . -1 0 o -1 /h a o 0 h -a ir 0 j o ^ i ^ r A l \ o - i / -1 -X \ 0 -1 / — / " * q+i (-l) 1(q+l) a i , - a i ;E +e 0 1 e. i q-1 # ( - l ) 1 ( q - l ) 0 . B i ^ - g i . -(to 4 to ) -1 x i 1 1 1 1 1 q . q 1 -1 0 X 3 q+1 2 c .4+1 2 ( - 1 ) ' ° 0 l - / c q 2 x 4 q+1 2 q+1 C ' 2 (-1)K 0 l+/cq 2 x 5 q-1 2 q-1 0 -<^1)B -l+/cq 2 X 6 q-1 2 -c 3=1 2 0 - ( - DB - l - Z c T 2 1 -1 1 0 1 - / c i Where e e C i s a p r i m i t i v e q - 1 s t root of u n i t y , to e <E i s a p r i m i t i v e q + 1 s t root of u n i t y , and c = ( 1 i f 4 | q - 1 ' ( - D • ( - D : 1 l+/cq~ c- l-/cq~ I 1 j i 0 i i c. l+/cq c- l+/cq 2 - l - / c q j c- l+/cq -l+/cq 2 c" i + / c q 2 2 1--/cq 2 1-•/cq 2 1- /cq~ -1 i f 4 \ q - 1 33. IV The Modular I r r e d u c i b l e Characters of SL(2,q) . In t h i s s e c t i o n we s h a l l d e scribe q modular r e p r e s e n t a t i o n s of SL(2,q) and use Brauer's proofs (see [ 2 ] ) to show th a t these r e p r e s e n t a t i o n s are i r r e d u c i b l e and d i s t i n c t , and t h e r e f o r e give a l l the modular 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 of SL(2,q) . (r) For each g = a b \ e SL(2,q) , denote by p (g) the v c d ; (r) r t h symmetric product of g , i . e . p (g) i s the induced s u b s t i t u t i o n r r — 1 r - l r on x , x y , ... , xy , y where g - P ( 1 ) ( g > i s regarded as a s u b s t i t u t i o n x -> ax + by y ->• cx + dy . We note that f o r r = l , 2 , . . . , p - l t \ = exp t E I 1 ! 0 • = exp t F 1 where (r) 1 .0 (r) / 1 \ t 34. E = o r o r - l 0 o r - 2 0 1 o F = o 1 o 2 0 r o and ( r ) h 0 \ • -1 0 h / d i a g { h r , h r 2 , h" r} L e t p ^  d enote the 1 - r e p r e s e n t a t i o n . L e t 6^  , i = 0 , 1 , 2 , ... , n - 1 , denote t h e r e p r e s e n t a t i o n i o f SL(2,q) a f f o r d e d by a p p l y i n g the F r o b e n i u s automorphism a -»- a P t o the e n t r i e s o f g = / a b\ e SL(2,q) . That i s , c d j g e. f x i \ X - / * P J I \ c p d p / F o r 0 < r . < p - 1 , we f o r m the r e p r e s e n t a t i o n s H ( r , r , , . . . , r ,) — x — o 1 n-1 g i v e n by 35. g - P ° (g) * P 1 (g ^ x .... x P n _ 1 (g n _ 1 ) where * denotes the Kronecker product. Brauer [1] has shown th a t the q r e p r e s e n t a t i o n s thus obtained are i r r e d u c i b l e and d i s t i n c t . For the sake of completeness we give the proof i n [1] . Lemma 3. L e t 0 < r . < p - 1 . Then the represenation H(r , . . . , r ..) — — i — o n-1 of SL(2,q) as described above i s i r r e d u c i b l e . Proof. Regard g e SL(2,q) as a s u b s t i t u t i o n on the indeterminates e. x , y and f o r i = l , 2 , . . . , n - l regard g as a s u b s t i t u t i o n 0 o on the inde terminates x. , y. . Then the r e p r e s e n t a t i o n H(r , . . . , r ..) l l o n-1 d e f i n e s the a c t i o n of SL(2,q) on the module B c o n s i s t i n g of a l l p o l y -nomials i n x y y , ... , x , , y , which are homogeneous of degree o ^o n-1 n-1 r . i n the indeterminates x. , y. . To show that B i s i r r e d u c i b l e , 1 i i we show th a t any non-zero element generates the whole module. Let f E B , f ^ 0 . Then f i s of the form r r - l r f = ( a x + a , x y + .. . + a y ) oo o o l o o or o o x ( a l Q x 1 1 + a n y± + ... + y / ) r i r i - 1 r i , n-1 , n-1 . , . n - l N x ( a , ~ x , + a , , x n y . + ...+ a , y , ) v n-1,0 n-1 n-1,1 n-1 J n-1 n - l , r .. n-1 ' ' n-1 36. where a.. e F and f o r every i , a.. ^ 0 f o r some j , 0 < j < r . Let M(f) be the module generated by f . Then gf e M(f) V g e SL(2,q) . Let t be a generator of F^ and l e t f. = 1 i o t \ f l Then f i s obtained from f by ap p l y i n g the s u b s t i t u t i o n s x. ->• x. + t y. y i " y i We expand the product i n f and c o l l e c t terms w i t h the same t - power. Then 2 A P n - 2 f t = g o + t g l + t g 2 +,.. + t g P ^ where each g. i s a polynomial i n x , y , °i o o V l ' y n - l ' L e t f . tJ We have / 1 t j \ f 0 1 j = 2 , 3 , . . . , q - 1 . / \ 1 1 t t t 2 t 4 1 1 \ 1 1 .P n"2 k2(p n-2) \ S n 0 p -2 37. Now the m a t r i x of t - powers above has the Van der Monde determinant i " i + 7r ( t - t ) which i s non-zero, hence each g. can be expressed l<i<j£q-l 1 . as a l i n e a r combination of the f . . Therefore g. e M(f) , i = 0 , ... , p B t J 1 , r o r l r n - l The l a s t g. which i s non-zero must be of the form a y y_ • y . , 6 i o 1 n-1 r r i r -1 t h e r e f o r e y ° y ... y ^ ^ e M(f) . Denote t h i s p o lynomial by f and l e t f't-[l o\ f t 1 Then V i s obtained from f by applying the s u b s t i t u t i o n s x . ->- x . l l i y. t p x. + y. , Jx x Jx ' that i s , f ; - ( t x ^ ) ' 0 ( t ^ y / 1 .... (t P°" V ^ . ! ) ' 1 1 - 1 As b e f o r e , we expand the product and c o l l e c t terms w i t h the same t - power, then P _2. f ; - g ; + t 8 i + . . . . + t- - g , p -2 where each g! i s a polynomial i n x , y , . . . , x , > y -, • & i o o n-1 n-1 38. U s i n g e x a c t l y t h e same argument as b e f o r e we o b t a i n g_| e M ( f ) i = 0 , ... , p n-2 . B u t now i t i s easy t o see t h a t g' , g' , ... , g' p -2 g e n e r a t e a l l o f B , s i n c e g!^  f 0 i m p l i e s i = S q + s^p + ... + s ^ _ ^ p n 1 s r -s s. r ^ s , s .. r -s .. , , o o o l l l n-1 n-1 n-1 where 0 < s. < r . and g ! = x y x , y , ... x .. y .. — I — x I o o 1 1 n-1 n-1 Thus e v e r y s i n g l e power p r o d u c t i n B i s e q u a l t o some g^ . Lemma 4. I f H ( r , . . . , r .) = H(s ,...,s ,) the n r . = s . , i = 0 , . . . , n o n-1 o n-1 l l P r o o f . Assume the c o n t r a r y , i . e . suppose <*> ^ V - ' V i ) ° ^ v - V i * and f o r some i , r . ^ s. . x x We o r d e r t h e H ( r , . . . , r ,) l e x i c o g r a p h i c a l l y by s e t t i n g o n-1 H ( r , . . . , r n ) < H ( r ' , . . . , r ' -) when t h e f i r s t n o n - z e r o d i f f e r e n c e o n-1 o n-1 r ! - r . i s p o s i t i v e . Assume t h a t H ( r , . . . , r .) i s t h e l o w e s t r e p r e s e n t a t i o n x x o n-1 w h i c h c o n t r a d i c t s the s t a t e m e n t o f the lemma. C l e a r l y n e i t h e r o f H ( r , . . . , r n ) , H(s ,...,s ..) can have the m i n i m a l degree ( i . e . , a l l o n-1 o n-1 r ^ = 0 o r a l l s ^ = 0) o r the maximal degree ( i . e . , a l l r ^ = p - 1 o r a l l s = p - 1) . Assume t h a t r Q = r 1 = • • = r ^ . ^ = 0 , r k ^ 0 , k > _ 0 . (1) ^ i M u l t i p l y ( K r o n e c k e r p r o d u c t ) (*) by t h e r e p r e s e n t a t i o n g •> p (g ) • 39. We can f i n d the i r r e d u c i b l e constituents of the products by applying the re l a t i o n s P W ( S V P ( 1 ) (gS^- 1 ' <,VP<*"> (gS , 1 < r < p - 2 , P <• - " ( g 6 i ) x P ( 1 > C g \ ~ P ( " + 2 P ( P - 2 ' to'1) Since r 4 0 , H(r , . . . , r . r . - l , r r ..) w i l l appear as an i o l - l l i + l ' n-1 i r r e d u c i b l e constituent of the l e f t hand side of (*) a f t e r the m u l t i p l i c a t i o n . By the assumption that H ( r 0 > • • • > r n _ i ^ l s t n e l ° w e s t representation c o n t r a d i c t i n g the lemma, H ( r Q > • • • > r i - i ' r i ~ ^ > r i + i ' ' ' * ' r n - l ^ C a n ^ e s^ m l-*- a r to a representation H(r',...,r' .. ) only i f r! = r . - 1 , r ! = r . , j 4- i • o n - l i i 3 3 But t h i s representation cannot appear on the r i g h t hand side of (*) a f t e r the m u l t i p l i c a t i o n , because H(r , . . . , r .) < H(s ,...,s ) o n - 1 o n - 1 Hence we have a c o n t r a d i c t i o n of the uniqueness of the i r r e d u c i b l e constituents of a representation. In what follows, p(s ,...,s -) w i l l denote the character of o n-1 (m) the modular i r r e d u c i b l e representation H ( S D > • • • > s n _ i ^ > p w i l l 40. denote the character of the m . th symmetric product r e p r e s e n t a t i o n , (m) 6 i r g P (g) 9 0 _< m <_ q - 1 ; and (p ) w i l l denote the charac t e r of the C r) i r e p r e s e n t a t i o n g ^ p ( g ) , 0 < _ i < _ n - l . We have the ch a r a c t e r t a b l e : (m) »8n-l> 1 0 0 1 m+1 ( P S " r+1 n-1 n ( s . + i ) j = o 3 0 -1 h 0 0 \ h j (-l) m(m+l) ( - l ) r ( r + l ) m > h i k=0 T-a (m-'2k) e r i r —ap (r-2k) k=6 k K 0 0 k" m k=0 3(m-2k) r - 6 p 1 ( r - 2 k ) k=0 ( - i ) R IF (s.+i): n ( r . e ' J ') : II ( E n-1 ;h-l r j _ap : i(r.-2k) n-1 r j _Bp J(r.-2k) : j s 0 J !j=0k=6 j = o k=6 where R = s + s.p + ... + s o 1 n-1 n-1 P , e. . i s a. p r i m i t i v e q - 1 s t root of u n i t y i n "F co i s a p r i m i t i v e q + 1 s t root of u n i t y i n F V. The Decomposition M a t r i x f o r SL(2,q) . In t h i s s e c t i o n , we determine the decomposition m a t r i x of SL(2,q) , that i s , the c o e f f i c i e n t s of the l i n e a r combinations which give the ordinary i r r e d u c i b l e characters ( r e s t r i c t e d to p-reg u l a r c l a s s e s ) i n terms of the Erauer . 4 1 . i r r e d u c i b l e c h a r a c t e r s ([2] p. 588). In our case, the t a b l e of Brauer c h a r a c t e r s can be obtained from the t a b l e a t the end of S e c t i o n IV by r e p l a c i n g e and co by e and to , where e , to are as i n I I I . 5 . 4 . We show t h a t the e n t r i e s of the decomposition m a t r i x are 0 or 1 , and i n d i c a t e as e x p l i c i t l y as p o s s i b l e where the non zero e n t r i e s l i e . I n the passage from the Brauer i r r e d u c i b l e characters to the o r d i n a r y i r r e d u c i b l e c h a r a c t e r s , we f i n d i t convenient to use the Brauer c h a r a c t e r s of the symmetric product r e p r e s e n t a t i o n s as i n t e r m e d i a r i e s . Therefore, we begin by determining 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 , p <_ i <_ q - 1 . Lemma 5. L e t i = r + r,p + ... + r p™ , 0 < m < n - 1 , be the p-adic o 1 m — — expansion of i and suppose that r ^ 4 0 . Then .... (r +r.,p+. . .+r , p m 1 ) 6 r ( l ) o r m-1 o / m. m P = P 0 (p ) m-1, 6 r -1 p ( p - 2 - r o + ( p - l - r 1 ) p + . . . + ( p - l - r m _ 1 ) p ) & ( p m} m 4* Proof. We show f o r any z e C that . m l - r p m r m. „„. 1 ( i - 2 k ) _ / ( l" rm P " 2 j ) v m P ( rm- 2 / j ) k=0 j=0 £=0 z m „ . , m -p -2-i+r p / m „ ., m ... r - l m, ., v m^  (p -2-i+r p - 2 i ) m p (r -1-21) p m r m + I Z • • 1 Z j=0 1=0 Then s e t t i n g z = 1 , -1 , e a , to6 , 1 <_ a <_ , 1 <_ B £ v e r i f i e s the statement of the lemma at each of the q p - r e g u l a r c l a s s e s of SL(2,q) . 42. m i - r p ,. m ... r m. ™ ( x - r p -2j) m p (r -21) r m r m z z • Z z £=0 m m P - 2- 1 + rm P ( p m - 2 - i + r p ^ j ) V 1 P m ( r -l-2l) + I Z M ' I z m j=0 r -1 m, „ n N m P ( r -2-t) v m I z 1=0 m 1=0 m . . . . m „ . . m i - r p m . p -2-i+r p- , m ... ( i - r p - 2 j ) r mr (-2+x+rp -2j) V m . r m Z 2 + Z z j=0 j=0 (-2r P m + i - 2 j ) + I- Z 3=0 I n the second sum i n [ ] , l e t m s = i - r p + 1 + j m and i n the l a s t sum, l e t t = j + r p J mr m Then the r i g h t hand s i d e of our equation becomes i m -r - l m, „„ s p -1 m m p (r -21) v i - r p -2j Z m v ni z I z £=0 j=0 i + I m t=r p i - 2 t m 43. m V 1 P - 1 i-2(£pm+j) i i-2t I I z + I -£ = 0 J = 0 t = r p m m i i-2k J z QED. k=0 Since the product ( 5 ) distributes over the addition of characters, that i s , (9 + x ) © < | ) = e@<i> + x ® f we can apply Lemma 5 repeatedly to find the irreducible constituents of any p^^ . Investigating the manner in which each application of Lemma 5 application of Lemma 5 alters the coefficients r , r . . . . . . . r , , o 1 n-1 we conclude Lemma 6. Let i = r + r,p + ... + r n p n be the p - adic expansion o 1 n-1 of i . Then p(s ,...,s ,) is an irreducible constituent of p i f o n-1 and only i f the coefficients S Q > s ^ ' ••• > s n _ i satisfy the following conditions: (i) s = r or p - 2 - r 0 0 0 ( i i ) for each 1 _< i £ n - 2 , s. = r. , r. - 1 , p - 1 - r. , or p - 2 - r. 1 1 1 1 1 ( i i i ) (a) i f s. = r. or r. - 1 , 1 1 1 then s.,. = r. or p - 2 - r l+l l+l l+l 44. (b) i f s. = p - 1 - r . or p - 2 - r . , then s. ,. = r.. - 1 or p - 1 - r . ... . l + l l + l r l + l ( i v ) s , = r ' or r - - 1 . n-1 n-1 n-1 Remark. In the case r ^ = , we have to modify ( i i i ) as f o l l o w s : ( i i i ) (a) 1 i f s± = x± ~ ^~2^ ' then . s. .. = r. , or r . , - 1 i m p l i e s i - l l - l l - l S i + 1 = r i + l ° r P " 2 " r i + l and s . 1 = p - l - r . _ or p - 2 - r. , i m p l i e s i - l i - l i - l S i + 1 = r i + l ~ 1 ° r p - 1 " r i + l ( b ) ' i f s. = r. - 1 = ^ - 1 , then i i 2 s. , =• r. , or r . , - 1 i m p l i e s i - l i - l i - l s... = r . j , - 1 or p - 1 - r . . . l + l l + l l + l and s. _ = p - l - r . _ or p - 2 - r . , i m p l i e s i - l i - l i - l r S..T = r . . - or p - 2 - r . L 1 . l + l l + l v l + l (In other words, there are always only 2 p o s s i b i l i t i e s f o r s ^ + ^ terms of r . . , not 4 as i t may look i n t h i s case). 45. Lemma 7. 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 , 0 <_ i <_ q - 1 , appear w i t h m u l t i p l i c i t i e s 0 or 1 . Proof. Assume the cont r a r y . L e t i be the s m a l l e s t i n t e g e r such that contains an i r r e d u c i b l e c o n s t i t u e n t w i t h m u l t i p l i c i t y > 1 . Let i = r + r.p + ... + r p m be the p - a d i c expansion of i . o 1 m C l e a r l y m >_ 1 , because p^°^ , ... , p ^ P ^ are i r r e d u c i b l e . By Lemma 5, we have ,.N (r+...+r , p™ 9 r ( l ) o. m - l r - / m. m P = P OP (p ) (p-2-r +. . .+(p-l-r J p " 1 " 1 ) 6 r -1 , o r m-1 r . , m, m + P ® (P ) •Now 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 the f i r s t summand w i l l have the m + 1 s t c o e f f i c i e n t s = r , w h i l e 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 m m the second summand w i l l have the m + 1 s t c o e f f i c i e n t s' = r - 1 . m m Hence by Lemma 4, the two summands cannot have an i r r e d u c i b l e c o n s t i t u e n t i n common. But t h i s i m p l i e s the i r r e d u c i b l e c o n s t i t u e n t of p ^ ^ which appears w i t h m u l t i p l i c i t y > 1 must appear w i t h m u l t i p l i c i t y > 1 i n e i t h e r the f i r s t summand or the second summand. Suppose p(s ,...,s _ , r , o m-1 m appears with m u l t i p l i c i t y > 1 i n the f i r s t summand. Then ( i - r m p m ) p(s , ...,s ^,0,0,...) appears w i t h m u l t i p l i c i t y > 1 i n p C o n t r a d i c t i o n . S i m i l a r l y , i f p ( s ' , . . . , s ' _ , r -1,0,...) appears w i t h J o m-1 m m u l t i p l i c i t y > 1 i n the second summand the p ( s ' , . . . , s ' -,0,0...) o m-1 _ -( p - 2 - r o + ( p - l - r 1 ) P + . . . + ( p - l - r m ^ 1 ) p m L ) appears w i t h m u l t i p l i c i t y > 1 i n p 46. C o n t r a d i c t i o n . Next, we determine which p ' s are c o n s t i t u e n t s of a given o r d i n a r y i r r e d u c i b l e c h a r a c t e r . The f o l l o w i n g r e l a t i o n s are r e a d i l y v e r i f i e d : (IV.1) B* = p(i> + P ^ 1 - i ) 1 < i <*=3 V = p ^ Q ± = p ( i - 2 ) + p(q-l-i) 2 < i <£=! x l = p (o) X 2 (q-D /ill } x = x. = p v 2 J 3 ( S l l ) x = x = p v 2 ' 5 6 (Note: References to or d i n a r y characters i n t h i s and succeeding s e c t i o n s mean o r d i n a r y characters r e s t r i c t e d to p - regu l a r c l a s s e s ) . I t remains to show that the modular i r r e d u c i b l e c o n s t i t u e n t s of any o r d i n a r y i r r e d u c i b l e character have m u l t i p l i c i t y 0 or 1 . By Lemma 7 and (IV.1) i t s u f f i c e s to prove: Lemma 8. (a) For 1 £ i <_ } p ^ and 1 have no common i r r e d u c i b l e c o n s t i t u e n t s . x ^ . q-1 ( i - 2 ) , ( q - l - i ) , (b) For 2 <_ l _< -^y- , p and p have no common i r r e d u c i b l e c o n s t i t u e n t s . 47. Proof. (a) Suppose that and p^ q 1 have a common i r r e d u c i b l e c o n s t i t u e n t , say D ( s ,...,s .) . p o n-1 Let 1 = r Q + r^p + ... + r n_-j_P q - 1 - i = t + t^p + ... + t , p n 1 ^ o 1 n-1 where 0 < r . , t < p - l f o r a l l 0 < i < n - 1 — i i — — — Note t h a t t . = p - l - r . , 0 < i < n - 1 . x l — — Applying Lemma 6 ( i ) , we must have s = r or p - 2 - r o o o and s = t (= p - 1 - r ) or s = p - 2 - t ( = r - 1 ) . o o o o o o This can only occur when ( i ) s = r = t (= p - 1 - r ) o o o o or ( i i ) s = p - 2 - r = p - 2 - t (= r - 1 ) o o o o and r = -^77^  i n both cases, o 2 I f ( i ) h o l d s , then by Lemma 6 ( i i i ) , S l = r l o r P " 2 " r l and s 1 = t 1 (= p - 1 - r 1 ) or s 1 = p - 2 - ^ (= ^  -Then we again o b t a i n 48. 8 l - r x - t x (- p - 1 - r±) or s 1 = p - 2 - r 1 = p - 2 - (= r 1 - 1) and r ^ = l n both cases. The c o e f f i c i e n t can now be determined by the same argument as i s used to determine r ^ . I f ( i i ) h o l d s , then by Lemma 6 ( i i i ) , 81 " r l ' 1 ° r P " 1 " r l and s = t1 - 1 (= p - 2 - r 2 ) or s^ ^ = p - 1 - t± (= r 1 ) This can only occur when ( i ) ' S ; L = r1 - 1 = t± - 1 (= p - 2 -or ( i i ) 1 s 1 = p - 1 - r 1 = p - 1 - (= r ^ and r ^ = P~^" i n both cases. Now Lemma 6 ( i i i ) a p p l i e s to ( i ) ' ( ( i i ) ' ) i n the same manner i t a p p l i e s to ( i ) ( ( i i ) ) t h e r e f o r e the c o e f f i c i e n t can be determined i n t h i s case a l s o by the same argument as i s used to determine r ^ . We conclude t h a t r . = ^ ~ f o r a l l 0 <_ j _< n - 1 , the r e f o r e i = . This proves J 2 z ( a ) . 49. /, \ -• , i_ (i-2) • , ( q - l - i ) (b) We use an analogous argument to snow that i f p and p have a common i r r e d u c i b l e constituent, then i = • VI The Cartan Matrix for SL(2,q) . The e n t r i e s i n the Cartan matrix are the c o e f f i c i e n t s of the l i n e a r combinations which give the characters of the p r i n c i p a l indecomposable modules i n terms of the modular i r r e d u c i b l e characters ([2], p. 593). We have the following theorem of Brauer ([2] p. 593): The Cartan matrix C and the decomposition matrix D are r e l a t e d by the equation T C = D D . We s h a l l determine the diagonal entries of C and i n d i c a t e where the remaining non-zero entries l i e . We w i l l use the following notation throughout t h i s section: (i ) d ^ = i , j th entry of the decomposition matrix D . The i th row of D corresponds to the i th ordinary i r r e d u c i b l e character of SL(2,q) . Because of the theorem above, the order of the rows i s i r r e l e v a n t to out discussion. The j th column of D corresponds to the modular i r r e d u c i b l e character p(s ,...,s .) , where o n - i , , , n-1 1 = S + S - . P + . . . + S i P J o 1 n-1 ( i i ) c „ = i , j entry of the Cartan matrix C . The i th row of C corresponds to the character of the p r i n c i p a l 50. indecomposable module a s s o c i a t e d w i t h the i r r e d u c i b l e module a f f o r d i n g the chara c t e r p ( r , . . . , r ..) , a o n-1 where i = r + r n p + ... + r ,p n . (see C & R p. 590) o 1 n-1 The j th column of C corresponds, as i n D , to the modular i r r e d u c i b l e c h a r a c t e r p ( s ,...,s ..) , where o n - 1 i . ' , n-1 1 = s + s,p + ... + s ..p J o V n - l r ( i i i ) For i = s + s.p + ... + s -p , 0 < i < q - l , o 1 n-1 — — l e t P. = { p ^ I p ( s ,...,s .) i s a c o n s t i t u e n t of p ^ l o n-1 §VT.l. The d i a g o n a l elements of the Cartan matrix. n _ 1 Lemma 9. Let i = s + s,p + ... + s _p be the p - a d i c expansion o 1 n-1 of i , 0 < i < q - 1 . _ _ »n , i . r p-1 p-3 2 + 1 • l f s j = 2 " 2 Then c. . i i f o r a l l j . „t . c . p-1 p-3 c 2 i f S j T 2~ o r 2~ some j where t i s the number of c o e f f i c i e n t s s. which are not equal to p - 1 J Proof. By Brauer's Theorem we have q+4 c. = T d. . d . 51. I n the p r e v i o u s s e c t i o n we have shown t h a t d, . = 1 i f p ( s ,...,s .,) i s a c o n s t i t u e n t To. j o n-1 o f the k t h o r d i n a r y i r r e d u c i b l e c h a r a c t e r 0 o t h e r w i s e Hence, C . = number o f o r d i n a r y i r r e d u c i b l e c h a r a c t e r s o f w h i c h i i p ( s ,...,s n ) i s a c o n s t i t u e n t , o n-1 Now, by ( I V . 1 ) we see t h a t 1) , 0 _ < m _ < q - 2 , m ^ 3=1 > > 1 S a c o n s t i t u e n t o f 2 o r d i n a r y i r r e d u c i b l e c h a r a c t e r s . (3=1, ,3=1) 2) p 2 , p 2 a r e each c o n s t i t u e n t s o f 3 o r d i n a r y i r r e d u c i b l e c h a r a c t e r s . 3) p ^ q i s a c o n s t i t u e n t o f 1 o r d i n a r y i r r e d u c i b l e c h a r a c t e r . By Lemma 6, we have ,3=1) (3=1) 1) i f p v 2 ' o r p v 2 e P , t h e n a l l t he c o e f f i c i e n t s s. i n i = s + s_p + ... + s , p n 1 must be ^ — o r . C o n v e r s e l y , j o 1 n-1 2 2 i f a l l t he c o e f f i c i e n t s s. a r e e q u a l t o ov ^-y3- > t h e n one o f 3 z z p (3=1) p(3=l) b e l o n g s t o P^ . S i n c e j-1 0 \ ' i s i n the k e r n e l o f o n l y I 0 -1 (3=1) (3=1) one o f p 2 , p 2 , o n l y one o f t h e s e c h a r a c t e r s can b e l o n g t o P_^  2) i f p ^ q _ 1 ^ e P , t h e n i = q - 1 and V i " t " l , " 1 ) j -52. I t remains to determine | P . J , 0 <_ i <_ q - . 2 . Again using fm) Lemma 6, we have p e , w i t h p - ad i c expansions , , n-1 X = s + s . p + . . . + S TP o 1 n-1 , , , n-1 m = r + r n p + . . . + r ..p o 1 n-1 i f and only i f the c o e f f i c i e n t s r.. s a t i s f y the f o l l o w i n g c o n d i t i o n s : ( i ) r = s o r p - 2 - s v ' o o o ( i i ) f o r l < _ j < _ n - 2 , r . = s. , s. + l , p - l - s , o r p - 2 - s 3 3 J 3 3 ( i i i ) (a) i f r . = s. or s. + 1 , J J 3 then r . ." - = s. or p - 2 - s -j+1 j + l K J + l (b) i f r . = p - 1 - s. or p - 2 - s. , J 3 3 then r . , - = s... + 1 or p - 1 - s. J + l J + l . J + l ( i v ) r , = s .. or s . + 1 . n-1 n-1 n-1 Thus, f o r each c o e f f i c i e n t r _ . , 0 j < j _ < n - 2 , w e have 2 c h o i c e s , unless s. = p - 1 i n which case we have only 1 choice. There i s only 1 choice f o r r , , unless s . = p - 1 i n which case i f r = p - l - s „ n-1 n-1 n - 2 n-z or p - 2 - s 2 w e n a v e n o choice. Hence,"if t i s the number of c o e f f i c i e n t s s^ , 0 <_ j <_ n - 1 , which are not equal to p - 1 , we o b t a i n 53. C o l l e c t i n g our r e s u l t s , we o b t a i n the statement of the lemma. §VI.2. The e n t r i e s c . of the Cartan ma t r i x , i 4 j • i j ! •L Let i and j have the p - a d i c expansions i = s + s , p + . . . + s ,p o 1 n-1 • t i f i • t Tl X 1 = s + s'p + . .. + s ,p J o 1 n-r Repeating the argument at the beginning of Lemma 9, we have c.. = number of or d i n a r y i r r e d u c i b l e characters of which p(s ,...,s ' ) and p ( s ' , . . . , s ' ..) are both c o n s t i t u e n t s , o n - 1 o n—1 By (IV. 1 ) , we o b t a i n c „ 4 0 i f and only i f one of the f o l l o w i n g occurs: ( i ) P. I P. 4 J ( i i ) p(m) e P. such that p(q-l-m) £ ( i i i ) p (m) e P. l such that p(q-3-m) e J ? , m s « q - 2 , q - l We o b t a i n f o r i , j not both equal to q - 1 c^j = 2|P^P\P^| + number of p ^ ' s s a t i s f y i n g ( i i ) Cm) + number of p s s a t i s f y i n g ( i i i ) . For the sake of completeness, we determine r e s t r i c t i o n s on the c o e f f i c i e n t s s , . . . , s , , s' . . . . . s' , such that c.. 4 0 o n-1 o n-1 i j These r e s u l t s w i l l not be used i n the next s e c t i o n . 54. (a) L e t m have the p - a d i c e x p a n s i o n n-1 m - r + r^p + o 1 + r -p n-1 , and suppose t h a t (m) p e P_^ P| P_. . The p o s s i b l e v a l u e s f o r s' , i n terms o f s , n-1 o , .s ., a r e n-1 shown by the f o l l o w i n g t a b l e : (see Lemma 6 ( i i ) ) r . ... j 1 S3 1 3 s'. + 1 J p - 1 - s! J P -- 2 - s\ 3 s. 3 s\ = s. J 3 s! = s. - 1 J J s\ = p - 1 - s. 3 3 s*. = 3 p - 2 -s. + 1 J s ! = s. + 1 3 3 s! = s. J 3 s! = p - 2 - s. J 3 3 p - 3 -p - 1 - s. 3 :• s'."= P - 1 -J s. J s\ = p - 2 - s .' 3 3 s\ = s. 3 3 s\ = 3 s. - 1 3 P - 2 - s. 3 s'. = p - 2 -J s. J s! = p - 3 - s. I 3 s\ = s. + 1 3 3 s\ = 3 s . 3 By Lemma 6 ( i ) we must have s ' = s o r p - 2 - s . By J v ' O O o Lemma 6 ( i i i ) we have: ( i ) i f s! = s. , s. - 1 , o r s. + 1 , J J J J t h e n j + l j + l r j + l ( i i ) i f S j = p - l - S j , p - 2 - s ^ . , o r p - 3 - s , , t h e n ' j + l j + l - 1 , p - 1 - s , p - 3 - s o r s + 1 j + l j + l j + l By Lemma 6 ( i v ) we must a l s o have s ' = s , , s . - 1 , o r s . + 1 . J n-1 n-1 n-1 n-1 I f we i n c l u d e a remark a n a l a g o u s t o the one f o l l o w i n g Lemma 6 about the TP "** 1 O*"* 3 c a s e s s. = *-~r- o r , then the above g i v e n e c e s s a r y and s u f f i c i e n t 3 L A 55. c o n d i t i o n s f o r P^ f\ P to be non-empty. (b) L e t m have the p - a d i c expansion m = r + r.p + ... + r ,p n 1 , m ^ 0 or q - 1 , o 1 n-1 j (m) ^ (q-l-m) _ and suppose that p e P^ , p e P.. . The p o s s i b l e values f o r s' , ... , s' i n terms o n-1 of s , ... , s , are given by the f o l l o w i n g t a b l e o n—1 (see Lemma 6 ( i i ) ) 3 \^ J s'. + l 3 p - l - s ' . ' 3 P - 2 - s s. 3 s'. J = p-l-s. 3 s! = p-2-s. 3 3 s\ = s. j 3 s' = s . - l 3 s.+l J s'. J = p-2-s. 3 s! = p-3-s. 3 3 s'. = s.+l 3 3 • s' = s . 3 p-l-s. J s'. J - s . J s'. = s . -1 3 3 s' = p-l-s. 3 3 s'. J = p-2-p-2-s. J s! J = s.+l J s! = s . 3 3 s\ = p-2-s. J J s'. J = p-3-Again u s i n g Lemma 6 we have a) 8 ; = p - 1 - s , s o o - 1 , p - 3 - s o or s + 1 0 ( i i ) I f s j = p " 1 "  si • p - 2 - s. , or 3 P - 3 - s. , then s1,.. J+1 p - 1 - s j + i • S j + 1 - 1 . P - 3 " > °r 8 j + 1 + 1 I f s! = s. , s. + 1 , or s. - 1 , 3 3 3 3 then s' = s . ... or p - 2 - s . ,.. j+1 j+1 r j+1 . ( i i i ) s ' = p - l - s , , p '- 2 - s , , or p - 3 - s _ . n-1 r n-1 n-1 n-1 56. W i t h a n o t h e r remark about the c a s e s s. = ^ ~ o r , t h e s e g i v e n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s f o r t h e e x i s t e n c e o f a p ^ e P. s u c h t h a t p ^ - 1 - * ) P . . J ( c ) I f t h e r e e x i s t s p ( m ) e P ± s u c h t h a t p t p ^ ^ we o b t a i n e x a c t l y t h e same c o n d i t i o n s on s' , ... , s' , a s f o r ( b ) . o n-1 V I I C h a r a c t e r s o f t h e P r i n c i p a l Indecomposable Modules L e t n ( s o , . . . , s n _ j ) be the c h a r a c t e r o f t h e p r i n c i p a l i n d e c o m p o s a b l e module a s s o c i a t e d w i t h the i r r e d u c i b l e module a f f o r d i n g t h e c h a r a c t e r p ( s ,...,s ..) . I n t h i s s e c t i o n we d e t e r m i n e the v a l u e o f o n - 1 n ( s Q , . . . > s n _ ^ ) a t each o f the q ,p - r e g u l a r c l a s s e s o f SL(2,q) . F o r c o n v e n i e n c e o f n o t a t i o n , we w i l l n o t c o n t i n u a l l y d i s t i n g u i s h between modular c h a r a c t e r s and t h e i r a s s o c i a t e d B r a u e r c h a r a c t e r s . We m e r e l y r e m i n d t h e r e a d e r t h a t t h e modular c h a r a c t e r s have v a l u e s i n F ^ , q w h i l e B r a u e r c h a r a c t e r s a r e c o m p l e x - v a l u e d . R e p l a c i n g e , to by e , OJ o r v i c e v e r s a a c c o m p l i s h e s the t r a n s i t i o n f r o m one s e t o f c h a r a c t e r s t o the o t h e r . U s i n g B r a u e r ' s Theorem and Lemmas 7 and 8, we have n ( S q , . . ., s n_-^) = £ ( o r d i n a r y i r r e d u c i b l e c h a r a c t e r s o f w h i c h p ( s ,...,s .) i s a c o n s t i t u e n t ) , o n-1 57. By ( IV .1) t h i s becomes / \ V 9 ( m ) _i_ ( q - l - m ) , ( q - 3 - m ) n(s = L 2 P + P + P P( m )e p . where i = s + s . p + . . . + s , p n , i ^ 0 o r q - 1 , o 1^ n-1 n n ( o , o o ) - I 2 p w + p < i - 1 ^ , ) + p ( < 1 - 3 - m ) - p ^ - " , O and n ( p - l , p - l , . . . , p - l ) = p ( p - l , . . . , p - l ) = x 2 . The c h a r a c t e r s n ( p - 1 , . . . , p - l ) , p ( p - 1 , . . . , p - l ) , and X 2 b e l o n g i n g to the b l o c k o f - d e f e c t 0 w i l l be e x c l u d e d f rom f u r t h e r d i s c u s s i o n . C h a r a c t e r v a l u e s a t s p e c i f i c c l a s s e s w i l l be.-.denoted i n the same manner as i n S e c t i o n I I I ( see I I I . l , n o t e s ( i i i ) and ( i v ) ) . § V I I . l . C o m p u t a t i o n o f n ( s . . . s n ) a t the c l a s s e s / l 0 \ and I-l 0 \ ° " - 1 ( o i ) \ o J T i i i n-1 L e t l = s + s .p + . . . + s , p o 1 n-r ( i ) A t the c l a s s / 1 0 \ we have 0 i l ( 2 p ( m ) + p ^ - l - i ) + p(q - 3-m> ) ( 1 ) , 2 q Thus , 58. n(s ,...,s ..)(1) = 2q |P.| i f s. 4 0 f o r some i o n-1 I 1 i 1 j 2q P - q i f a l l s. = 0 1 o 1 j I n the proof of Lemma 9 we have shown that |P j = 2*" 1 , where t i s the number of S j ' s n o t e a . u a x t o P ~ 1 > hence n(0,...,0)(l) = 2 q - q n ( s o , . . - . . s ^ ) (1) = 2 q ( i i ) At the c l a s s f-1 0 \ , we have 0 -1 ( 2 p ( m ) + p ^ - 1 " ™ ) +. p ^ - ^ X - l ) = 2q (-1) m Since p ^ e P^ i m p l i e s (-1)™ = (-1)^ > we o b t a i n i l ( 0 , . . . , 0 ) ( - l ) = 2 q - q n(s . . . . . s ^ = 2tq (-1) 1 . o n-1 §VII.2. Computation of r'( s 0» • • • ' s n _ i ^ a t ^ e c l a s s e s f h a 0.: A and / k B 0 0 h -a At the c l a s s ha 0 \ we have 0 ti 59. ( 2 p ( m ) + p ^ " 1 - ^ + p ^ - 3 - m ) ) ( h a ) - = e a m + £ - a m and at the c l a s s / k 0 \ we have 0 k"f (2p (m) + p(q-l-m) + p (q-3-m) ) ^  = _ ( w3(nH-2) + a > - B ( m f 2 ) ) We o b t a i n a r e c u r s i v e formula which w i l l enable us to compute I e a m + s - a m and £ V ( m + 2 ) + ^ ^ 2 ) ) , f o r » . , n-1 i = s = s-p + ... + s ,p o r n-1 We have by Lemma 5, ( r + . . . + r p m ) (r+.'..+r -.p"1"1) .' 9 . r p ° m = p ° ^ ® (P m ) m ( p - 2 - r o + ( p - l - r 1 ) p + . . . + ( p - l - r m _ 1 ) P m - 1 ) 6 V l + p ® (P ) Hence, ( i ) i f p(s ,...,s .,) i s a c o n s t i t u e n t of v o n-1 p< ro +- • " + V l " ^ ( p •») » @ .. . g (p , then I t i s a l s o a c o n s t i t u e n t of p ^ ' ' ^  ® toW"" & ... ® ( P V 1 ) ' - 1 and ( i i ) i f p(s ,...,s -) i s a c o n s t i t u e n t of o n-1 60. ( r +...+r p ) 8 r 8 ., r o m , m+1. m+1 ,_ _ - n-1. n-1 . ( i ) _ P © (P ) © ... ® (P ) t h e n p V J / e P ± , n~~l n~1 where i = r + r.p + . . . + r ,p and i = s + s , p + . . . + s .p o l r n - l r o 1 n-1 W i t h t h e s e comments i n mind, we d e f i n e ( s +s p+. . .+s p n -1) P t N o 1 n-1 V ^ ' - ' V i ' = p (s +s,p+...+s , p n " S (p-2-s + ( s . + l ) p + . . .+s , p n " S „ z v N o 1 n-1^ , r o 1 n - l r S l ( s o " - " S n - l ) = p + P V ^ ' - ' V ^ = V l ^ o ' - ' V l ) + S k - l ( S o " - " P * 2 - S k - l ' S k + 1 " - " S n - l ) where 1 < k < n - 1 . We p r o v e : n—1 Lemma 10. L e t i have t h e p - a d i c e x p a n s i o n i = s + s.p + ... + s .p o 1 n-1 L e t t be the number o f S j ' s n o t e 1 u a ± t o P ~ 1 • Then P W e P . 1 P r o o f . ( i ) i f p ^ e P. t h e n p ^ i s a summand o f S .. (s , . . . ,s .) I n-1 o n-1 We r e c a l l the c o n d i t i o n s g i v e n : i n t h e p r o o f o f Lemma 9 f o r ( r o + r i P + " ' * + r n - l p n - 1 ) p t o b e l o n g t o P^ . C l e a r l y t he summands S ( r , . . . , r ,) o f S . (s , . . ., s ., ) e x h a u s t a l l p o s s i b l e sequences o o n-1 n-1 o n-1 rr , ... , r . w h i c h s a t i s f y t h e s e c o n d i t i o n s , o n-1 J 61. ( i i ) i f S ( r , . . . , r ,) i s a summand of S .. (s ,...,s ,) , v ' o o n-1 n - l v o n-1 then S ( r , . .., r ) e P . . o o n-1 1 Let k be the s m a l l e s t index such that S. (s ,...,s ..) contains k o n - l a summand S ( r , . . . , r ,) which does not belong to P' . o o n -1 i Then k 4 0 , and by the m i n i m a l i t y of k , S Q ( r ,. . . »rn_-|_)' i s a summand of S, , (s ,...,p-2-s, , ,s,+1, ...,s ..) . Hence there e x i s t s a summand k-1 o k-1 k n-1 S ( r r ' .,) of S. n(s ,...,s ..) such that o o n-1 k-1 o n-1 V V ' - ^ n - l ) = S 0 ( p - 2 - r ; f p - l - r ' f . . . , r i + l , . . . , E ; _ 1 ) • By the m i n i m a l i t y of k , S ( r ^ , . . ., r^_^) e P.^  , th e r e f o r e by the remarks ( i ) and ( i i ) preceding the Lemma, S o ( p - 2 - r ^ , p - l - r ^ , ... , r^+1 , ... , r ^ ) e P. . C o n t r a d i c t i o n . ( i i i ) i f s^ = p - 1 , then sk+i ( so = 2 W ^ ' V i * We note that h a l f the summands of S, (s ,...,s ..) have the form k o n -1 62. o o 1 k-1 k n-1 and the other h a l f have the form S o ( r o ' r i ' ' ' ' ' r k - l ' S k + 1 ' " * *' V P f o r some r. , r ! . 1 x Now each summand of S, (s ,...,s .,) generates e x a c t l y two summands of k o n—1 S, .. (s , ...,s ..) by our r e c u r s i v e formula. Namely, «C"rj_ o n x S ( r , . .. ,r, .. , s, , . . . ,s ..) generates the summands o o' k-1 k' ' n-1 6 V v • • • ' rk-rV• • -en-i> a n d V v • • •»rk-l'p-2-Sk'Sk+l+1'• • '' V P and S ( r r , ' ,,s,+1, ...,s ,) generates the summands o o' k-1 k n-1 SoK'' • • ' rk-i' sk + 1' •'" V i } a n d s o ( r o rk-r p- 1- sk' sk+i + 1' • • • • V i } I f s^ = p - 1 , then ^ j . j . k-1 , k n-1 r + r,p + . .. + r. ,p + s, p + ... + s -p o 1 k-1 k n-1 r o + r l P + *•* + r k - l p k _ 1 + ^ ~ 2 ~ \ ^ k + ^ s k + l + 1 ^ p k + 1 + ••• + s n - l p n - 1 ' r o + r i p + •;• + r k - i p k _ 1 + < s k + 1 ) p k + \ + i p k + 1 + ••• + s n - i p n _ 1 r o + r l P + + r k - l p k _ 1 + ( P " 1 _ s k ) P k + ( s k + l + 1 ) p k + 1 + ••• + s n - l p n _ 1 " 63. Hence no 'new' summands occur i n S, ... ( s ,...,s ..) , and k+1 o n - i Sk+l ( so , ,**» Sn-l ) " 2 S k ( s o » ' , , , S n - l ) Now suppose t i s the number of s_. ' s note equal to p - 1 Let v = n - t and suppose s, = s, = ... = s, = p - 1 k 1 k 2 k v k < k„ < ... < k 1 2 v We expand S , ( s , . . . , s . ) 'downwards'. n-1 o n-1 S _ ( s , . . . 5 s ) = S ( s , . . . , s ) + S ( s , . . . , p - 2 - s , s - , + D n-1 o n-1 .n-2 o n-1 n-2 o n-z n-1 * - r - k v - 2 2 Y o / U) U) s ~ k + l < - S o ' * * " S k ' r k + l ' " " r n - r -t=l v v v n-k -2 2 V U) CO = 2 . S k ( s o " " > S k ' r k + l > , - - ' r n - l ) , Z=l v v v • n-k -3 - 2 2 7 ^ S (s s r ( £ ) r ( l ) ) 1 \ . + i l 8 o " " , s k / r k 1 + i * - * " r n - r 1=1 v-1 v-1 v-1 n-k -3 ? v-1 V ^ U ) v _ 2.'2 / S k 1 ( S o ' " - ' S k / r k .+I'"*» rn-1 ; , Z=l v-1 v-1 v-1 &1 ° ° n"1 64. = 2 n - t 2 T s <r<*\....r<*>) (the sequences , ... , ^ are j u s t those which occur a f t e r - t h e a p p l i c a t i o n of the r e c u r s i v e formula the i n d i c a t e d number of times). 2 t _ 1 r (£.") ( Now ) S ( r , . . . , r ,) contains a l l the d i s t i n c t summands of S n - 1 ( s o > • • ., sn_-^) • Since | P.J = 2t X (Lemma 9)p by ( i ) and ( i i ) we have Hence, c / \ o n - t ) ( m ) S n - l ( S o " - ' S n - l ) = 2 P Now we are ready to compute I (e a mH e- a m) and J ( c ^ 2 ^ " ^ 2 ) ) P ( m ) e P . (m) y) 1 p eP. l n—1 ( i ) Let i = s + s.p + ... + s ..p o 1 n-1 Let E (s ,...,s ..) = e a l + e a l o o n-1 E (s , . . ., s .) = E. . (s , . . . ,s T ) + E .. (s , . .. ,p-2,-s, . ,s. +1, k o n-1 k-1 o n-1 k-1 o c k-1 k 6 5 . Let t be the number of 's not equal to p - 1 . Then by Lemma 10, we have J (e^+e-™) = I-**-* E n 1 ( s , . . . , s n ) vm; _ n - 1 o n - 1 p eP. I We show by i n d u c t i o n that n-1 a p 3 ( p - l - s . ) - a p 3 ( p - l - s . ) E (s ,...,s ) = n (e J + e. J ) n - 1 o n - 1 . . j=0 We have a ( p - l - s ) - a ( p - l - s ) a ( i + p - l - s ) - a ( i + p - l - s ) E.(s ,...,8ri .) = ( e ° + e ° ) ( e ° + e ° ) 1 o n - 1 Assume that k - 2 a p 3 ( p - l - s . ) - a p 3 ( p - l - s . ) E, ,(s , . . . , s n ) = n (e 3 + -e 3 ) k - 1 o n - 1 . _ j=0 a ( i + p k _ 1 - l - ( s k _ 2 p k ~ 2 + . . . + S q ) ) -a(i+p - l - ( s k _ 2 p x ( e + £ + + s ) ) ) o Then . . k - 2 a p 3 ( p - l - s . ) - a p 3 ( p - l - s . ) E k _ 1 ( s o , . . . , p - 2 - s k _ 1 , s k + l , . . . , s n _ 1 ) - n (e 3 + t 3 ) 1=0 '. a ( i - p k ~ i l + 2 p k - 2 s k _ 1 p k 1 - ( s k _ 2 p k 2 + . . . + S Q ) >c ( £ , . k - 1 , , „ k „ k - 1 , k - 2 x\ -a ( i-p -l+2p -2s, .jP _ ( s k _ 2 P +..-+S )) + e ) 66. Hence . k-2 a p 3 ( p - l - s . ) - a p 3 ( p - l - s . ) E ( s ,. ..,s ) = n (e 3 + e 3 ) k o n-1 . ~ a ( i + p k - l - ( s p k 2+...+s ) ) - a ( i + p k -l-(s. 0 p k 2 x (e k ~ 2 ° + e k " 2 + . . . + S Q ) ) a ( i - p k - l + 2 p k - 2 s , n p k "*"-(s 0 p k 2+...+s ) ) -a(same)) + e k-1 k-2 ° + e ) k - 1 a p 3 ( p - l - s . ) - a p 3 ( p - l - s . ) = Jl (e 3 + e J ) j = 0 a ( i + p k - l - ( s p k "*"+..,+s ) ) - a ( i + p - l - ( s , p k ^ " + . . . + 3 x ( e k i ° + e k l c T h i s p r o v e s t h e a s s e r t i o n . , T h e r e f o r e , we have n-2 a p 3 ( p - l - s . ) - a p 3 ( p - l - s . ) ? n-l (V--" 8 n-l ) = 11 ( £ 3 + e 3 > j=0 a ( i + p n ~ - l - ( s 0p n~ 2+...+s )) -a(same) x (e n _ z 0 + e ) n - 1 a p 3 ( p - l - s . ) - a p 3 ( p - l - s . ) = n ( e J + e 3 ) , n - l s i n c e = 1 ( i i ) An ana l o g o u s p r o c e d u r e shows t h a t I _ ( t 06(m +2) - B ( m f 2 ) . _ - ( n - t ) n ~ 1 , epJ ( p - l-s ) -Bp 3 ( p - l - s j (m) W + u ) - -2 II (co J + co J ' p W e P . j = 0 T h i s c o m p l e t e s o u r d e t e r m i n a t i o n o f n ( s ,...,s .) a t the c l a s s e s o n-1 67. / h° 0 \ andl k" 0 We have at the c l a s s / h 0 \ 0 h -a 0 k 0 h -a J n-1 a p J ( p - l - s . ) - a p 3 ( p - l - s ) r,(0, ...,0)(ti ) = n (e j=0 + e ) - 1 n(s , . . . ,s ) (h o n - 1 j=o at the class / k 0 n-1 Bp 3(p-l-s.) -Bp 3(p-l-s ) n(o, . . . ,o) (k M ) = - n (to j=o + 03 ) + 1 n(s , ...,s )(k ) = - 2 o n — l , n-1 Bp j(P-l-s.) -Bp 3(p-l-s ) n (to j=o We summarize w i t h a t a b l e : n(0,...,0) 1 0 0 1 (2 n-l)q n(s ,...,s ) 2 q o n - 1 Not a l l s. =0 3 -1 0 V h Q 0 0 -1 /( 0 h k h 0 0 V + (2^1) q -1 jn-1 apJ (p-l-s.)+ i n-L Bp J(p-l-n (e . 3 • j - II (to ^ = 0 -ap 3(p-l-s.) } 2 t q ( - D 1 j=0 e - 1 -Bp J(p-1-to + 1 / 4-\ 1 1 - 1 2 - ( n _ t ) n i j = 0 a p J ( p - l - s ) -ap 3 ( p - l - s ) 3 ) Where t = number of s.. ' s not equal to p - 1 , _ 2 - ( n - t ) n Bp 3(p-l-s.) -Bp 3(p-l-s. + ar -and i = s + S - P + . . . + S np o 1 n - 1 n-1 68. B i b l i o g r a p h y 1. R. Brauer and C. N e s b i t t , "On the modular characters of groups", Annals of Math. 42(1941) 556-90. 2. CW. C u r t i s and I . Reiner, Representation Theory of F i n i t e Groups' and A s s o c i a t i v e Algebras, John Wiley and Sons, New York, 1962. 3. H. Jordan, "Groups characters of v a r i o u s types of l i n e a r groups", American J . Math. 29(1907) 387-405. 4. I . Schur, "Untersuchungen uber d ie D a r s t e l l u n g der end l i c h e n Gruppen durch gebrochene l i n e a r e S u b s t i t u t i o n e n " , J . Reine Angew. Math. 132(1907) 85-137. 5. Bhama S r i n i v a s a n , "On the modular characters of the s p e c i a l l i n e a r group S L ( 2 , p n ) " , Proc. London Math. Soc. I l l 14 (1964) 101-114. 6. R. S t e i n b e r g , Lectures on Chevalley groups, Yale Lecture Notes (1967). 

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