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Bruhat decompositions and Schubert varieties Li, Qing 1994

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B R U H A T D E C O M P O S I T I O N S A N D S C H U B E R T VARIETIES By Qing Li B. Sc. (Mathematics) Hunan Normal University, P.R.China, 1984 A THESIS SUBMITTED IN PARTIAL FULFILLMENT O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F S C I E N C E in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF MATHEMATICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA January. 1994 © Qing Li, 1994 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of IM (A T M;Lwa:^ ~V''C-^ The University of British Columbia Vancouver, Canada Date J DE-6 (2/88) A b s t r a c t This paper is divided into two parts. In first chapter we give a scecial C* action on G/ P and show that the associated plus decomposition of this action is the Bruhat decompo-sition of G/P. Moreover we prove that the set of Schubert varieties of Gi,(C") is in one to one correspondence with the plus cells of a certain C* action on Gfc(C"). In chapter two we relate partitions to Schubert varieties. Also we relate the ordering of Schubert varieties by inclusion GA:(C") to a quotient of the Bruhat ordering on the Weyl group Sn. Several examples have been included. Table of Contents Abstract ii Acknowledgement v 1 Bruhat decompos i t ion , B - B decomposi t ion , Schubert varieties 1 1.1 The Bruhat decomposition and B-B decomposition of G / P 1 1.1.1 S<"hubert varieties 1 1.1.2 C*-a(-tions 5 1.1.3 The B-B decomposition 6 1.1.4 The Bruhat decomposition 7 1.1.5 The root system 11 1.1.6 G/P and GA-(C") 12 1.1.7 A C*-action with isolated fix(xl points (m G / P 13 1.1.8 The minimal cos(^t repres(nitatives 15 1.1.9 The Bruhat de<:omposition and the B-B decomposition 16 1.2 Schub(!rt vari(>ti(;s and th(; B-B decomposition 20 1.3 Bruhat d(x-<)mposition and th(^ Schub(;rt varieti(^s 23 2 Part i t ions and Schubert Varieties 26 2.1 Partitions and Schubert Vari(?ti(;s 26 2.1.1 Partition of positive integer 26 2.1.2 Schub(^rt Vari(?ti(;s 27 2.1.3 The Basis Th(!or(mi 28 111 2.1.4 Partitions and S('bub(;rt vari()ti(;s 29 2.1.5 Unimodality 29 2.1.6 Uniniodality of S<;hnbcrt variety 30 2.2 The Briihat orcicn- and Schubc^rt vari(!ti(>s 32 Bibliography 35 IV Acknowledgement It is a special pleasure to be able to thank here my supervisor Dr. J. Carrell. Also I would like to thank my husband Liang for his support. Chapter 1 The Bruhat decomposition, B-B decomposition and the Schubert varieties 1.1 The Bruhat decomposition and B-B decomposition of G / P 1.1.1 Schubert varieties Let V be a finite dimensional complex vector space, C* the complex multiplicative group, Z the set of integers and R the field of real numbers. The space of n-tuples (a(l),..., a(n)) of complex numbers is called afRne space and denoted by A". Projective n-space P " may be defined to be the set of equivalence classes of C"+^ — {(0 , . . . , 0)} relative to the following equivalence relation: {p{0),...,p{n))^{q{0),...,q{n)) if and only if there exists a c in C*, such that p(i) = cq{i) for all i. Intuitively, P" is just the collection of all lines through the origin in C""*"^ . It is convenient, when working with V of dimension n + 1, to identify the set of all 1-dimensional subspaces of V with P"; we write P{V) for P " in this case. Each point in P " can be described by homogeneous coordinates {p{0), • •. ,p{n)), which are not unique but may be multiplied by any nonzero scalar. We topologize P " by taking a closed set to be the common zeros of a collection of homogeneous polynomials. Every closed subset of P " is called a projective variety. A linear space L in P" is defined as the set of points P = {p{0),...,p{n)) of 1 Chapter 1. Bruhat decomposition, B-B decomposition, Schubert varieties 2 P " whose coordinates p{j) satisfy a system of linear equations X)"=o ^ijPU) = 0 with i = 1,... ,{n — d). We say that L is d-dimensional ( or rf-plane for short) if these (n — d) equations are independent , in other words, if the [n — d) x (n + 1) matrix of coefficients (bij) has a nonzero {n — d) x {n + l)-minor. Define Gd(C") be the set of all d-planes in P" . Following we want to show that can always obtain Gn,d as a closed subset of projective space P^ where we put once and for For convenience let us make the following convention. For any (<^+1) x (n + l)-matrix [p«(i)] with i = 0,... ,d and j = 0,... ,n, and any sequence of {d + l)integers jo, • •. ,jd with 0 < jk < n, let us denote by p{jo, • • • ,jd) the determinant of the {d+ 1) x {d + 1)-matrix [pt(i*:)] with i,k = 0,... ,d. Fix a of-plane L in P" . Pick {d + 1) points P, = {pi{0),... ,Pi{n)) with i = Q,... ,d which span L and form the {d + I) x {n + l)-matrix [pi(i)]. Note that the number of sequences jo,... ,jd with 0 < jo < • • • < id < " is exactly {N + 1). By linear algebra at least one of the [N + 1) determinants p{JQ,... ,jd) with 0 < jo < • • • < irf ^ " must be nonzero. So, when ordered lexicographically, these determinants define a point ( • • • ,p ( io , . - . , id ) , . . . )o fP^ . Let Qi = (9t(0),.. . , qi{n)) for i = 0 , . . . , rf be another {d + 1) points spanning L. Then linear algebra yields a nonsingular (o? + 1) x (d + 1)-matrix A which carries the P,-into the Qi; in other words , we have [qi{j)] = A • \pi{j)] where the dot denotes matrix multiplication. Clearly we then have q{jo, • • • ijd) = d6't{A)p{jo,... ,jd), where det{A) denotes the determinant of A. So the points Qi give rise to the same point of P^. The coordinates P{JQ, ... ,jd) of this point are called the Plucker coordinates of L. Not every point of P ^ arises from some rf-plane in P" . It can be proven that the Plucker coordinates P{JQ, ... ,jd) of any rf-plane in P " always satisfy the following Chapter 1. Bruhat decomposition, B-B decomposition, Schubert varieties 3 quadratic relations (QR for short): ^ ( - 1 ) V( io , . . . Jd~ikx)p{ko, ...,kx,..., fcd+i) = 0 where jo,... ,jd-i and ko,.. •, kd+i are any sequences of integers with 0 < _7^ , k^ < n arises from a c?-plane L in P" . Here kx means that the integer kx has been removed from the sequence. On the other hand, we are now going to prove any point (... ,p{jo,... ,jd), • • •) of P^ whose coordinates satisfy the QR arises from a unique rf-plane L in P" . First, assume p{ko,... ,kd) is not zero, now we will show that the {N + 1) coordi-nates p{jo,... ,jd) are already determined by the [{d + l){n — d) + 1] coordinates of the form p{ko,... ,kx,...,kd,ja), that is by the coordinates p{io,...,id) with at most one of io, ...,id not among ko,..., kd. Let jo,... ,jd be a sequence of integers of which exactly m integers are not among the integers ko,... ,kd and let jp be one of these m. The QR corresponding to the sequences io, • • • ijfi,- • • ijd and ko,..., kd,j/3 obviously yields the equation PUO, •••,//?,• ••jdjfi)piko, ...,kd) = X)(~l)V(io, • • • ,i/3, • ••jdkx)p{ko, ...,kx,---, kd,jp) Now if kx is among j o , . . . ,jd, then p{JQ,... ,jfi,.. .jdkx) is zero; if kx is not among jo,..., jd, then exactly m — 1 of jo, • • •, j/3, • • • jdkx are not among ko,..., kd- So, if m > 2, we can express p(jo, • • • ,jd)p{kQ,... ,kd) in terms of the coordinates p{io,... ,id) with at most m — 1 of io,... ,id not among ko,... ,kd. Continuing this process of multiplying by p{ko,... ,kd) and of using a QR, we can express p{jo,.. • ,jd)p{ko, • • •, kd)"^~^ as a polyno-mial in the coordinates p{io, • • •, id) with at most one of «o, • • •, id not among ko,...,kd,. Since p{ko,..., kd) 7^  0 we conclude that these [{d -|- l)(n — rf) -f-1] coordinates determine the others. Without loss of generality, assume p{kQ,... ,kd) = 1. We are going to construct a c/-plane L in P " whose Plucker coordinates are equal to the coordinates p{jo,... ,jd) of Chapter 1. Bruhat decomposition, B-B decomposition, Schubert varieties 4 the given point in P^. For i = 0,... ,d and j = 0,... ,n put PiiJ) = Pih • • • ki-ijki^i •••kd) The vectors (pi(0),... ,Pi{n)) with i = 0,.. .,d are linearly independent because we have Pi{k-y) = 0 for i ^ j and Pi{ki) — 1. So, these vectors span a d-plane L in P " . Now the Plucker coordinate p'{jo,... ,jd) of L is defined as the determinant of the matrix \pi{j^)] with i,/3 = 0,... ,d. So, if jjj = k^ for (5 ^ \, this matrix coincides with the identity matrix outside the A-th column. Hence p'O'o, •••,3d) = p{j>) = p{jo, • • • ,jd) whenever at most one jx of JQ, ... ,jd is not among ko,..., kd. Because these coordi-nates determine the rest as it is shown above, we have p'{jo,... ,jd) = pijo, • • • ,jd) for all sequence jo, •••,Jrf • Thus the point (... ,p{jo,... ,jd),- • •) of P^ arises from the d-plane L. Finally, we can show that this d-plane L is unique. The above shows that we can always get Gn,d from the closed subset of P ^ which satisfying the QR. In the course of the proof, the closed subset of P^ satisfying QR is covered by (N-Hl) copies of affine (d + l){n — d) space , so this set of points is a submanifold of P^ of dimension (d-f- l)(n — d). Thus we call Gn,d the Grassmann manifold, (cf. [KL]). Example 1 . The lines in P^ are represented by the points of the 4-dimensional Grassmann manifold G^^i , which can be described as the points of P^ ( N = (f\-l = 5) whose coordinates satisfy the quadratic relation. Let AQ C Ai G • • • C Ad he a strictly increasing sequence (or flag) of (d -|- 1) linear spaces in P " . A d-plane L in P " is said to satisfy the Schubert condition defined Chapter 1. Brubat decomposition, B-B decomposition, Schubert varieties 5 by this flag if dim{Ai f]L) > i for all i. The set of all such rf-planes L corresponds to a subset of Gd(C") , which is denoted by ft{Ao, • • •, Ad) . The subsets fl{Ao, • • • ,Ad) are subvarieties of Gn,d, called Schubert varieties . Particularly, for the strictly increasing sequence C"" C C"^ C • • • C C " , here ao , . . . , a„ G Z, we always use fi(ao) • • •, ^d) to denote the Schubert variety 17(0"°, • • •, C""). 1.1.2 C*-actions By a holomorphic C*-action on a complex projective variety X, we mean a holomorphic map fj. : C* x X —^ X satisfying/i(l,a;) = x and fi{XiX2,x) = fi{Xi, fx{X2,x)). Denote by GL{V) the set of all invertible endomorphisms of V, i.e. the group of all invertible n x n matrices with complex coefficients. Then a holomorphic mapping f : C* -^ GL{V) is a one parameter subgroup of GL(y) ifit satisfies/(si) = f{s)f{t), for s,t in C*. It is well known that any holomorphic action of C* on P" arises through a one parameter subgroup A : C* —>^  PGL{V). Here , PGL{V) is the projective general linear group. Up to conjugation by a projective transformation, a C*-action on P " is of the form A • [z i , . . . , ^„+i] = [A"i^i,..., A«"+>2:„+i] (1.1) where o i , . . . , a„+i 6 Z. A point X in X is called fixed if it satisfies X • x = x. Use X^* to denote the set of fixed points , i.e. X*^' ={xeX\X-x = x} (1.2) For any C*-action on F " of the form (1.1), [0 , . . . , 0 ,1 ,0 , . . . , 0] is obviously a fixed Chapter 1. Bruhat decomposition, B-B decomposition, Schubert varieties 6 point since we can always find some i between 1 and n, such that a, ^ 0. This means any C*-action on P " has fixed points. It is easy to see that if oi, • • •, a„+i are distinct, the fixed point set of (1.1) is X^* = {[0 , . . . ,0 ,1 ,0 , . . . ,0] |a . -^0} (cf. [CJl]). Example 2. Let V = C .^ Then X = P{V) is a smooth projective variety. Let A • [a;i,a;2,a;3] = [A^'xi, A"2a;2, A^'xa] be the C*-action on X, for 01,02,03 E Z. Let ei, 62,63 denote the standard basis of C^. If Oi = 02 = 03, since A • [xi, a:2,2:3] = [a;i,X2,X3], the set of fixed points is P{V). If oi = 02 > 03, A • [xi,X2,X3] = [xi,X2,\''^~"^X3] , SO the fixed point set is P(Cei + Ce2). Similarly, since A • [rri,a:2,a;3] = [\''^~"^xi,X"^~'"^X2,X3] , Ce^ is also a fixed point and the fixed points set is P(Cei + Ce2) [j P{Ce3) . If oi > 02 > 03, then A • [a;i,a;2,a:3] = [a;i, A°2-"ia;2, A"'"''ix3] , so Cei is a fixed point. Similarly, Ce2 and Ce^ are fixed points and the fixed points set is P(Cei) UP(Ce2) U-P(Ce3). 1.1.3 The B-B decomposition If X is a smooth complex projective variety with C*-action having only finitely many fixed points, then one can obtain the Bialynicki-Birula (or B-B) decomposition as follows (cf. [A]): Theorem 1 Assume X has a C*-action with fixed point set X^' — {xi,... ,Xr}. Let X^ = {x £ X\ limf^o t • X = Xi). Then, for each i = 1 , . . . , r, X,^ is an affine space and x = \juxt. Chapter 1. Bruhat decomposition, B-B decomposition, Schubert varieties 7 We call this decomposition a plus decomposition of X and say X^' is a plus cell. Let Tx.{X) denote the tangent space of X at Xi and Ta;.{X)'^ be the tangent space of X+ at Xi. The C*-action // : C* x X -^ X on X induces a C*-action a : C* x T^^iX) -> Tx- (X) on Tx- (X) which sending (A, t) to dfj,x{t) , here dfj,x{t) denotes the differential of the map fix '• X -^ X at X, where fj.x{x) = ^-x- Since Tx^{X)'^ C Tx.(X), a induces C*-action on n,{X)+ by restricting a on Tx,(X)+, that is a\T^.^x)+ : C* x T,,(X)+ ^ Tx,{X)+ given by {\,t) H-)> dfix{t). (cf. [A]). Example 3 . Let C* act on F^ by ^- [Z1,Z2,Z3\ = [zi,tZ2,t'^Z3] Clearly the fixed points are [1,0,0], [0,1,0], and [0,0,1]. [1,0,0]+ = {z e P^|limt_vot • z = [1,0,0]} , since limt_^o*- z = \imt^Q[zi,tz2,t^Z3] = [^1,0,0], thus, [1,0,0]+ = {ze P^\zi # 0} = P2_ y(^^). [0,1,0]+ = {z e P^|limt_^o* • z = [0,1,0]}, since lim(_>o* • z = limt^o[zi,tz2,t'^Z3] = limt^o[0,i^2,«^^3] = [0,2^2,0], if zi = 0. So, [0,1,0]+ = {z E P^\zi = 0 and Z2 j^ 0} = Viz,)-{[0,0,1]} [0,0,1]+ = {z e P'^\limt^ot • z = [0,0,1]}. since lim(_^o<--2^ = limt^o[zi,tz2,t'^Z3] = limi^o[0,0, t^zs] = [0,0, Z3], if ^1 = 0 an d 2^ = 0. So, [0,1,0]+ = {^  G P^\zi = 0,Z2 = 0 and ^3 7^0} = {[0,0,1]}. Obviously, P^ = [1,0,0]+U[0,1,0]+U[0,0,1]+ and in each case , the plus cell is an affine space. 1.1.4 The Bruhat decomposition First, we introduce some concepts. Chapter 1. Bruhat decomposition, B-B decomposition, Schubert varieties 8 We topologize an affine space by decreeing that the closed sets are to be precisely the affine varieties. This is called the Zariski topology. A linear algebraic group is defined as a Zariski closed subgroup of GL{V). A linear algebraic group is semisimple if the center of its identity component is 0-dimensional. Let G be a connected semisimple linear algebraic group; A subgroup B of G is said a Borel subgroup if 5 is a maximal closed connected so Ivable subgroup. A connected algebraic group is a torus if it is isomorphic to some group of diagonal matrices. Let T be a maximal torus contained in B. Let D denote the group of all upper triangular n x n matrices. We define the additive group Ga as the affine line A^ with group law f{x,y) =x + y. Example 4. The special linear group SLiV) consists of all the matrices of deter-minant 1 in GL{y); it is clearly a group ( because of the product rule for determinant) and is closed ( being the set of zeros of det(Tij) — 1 ). Let B be the upper triangular subgroup of SLiy) . Obviously, B = {bij){i < j) is closed as the set of common zeros of all polynomials ZI"=i bij = 0,1 < i < n. We claim that .B is a Borel subgroup of SL{V). One can show that B is generated by subgroups Uij{i < j) where Uij consists of all matrices with I's on the diagonal, arbitrary entry in the {i,j) position and O's elsewhere. Uij is isomorphic ( as algebraic group ) to the additive group Ga, since multiplication of such matrices just involves addition of the {i,j) entries . As Ga is connected , so is Uij{i < j) . Since a group generated by a family of closed connected subgroups is also connected, we conclude B is connected. Let X and Y be arbitrary closed (e.g. finite) subgroups of an algebraic group G. Then Chapter 1. Bruhat decomposition, B-B decomposition, Schubert varieties 9 the commutator group {X,Y) is the subgroup generated by commutators xyx~^y'~^ for all X in X and y in Y. Now we are going to prove that B is solvable. Recall that an abstract group G is solvable if its derived series terminates in e (here, e is the identity), this series being defined inductively by V^G = G, V'+^G = {V'G,T>'G) for all i > 0. Let H denote the group consisting of upper triangular matrices with all diagonal entries 1 and e,j be the matrix having 1 in the {i,j) position and 0 elsewhere. Since the diagonal entries in the product of two upper triangular matrices are just the respective products of diagonal entries, it is clear that the derived group of B lies in H. On the other hand, H is obviously generated by the matrices hij{a) = 1 + ae,j, {i < j ; aEC).lftis the diagonal matrix with i-th entry 2 and all other diagonal entries 1, then a quick calculation shows that thijt~^{a) = hij{2a), hence that {t,hij{2a)) = hij{a). It follows that H is precisely {B, B). Now look at {H,H). Since eijCki = SjkBii, i < j,k <l. we get hij{a)hki{h)h-^{a)Ki^(b) = (1 + aeij){l + bcki)(1 - ae,-j)(1 - 6e«) 1 — abcii \i i < j = k <l 0 otherwise That is , {H, H) = {V^B, V^B) = V'^B is the linear span of all (1 + e,-j), for all j-i>l. Similarly, V^B is the linear span of all (1 + e,y) for all j — i> h — 1. Thus, it is clear that the derived series of B will terminates to e. That is, B is solvable. Chapter 1. Brubat decomposition, B-B decomposition, Schubert varieties 10 Therefore 5 is a connected solvable subgroup of SL{V). The Lie-Kolchin Theorem (cf. [Hul]) says that if G is a closed connected solvable subgroup of GL{V) and V^  7^  0 is finite dimensional , then G has a common eigenvector inV. So B has a common eigenvector in V. The theorem allows us to regard the solvable connected algebraic group as a subgroup of some upper triangular group D. This implies that D = B is a maximal closed connected solvable subgroup of SL{V). Thus S is a Borel subgroup of SL{V). Let T denote the group consisting of all the diagonal matrices in SL{V) . Evidently T is a torus and T contained in B, recall we define a torus to be a connected algebraic group which is isomorphic to some group of diagonal matrices, so T is a maximal torus. An opposite Borel subgroup of C? is a Borel subgroup B~ which satisfies B f] B~ = T. If G = GLiy), the upper triangular matrices form a Borel subgroup B and B" is the lower triangular matrices, (cf. [Hul] p.160). Thus, B~ is the lower triangular subgroup of SL{V). Now let us go back to G{= GL{V)). Let U~ denote the unipotent radical of B~ , that is U~ is the largest connected normal unipotent subgroup of B~. If G = GL{V), then U~ consists of lower triangular matrices with I's along the diagonal. Then there is a decomposition B~ = U~T and T normalizes U~ . Define the normalizer N(T) of T as N{T) = {x e G\xTx~^ = T). The group W = N{T)/T is called the Weyl group. It is a finite group, (cf. [Hu3] p.43). Chapter 1. Bruhat decomposition, B-B decomposition, Schubert varieties 11 Let T still be a maximal torus in G. We fix a Borel subgroup of G containing T. G has an associated decomposition called the Bruhat decomposition. Theorem 2 G = \Jwew Bn^B (disjoint union) where n^ G N{T) is a representative of w. The homogeneous space G/B is a disjoint union of double cosets Bn^B/B where n^ inN{T). 1.1.5 The root system A character a of T is called a root of G (with respect to T) if there exists a one to one homomorphism a:^  : C —> G such that txa{z)t~^ = Xa{a{t)z) with i in T and z in C. The root system is defined as the set $ of roots of G. (cf. [S] p77). The Lie algebra ^ of G is defined to be tangent space Te{G) with Lie bracket defined as in [Hu3] p.2. T acts on G via the differential at e of the mapping g y-> tgt~^, and we have the well known Cartan decomposition: g = h®Y.Q<^ into T stable subspaces. Here h and Q are the Lie algebras of T and G respectively. A Borel subalgebra is a maximal solvable subalgebra of ^. If J3 is a Borel subgroup of G, then B = Te{B) is a Borel subalgebra of Q. If 5 is a Borel subgroup of G such that T C B, then the Lie algebra B of B has a corresponding decomposition: where $+ C $. Chapter 1. Bruhat decomposition, B-B decomposition, Schubert varieties 12 The subset $+ of $ has the following properties: $ = $ + U $ - , 0 = $ + n o -where $ - = - $ + . (cf. [Hul] pl71). 1.1.6 G/P and Gfc(C") A closed subgroup of G is parabolic if it includes a Borel subgroup. A closed subvariety of a projective space is called a projective variety. For G = GL(C"), let P be the parabolic subgroup of G consisting of all the matrices of the form A B ^ 0 C J where A G GLk{C),C G GLn-k{C) and B is an arbitrary k x {n — k) matrix. G/P is a projective variety ( closed subset of P" . cf. [Hul] pl2) called an algebraic homo-geneous space. Since projective space P " is compact, it follows that every projective variety is compact. We now prove that G/P is homeomorphic to Gfc(C"). Let A = {aij)nxn be a matrix in GL(C"). Use Cj denote the column vector of A, (di,- • • ,ak) be the space spanned by k vectors ai,---,ak- Since ai , - - - ,a„ are linear independent, we can construct a map iPi : GL(C") -> Gfe(C") by sending A = (ai, • • •, a„) to the column space (ai, • • •, a^) of the first A; columns of the matrix A. Obviously ipi is onto. Since (fi{AB) and (pi{A) have same column space, for any B E P. ipi induces a map Chapter 1. Bruhat decomposition, B-B decomposition, Schubert varieties 13 <^:GL(C")/P-^Gfc(C") Easy to see that (p is also onto. Note that if (pi{A) = (pi{B) for any two matrices A, B in GI(C") , then A'^B e P. Therefore, the map ip : GL{C^)/P ^ Gk{C") sending AP to (ai, • • •, a^) is a bijection. Thus, we may identify Gfc(C") and GL{C^)/P. Because G/P is compact, hence Gfc(C") is compact also. 1.1.7 A C*-action with isolated fixed points on G/P Now we will give a special C*-action on an algebraic homogeneous space G/P and then show that the associated plus decomposition of this action is the Bruhat decomposition of G/P. Let A : C* —> T be a one parameter subgroup of T, so that X{siS2) = A(si)A(s2)- Let X{T) denote the set of all homomorphisms a : T -^ C* and Y{T) denote the set of all one parameter subgroups of T. There exists a perfect bilinear pairing: X{T) X Y{T) - i Z sending (a, A) to (a, A), where {a,X) is defined as follows: if a G X{T) and A G Y{T), then a • A G X{C*) and thus there exists a unique (a. A) G Z such that aX{s) = s^"'^\ since every 0 G X{C*) has the form <p{s) = s'' for some A; G Z. Now, we can choose A G Y(T) so that let (a. A) > 0 for all a G $+(cf. [Hu3] p.43). Clearly, a G $+ if and only if (a, A) > 0. So A defines a C* action on G/P Chapter 1. Bruhat decomposition, B-B decomposition, Schubert varieties 14 A* : C* X G/P -^ G/P given by X*{s,gP) = X{s)gP (1.3) Let U~ be the unipotent radical of B~, $~ = {/?i,..., /?„} and A : C* -> T be a one parameter subgroup of T, so that X{siS2) = A(si)A(s2)-Lemma 1 Assume u G U~. If \{s)u\{s)~^ = u for all s G C*, then u = 1. Proof . By definition of $, for each a in $~, there exists a one to one homomorphism Xa : C -¥ U~ with the following two properties: 1) For t £ T, tXa{z)t-'^ = Xa{a{t)z). 2) The mapping U/}^ x • • • x [/g„ -> U~ which sends (u i , . . . , «„) to ui • • • w„ is a homeomorphism onto U~, where t/g = x 13(C) We can write u = UiXo,{zi).{d. [Hul] p.lOO). Then X{s)u\{s)-' = \{s)l[xi3,{zi)^~\s) = l[\{s)x0,{zi)\-\s) Since u is a fixed point, That is , s^/^''^^Zi = Zi for all i. Because (^ ,-, A) < 0, Zi =0 for all i. Therefore « = 1. Q.E.D. Chapter 1. Bruhat decomposition, B-B decomposition, Schubert varieties 15 Lemma 2 (G/P)^* is finite. Proof . Let TT : G/B —> G/P be a map given by n{gB) = gP. Obviously n is continuous. Since Tr~^{gP) = gP/B is a closed subvariety of G/B, it suffices to assume P = B. Suppose there are infinitely many fixed points on G/P. Thus, there exist infinitely many fixed points on G/B. Let p-.U' ^ G/B send u to uB, and fi = p{U-) C G/B. Then ri is a Zariski open set and p is a homeomorphism. Now let T act on U~ by conjugation, / : U~ xT —^ U~, f{u, t) = tut~^. T acts on Q, by left translation, that is r^ : fi x T —>• Q.,g{uB,t) = tuB. and we get a commutative diagram: U- xT - A U-\. p X 1 \. p Q x r - ^ Q since tut~^B = tuB. This shows p is T-equivariant. From lemma 1, we see {G/Bf'\^n = {B}. In a similar manner we can show all points of (G/B)^' are isolated. Therefore (G/P)^' is isolated. Since (G/P)^' is compact, we get (G/P)^' is finite. Q.E.D. 1.1.8 The minimal coset representatives For an arbitrary linear algebraic group G, $"*" still denotes the positive roots . A linearly independent subset A of $+ is called a base if every root /? G ^"^ can be written additively as P = X)Q6A ^aO: with integral coefficients ka all nonnegative. The roots in A are called simple. Chapter 1. Bruhat decomposition, B-B decomposition, Schubert varieties 16 To every a G $, one can associate a reflection Sa €W in such a way that the s^ for a in A generate W. Let 5 be a set of simple reflections Sa, a E A. For any subset / C 5, define Wj to be the subgroup ofW generated by all Sa E I and let A/ := {a E A|sa G / } . Since the generators s E S have order 2'mW, each w ^ linW can be written in the form w = siS2- • • Sr for some s, (not necessarily distinct) in S. If r is the smallest such number, call it the length of w, written l{w). Define W^ : = {w E W\l{ws) > l{w) for all s E I}. Then, for every w in W, there is a unique w^ E W^ and a unique Wj E Wj such that w=w^wi. Their lengths satisfy l{w) = l{w^) + l{wj). W^ is called the set of minimal coset representatives, (cf. [Hu2] p.19). Now, for each I G S, set F/ := \JweWi Bn^B, where n^ is a representative of w. Theorem 3 1). If I G S, Pj is a parabolic subgroup of G and every parabolic subgroup of G which contains B is a Pj for some I G S. 2). There are generalized Bruhat decompositions of G and GfP: G= U Bn^P GIP= U Bn^P/P. We will explain this explicitly for Schubert varieties in §1.3. (cf. [Hul] p.177). 1.1.9 The Bruhat decomposition and the B-B decomposition Now we are ready to prove our main result. Let / G A and let P be the parabolic subgroup corresponding to / as in §1.1.8. Theorem 4 There exists a C*-action on G/P whose plus decomposition is the Bruhat decomposition G/P — \J Bw^P, where w^ are the minimal coset representatives. Chapter 1. Bruhat decomposition, B-B decomposition, Schubert varieties 17 Proof. Take a one parameter subgroup A : C* —>^ T of T with —{a, A) > 0 for all a £ $+ and the induced C*-action on G/P, A* : C* x G/P -^ G/P with X*{s,gP) = X{s)gP as ( 1.3 ). Let bnP E Bw^P where 6 in P , n a representative of w^ , since P = UT, we have b = ut with u in U and t in T, so \{s)bnP = X{s)unP = X{s)llx_,3i{zi)nP Because —(/?,•, A) > 0, we have limA(s)6nP = nP = w^P s—>0 Therefore w^P G X^'. In addition, Bw'P C X+,p Then G/P = \jBw'PG[jX+,j,CG/P As equality holds, and Bw^P C X^jp. Hence Bw^P = X+jj, w^ P Q.E.D. Corollary 1 {G/P)^' = {w^P\ w^ a minimal coset representative } and (G/B) = {wB\w e B} Proof. From the proof of Theorem 3, we have Chapter 1. Bruhat decomposition, B-B decomposition, Schubert varieties 18 (G/P)^* = {w^P\w^ a minimal coset representatives }. For G/B — {jBu^B where Uyj in N{T) ( N{T) is the normalizer of T ), let A : C* —>• T be a one parameter subgroup of T, so that A(siS2) = A(si)A(s2). A defines a C*-action on G/B At : C* X G/B -> GIB given by X{{s,gB)^\{s)gB. (1.4) Let hriy^B € Bn^B/B where h in B, since B = UT, we have b = ut with u in U and t in T, considered that t is in N{T) , so \{s)hnwB = X{s)unwB = \{s)Y[x-ii,{zi)n^B = Y[x_p,{s-^^^^>^hi)n^B. Because —{Pi,X) > 0, by lemma 1, So limA(s)6n^5 = n^B s-fO 5n.5 c x+o But G/B = [JBn^B C [jXls C G / S Obviously the equality should hold. Hence Bn^B = X+ That means (G/Bf = {n^B\n^ G N{T)} Chapter 1. Bruhat decomposition, B-B decomposition, Schubert varieties 19 We know that wB = n^B Thus {G/B)^' = {wB\w e B} Q.E.D Now applying Theorem 1 to G/P , we can get the following result: Theorem 5 / / G/P = \Jwew' Bw^P is the Bruhat decomposition of G/P, then each Bw^P is an affine space and dim{Bw^P) = l{w^) Proof. It is known that G/P is a smooth projective variety. Prom lemma 2, we know the C*-action on G/P has finite fixed points. So by Theorem 1, each X,^ is an afiine space. Prom above however, X^ip = Bw^P, that is, Bw^P is an affine space and Now we have T.u,lp{G/P) = ®ae<l>-\^iGw'{a) and it is possible to show T^ip{G/P)+ = e G^na) aG $ ~ \ $ / w^{a) > 0 Thus dimcT^ip{G/P)'^ is the number of roots a in $ ~ \ $ / , such that w^{a) > 0 , or the number of roots a in $'*'\$/, such that w\a) < 0. Since «;^($/f|$+) C$+ We conclude dimcT^ip(G/P)+ = l(w^) Q.E.D. Chapter 1. Brubat decomposition, B-B decomposition, Schubert varieties 20 1.2 Schubert varieties and the B-B decomposition Here we would like to show that for a Schubert variety J l(ai , . . . ,0^) in Gfc(C"), there exists a C*-action on ^^(C") such that Xj' = f ] (ai , . . . , a^) for some component Xj. First, let's look at a simple example: Example 5. Consider the action on X = G2(C^) induced by the action A-(2;i, Z2, Z3) = (A""^i, A^^ zg, A^^ a^) on C^ where ai > 02 > «3 and Oj in Z , i = 1, 2, 3. For a pair of independent vectors u,v E C^, let {u, v) denote the 2-plane they span. We will compute (62,63)"^ , where 61,62,63 denotes the s tandard basis of C^ . It suffices to consider limA-yo A • F for 2-plane of the form V - (ai6i + ^262, /?i6i + /?2e2 + Psez) where 0:2 • /?3 7^  0. Now X-V = (A"'aiei + A"^a2e2, A"'/?i6i + A"^ /32e2 + X^'Pses) = (A°'-«^aiei + ^262, A"i-°'/3iei + X"''"'(^262 + P^e^) Since ai > 02 > 03, it follows that lim A • y = (a2e2, heo^) = (62,63) A—^U Recall the definition of Schubert cycles in G2(C^). If 61,62 are integers so that 1 < 61 < 62 < 3, set f](6i,62) ^{V G G2{C^)\dimc{VC\&^) > i} where C^ C C^ C C^ is the standard flag in C^. Then we call ^(61,62) the Schubert cycle of G2(C3). Note that if ^2 7^  0, then V 0 0(1,3) . Thus (e2,e3)+ = f2(2,3)- l]( l ,3) . Chapter 1. Bruhat decomposition, B-B decomposition, Schubert varieties 21 It follows that {e2,es)+ = ni2,3). By a similar argument, (ej, 6 )^+ = Q,{i,j), if 1 < « < i < 3. Now we consider the C*-action on X = Gfc(C") induced by the action A.(^ i , . . . , z„) = (A°'zi,...,A""z„) on C" , where ai > • • • > a„ and a, in Z, «' = 1 , . . . , n. First we give a very useful lemma. Lemma 3 Let 0 < oi < • • • < ajt < n (resp. 0 < bi < • • • < bk < n ) be a sequence of strictly increasing natural numbers. n{ai, • • • ak) (resp. fl{bi, • • • bf.)) denote the associate Schubert varieties. Therefore, ^{bi, •••bk) Q fl{ai, •••ak) iffbi < ai for alii = 1,..., k. Let e i , . . . , e„ denote the standard basis of C", then we have the following result: Theorem 6 If ii < • • • < ik are k integers between 1 and n, then for each fi(«i,..., ik) in Gk{C-) (e i j , . . . , e i j+ = Q ( n , . . . , 4 ) Proof. For independent vectors ui,... ,Uk G C", let {u\,..., Uk) denote the k-plane they span. Now we would like to compute (e j j , . . . , e,,.)"'". It is sufficient that we consider limA->o A • V for k-plane of the form V — (ofiiei H \- ttiijCij,... ,Q;fciei H 1- aki^ei^) where ai,j • • • aki^. 7^  0. Now Chapter 1. Bruhat decomposition, B-B decomposition, Schubert varieties 22 , . . . , , . . . , Hence lim A • F = (Q;i,ije,i,.. .,ak,i,ei^) = (e^j,. . . ,6,-,) A-s-O Recall the definition of Schubert cycles Q{bi,- • • ,bk) in Gfc(C"). If 6 i , . . . , 6jt are integers so that 1 < 6i < • • • < 6jt < n, set n{b,,.-.M) = {V£ G,(C")|dimc(i^nc'') > 0 where C^ C C^ c • • • C C" is the standard flag in C". Note that if Q:I_,J ^ 0, and ii — 1, ^ 2 , . . . , ik, is a strictly increasing sequence of integers between 1 and n, then V ^ fi(ii —1,^2,... , 4 ) ; • • •; iictk,ik ¥" 0) ^^^ h, • • • ?4- i , «fc —1, is a strictly increasing sequence of integers between 1 and n , then V ^ ^{ii, • •., ik-i,ik — 1)-The above calculation shows that A; ( e , j , . . . , e,-J+ = n{iu...,ik)\ \J n{iu...,ij -l,...,ik) Therefore (eij, . . . ,ei,)+ cn{ii,...,ik) Chapter 1. Bruhat decomposition, B-B decomposition, Schubert varieties 23 It follows that (e i j , . . . ,e , J+ C n{ii,...,ik). On the other hand, for any IT G J^(«i,... ,ik), H is thus a limit of k planes n„ where every n„ G (si^,..., e,-^ )^"''. We omit the details here. Thus (ei,,...,e,-,)+ = n{ii,...,ik) Q.E.D. We thus see that any Schubert cycle of Gjt(C") is in one to one correspondence with a plus cell of some C* action on Gfc(C"). 1.3 Bruhat decomposition and the Schubert varieties In this part, we will show that the closures of Bruhat cells in Gfc(C") are the Schubert varieties. Now view Sn the symmetric group on n letters as a subgroup of GL„(C) consisting of permutation matrices by letting a E Sn permute the basis vectors via cr(ej) = e„(i) where e,- is the i-th standard basis vector of C", i = 1,... ,n. Let T denote the group consisting of all the diagonal matrices in GL„(C), and B the upper triangular subgroup of GLn{C). The Weyl group of GL„(C) is isomorphic with Sn viewed as the set of permutation matrices. We have the Bruhat decomposition GLn{C)=[JBaB where o runs through the Weyl group 5„ of GL„(C). For any t = diag{ti,..., f„) E T where / i , . . . , t„ G C , ofjj : t i-> titj^ is a character. Every root has this form. The corresponding Xaij{s) = I + sEij ( s G C, E^j has 1 in {i,j) position and 0 elsewhere) has following property: txai^is)t~^ = t{I + sEij)t~'^ = I + stEijt'^ = I + saij{t)Eij = Xaij{aij{t)s). Chapter 1. Bruhat decomposition, B-B decomposition, Schubert varieties 24 This gives all the roots of GLn (C) with respect to T. The simple roots are of the form Q i^.j+i) 1 ^ * ^ w "~ 1 and positive roots are the aij with i > j . For any k = 1,... ,n — 1, let / = A\{Q;fc}- Then the parabolic subgroup P/ = \JweWj Bn^B of GL„(C) consists of all block upper triangular matrices ' A B 0 C where A e GLk{C), C G GLn-k{C) and B is an arbitrary k x (n — k) matrix. Thus, P/ is the P of §1.1.6, and Gfc(C") has been identified with the coset space G/P = {gP\g G GLn{C)}. The corresponding Bruhat decomposition of G/P is: G/P = Uu>evr/-^^lu^-Moreover, it can be checked that corresponding to /(cf.[Al]): Wj = {{ii,--- ,ik,ji,--- Jn-k) e Sn\l <ii,...,ik<k,k + l <ji,...,jn-k < n} and W^ = { (« i , • • • , ikJu • • • Jn-k) e Sn\il < i2 • • • < ik,jl <J2---< jn-k} Theorem 7 Assume Gk{C") = [JBw^P is the Bruhat decomposition o/Gfc(C"), where B is the Borel subgroup of upper triangular matrices, w^ varies through the minimal coset representatives ofW/Wj and P is the Parabolic subgroup P/ . (cf [HulJ p.l77). Then Bw^P = n{ii,...,ik) Proof . Let b = (6 i , . . . , 6„) in P with 6 i , . . . , 6„ be the column vector of b. Recall (bi,... ,bn) denotes the space spanned by the column vectors 6i , . . . ,6„. For any a in 5„, ba is therefore obtained by rearranging 6 i , . . . ,b„. Since / = A\{«fc}, any element w of W^ has the form (ii, • • •, 4 , i i , • • • ,jn-k) where ii < 22 • • • < ikji < J2 • • • < jn-k- Hence bw = {bi^,---,bi^,bj^,---,bj^_^) Chapter 1. Bruhat decomposition, B-B decomposition, Schubert varieties 25 Since the isomorphism ip : G/P —> Gfc(C") sends bwP to the space spanned by the first k columns of bw, thus ip{bwP) = (6 i i , . . . ,6 i , ) Take a one parameter subgroup A : C* —> T of T with A(s) = diag{s"^,..., s°") where ai > . . . > a„. Therefore, for each ctj = titj^^, (cVj, A) > 0. It was proved in Theorem 3 that ip{Bw'P) = {ei„...,ei,)+. So, (p{Bw^P) = ( e i j , . . . ,e,j.)+. In Theorem 5 we have proved that ( e , j , . . . ,e,j^)+ = Q{ii,...,ik). Thus Q.E.D. Chapter 2 Partitions and Schubert Varieties 2.1 Partitions and Schubert Varieties 2.1.1 Partition of positive integer Let n be a positive integer, then a partition of n is a finite nondecreasing sequence of positive integers si,S2, •.. ,Sr such that ]C[=i Si = n . The Si are called the parts of the partition. Many times the partition (si, S2,..., Sr) will be denoted by s, and we shall write s \- n to denote "s is a partition of n". Example 1 . (1, 2, 3) is a partition of 6, (1, 1, 1, 3) is another partition of 6. i.e. (1,2,3) 1-6 and (1,1,1,3) h 6. For studying partitions, the graphical representation is an efi"ective elementary device. To each partition s is associated its graphical representation Tg ( or Ferrers dia-gram), which formally is the set of points with integral coordinates {i,j) in the plane such that if s = (si ,52,. . . ,s„), then {i,j) G Tg if and only if 0 > z > —n + 1, 0 < j < Si+i —1. Example 2 . The Ferrers diagram of the partition 1 + 3 + 5 + 6 is Note that the «th row of the Ferrers diagram of (sj, S2, . . . , s„) contains s,- points.(cf. 26 Chapter 2. Partitions and Schubert Varieties 27 [AG] p.6). 2.1.2 Schubert Varieties Recall the definition of the Schubert varieties. Given an integer sequence 1 < oi < • • • < a^ < n, let ^ i C • • • C ^A; be a sequence (or flag) of subspace of C" satisfying dimAi = o-i,i = 1,... ,k. Define Cl{Ai,..., Af,) to be the subset of Gfc(C") consisting of all A;-planes L satisfying dimAi > i for i = 1,... ,k. Thus, if dimAi = i ior i — 1,... ,k, then Cl{Ai,..., Ak) consists of the single A;-plane Ak] while if dimAi = n — k + i for i = 1,... ,k, then fl{Ai,..., Ak) = Gk{C"). It can be shown that 0,{Ai,... ,Ak) defines a homology class (cf. [KL]) (we call it Schubert class) in the homology H*{Gk{C"-)]Z). Although the variety ^{Ai,... ,Ak) depends on the choice of fiag Ai C • • • G Ak, As in [KL], we are now going to show that, the homology class of fl{Ai,..., Ak) depends only on the integers a, = dim,Ai, for i = 1,... ,k. Indeed, for any subvariety ft{Bi,..., Bk) with dimBi = a,-, i = 1 , . . . , A;, there obvi-ously is a linear transformation /\[aij\ of P^ into itself which carries Gfc(C") into itself and Vt{Bi,... ,Bk) into Vt{Ai,... ,Ak) (where A[a,j] denotes the linear transformation induced by the matrix [dij]). Now consider the continuous system of subvarieties {hM)Q,{Ai,..., Ak) parametrized by the nonsingular n X n-matrices M, here (AM) denotes the linear transformation of P^ into itself induced by the matrix M. This system clearly includes Q,{Ai,..., Ak) and it also includes every subvariety r i ( 5 i , . . . ,Bk) with Gi = dimBi, for « = 1 , . . . , A;, since all the subvarieties in a continu-ous system are assigned the same homology class, the homology class of fl{Ai,... ,Ak) depends only on the integers a, = dimAi. We therefore denote the class of f!,(Ai,... ,Ak) by [o i , . . . , ttk] and call it the Schubert class associated to 1 < oi < • • • < a^ < n. Chapter 2. Partitions and Schubert Varieties 28 If we take Ai to consist of all points in P " of the form (p( l ) , . . . ,p{i),0,..., 0), then we call Ai C • • • C Ak & standard flag. 2.1.3 The Basis Theorem The classical basis theorem of Schubert calculus says that the Schubert varieties form a homology basis for Gfc(C"), in other words, the Schubert classes completely determine the homology of Gfc(C"). Consider the standard flag C^ C C^ C • • • C C" in C". Then for any increasing A;-tuple ( a i , . . . , a/t) of integers so that 1 < ai < a2 < • • • < a*. < n, the r i (C"i , . . . , C"*) are projective varieties whose associated homology classes in H{Gk{C");Z) we denote by [ai,...,ak]. Let X be a smooth projective variety with a C*-action. If the C*-action of X has finite fixed points, then H{X; Z) is a free Z-module and X/" are a basis of H{X; Z). (cf. [CJ3]). In §1.2 we have proved that a Schubert variety f i (ai , . . . ,ak) is the closure of some plus cell Xi' for a C* action on Gk{C"). Hence we can now deduce the Basis theorem (Basissatz) (cf. [KL]) as following: Theorem 8 For each m with 0 < m < k{n — k) , the[ai,... ,ak] with X)j=i(oj —j) = m form a basis of H2m{Gk{C"');7j). Chapter 2. Partitions and Schubert Varieties 29 2.1.4 Partitions and Schubert varieties Now we can establish a one to one correspondence between partitions and Schubert varieties. We consider the partition of m with a finite nondecreasing sequence of positive integers bi < b2 < • • • < bk such that J2i=i bi = m and m is the integer between 0 and k{n — k). Let a, = bi + i with i = 1,2,... ,k, thus a i , . . . , a ^ is a strictly increasing sequence of positive integers between 1 and n, that is , 1 < ai < 02 < • • • < Ofc ^ ^- Hence [o i , . . . , Ofc] in H2miGk{C^)', Z). This shows that a partition bi,... ,bk of a positive integer m corresponds to a unique Schubert class [o j , . . . , ak] where Oj = bi+i in H2m{Gk{C"')', Z). Therefore, we conclude that there is a one to one correspondence between the parti-tions and the Schubert varieties. 2.1.5 Unimodality Now. let us look at a sequence of non-negative integers xi,X2,- • •. First of all, if there exists an integer A'' such that xi < X2< ••• <XN > x^^i > XN+2 >••• then this sequence of integers {xi} is called unimodal. A polynomial is called unimodal if its sequence of coefficients is unimodal. The generating function f{q) of a sequence xi,X2,... is defined as the power series f{Q) = E ^"?"-n>0 Next we introduce a very important lattice. Chapter 2. Partitions and Schubert Varieties 30 Let X = (rci, a;2,...) with xi < ^2 < • • • denote an infinite sequence of non-negative integers which is eventually zero. Define x < y if Xj < yi for all i > 1. We call the partially ordered set of such sequence the Young lattice y. (cf. [H] pl69) Suppose now r = (^i, • • •, r^) is a partition with 1 < ^i < r2 < • • • < rj.. We consider Young's lattice y^ of the partition r. For any integer n, let x„ be the number of partitions of n such that the Ferrers diagrams of those partitions lie inside the Ferrers diagram of r. Use G(X)(g) to denote the generating function for this sequence. We call a partition r unimodal if G{yr){q) is a unimodal polynomial. The fact is, not all partitions r are unimodal. The first non-unimodal partition is r = (4,4,8,8). G{yr){q) has coefficients 1,1,2,3,5,6,9,11,15,17,21, 23, 27,28,31,30, 31,27,24,18,14,8,5,2,1 Uimodality fails at the 15th number and the three offending values are 31, 30 and 31. For more details, see [D]. 2.1.6 Unimodality of Schubert variety We now return to Schubert varieties. Let X be a smooth projective variety. If X is non-singular, then the sequence of Betti numbers of X is unimodal and symmetric. These properties of a nonsingular projective variety follow from the Lefschetz theorem and Poincare duality. Example 3. It is well known that G2(C'*) is a non-singular projective variety. Let us consider the Betti number sequence of 6*2(0"*). It is easy to check that this sequence Chapter 2. Partitions and Schubert Varieties 31 is 1,1,2,1,1. Obviously, this sequence is unimodal and symmetric. Hence it is quite natural for us to ask that what happens if X is singular? We would like to show that the unimodality of the Betti number sequence fails at some Schubert varieties. Example 4. We consider the Schubert varieties in 04(0^"*). Let p be a positive integer between 10 and 90. Let (61,. . . bk) be a partition of p. By the definition of partition, it follows that 61 < 62 < • • • and 61 H + bk = p. As we have discussed in §2.1.4, any partition (61,.. .bk) of p is in one to one correspondence with a Schubert variety fl{bi + l,62 + 2,... ,bk + k). We only consider those partitions of p which lie inside (4, 4, 8, 8). The Schubert variety Q(5,6,11,12) is the one in G4(C^^) which corresponds to the partition (4, 4, 8, 8). The Betti number sequence of $7(5,6,11,12) is obtained from Theorem 7. By this result the sequence of Betti numbers of $7(5,6,11,12) is 1,1,2,3,5,6,9,11,15,17,21,23,27,28,31,30,31,27,24,18,14,8,5,2,1 which uses the calculation of [D]. This sequence is non-symmetric and non-unimodal. In other words, the Betti number sequence of 0(5,6,11,12) is non-unimodal and non-symmetric. From the above example, we conclude that the Betti number sequence of a projective variety may not be either unimodal nor symmetric. Chapter 2. Partitions and Schubert Varieties 32 2.2 The Bruhat order and Schubert varieties If Sn is viewed as the group of permutation matrices, the Bruhat order on Sn is defined by a > T <j=> BaB D BTB where a and r are in 5„(cf. [HM]). Let a (resp. r) in 5„ and let 71 < . . . < 7jt (resp. / / ! < . . . < Hk) be the k natural numbers in increasing order determined by a{e^.) e{ei,...,ek} i = l,...,k (2.5) (resp. '^ (^ M.) G {ei,---,ejt} i = l,...,k) Now we show the relations between the Bruhat order and the Schubert varieties. Let I = A \ K } . Theorem 9 For any a, r in Sn which are minimal coset representatives for Wj, T < a if and only ifQ,{fii,... ^JJ.^) C f2(7i,... ,7it) for all k = 1,... ,n, where fl{fii,... ,/J.k) and f2(7i,... ,7fc) are the associated Schubert varieties in Gk{C"')-Proof. First, we fix some notation. For any b E B, let aj denote the j'-th column vector of b, then ba is constructed by rearranging a i , . . . ,a„, that is ba = ( a i , . . . , a„ )a = (a^j, . . . ,a,„) Similarly, any b*T is of the form b*T= « , . . . , < ) r = (a*^,...,a*J Chapter 2. Partitions and Schubert Varieties 33 Therefore, BaB =\JbaB = [j{ai,,..., UijB beB and BrB=[jb*TB = \J{a*^,...,al)B beB For any A; between 1 and n — 1, let a{e^.) € {ei,...,efc} r(e^J e {ei,...,ek} where 7i < • • • < 7^ (resp. JJLI < • • • < Hk) are k natural numbers same as we defined in (2.5 ). So Now, we assume T < a. Suppose fXk = "^Q^Oi, • • • ,ifc} = jh, where I < h < k. Because BTB C BaB{ By the definition of r < cr ), then jh < ih- Say 4 = 7;, thus y^fc - jh < ih = li, but 7, < 7/+1 < . . . < 7fc and /i/ < /i/+i < ... < fik, so m < ji,... and /^ fc < 7A;-Similarly, if /i/_i = jm, we have _;Vn < m^ = 7p because of BTB C -Ba"^. Therefore fJ-i-i = im < «m = 7p > but 7p < 7p+i < . . . < 7;_i and Hp < fXp+i < ... < yu;_i, so, IJ'P < Ipi • • • ilJ-i-i < 7;-i- Repeated applications of this procedure produce //i < 7 1 , . . . and fik < Ik for all A; = 1 . . . , n — 1. By lemma 3, we can conclude that ri(r) C VL{a). Conversely, suppose Jl(r) C Q.{u). By lemma 3, /i^ < 7fe for all A; = 1 . . . , n — 1. Say ^J'l - jhx and 4i = 7/_i where I <hi<k. Since //i < 71, so j/,j = //i < 71 < . . . < 7/_i = 4 j , that is, jh^ < ih,. Similarly, H2 = jhi < I2 < • • • < 7/-2 = ih2 gives us jh^ < ih2 with 1 < h2 < k. Continuing this, we have jh, < ih,,- • • and jh,, < ih^- Now choose k = n — 1, Chapter 2. Partitions and Schubert Varieties 34 we get j i < ii,... and jn-i < in-i- This implies that b*TB G BaB. So BTB C BaB. It follows that T <a. Q.E.D. Bibliography [A] E. Akyildiz, Bruhat decomposition via Gm action, Bull. Acad. Pol. Sci., Ser. Sci., Math., 28(1980), 541-547. [Al] E. Akyildiz, Gysin homomorphism and Schubert calculus, Pacific. J. Math. 115(1980), no.2. 257-266. [AG] G. Andrews, The Theory of Partitions, Addison-Wesley, Reading, MA, (1976). [BB] A. Bialynicki-Birula, Some Theorems on Actions of Algebraic Groups, Annals of Math, 98(1973), 480-497. [BG] I. N. Bernstein, I. Gel'fand and S. Gel'fand, Schubert Cells and Cohomology of the Spaces G/P, Russian Math. Surveys, 28(1973), 1-26. [CJl] J. Carrell, Holomorphic C* Actions and Vector Fields on Projective Varieties, Topics in the theory of algebraic groups, 1-37, Notre Dame Math. Lectures, 10, Univ. Notre Dame Press, South Bend, Ind.-London, 1982. [CJ2] J. Carrell, Vector Fields and the Cohomology of G/B, Manifolds and Lie Groups, Papers in honor of Y. Matsushima, Progress in Mathematics, Vol. 14, Birkhausen, Boston, 1981. [CS] J. Carrell and A. Sommese, Almost Homogeneous C* actions on compact complex surfaces. Group actions and vector fields (Vancouver, B.C. 1981) 29-33, Lecture Notes in Math. 956, Springer, Berlin-New York, 1982. [DP] C. DeConcini and C. Procesi, Symmetric Functions, Conjugacy Classes and the Flag Variety, Inventions Math, 64(1981), 203-219. [H] H. Hiller, Geometry of Coxeter Groups, Pitman, (1982). [HM] M. Hazewinkel and C. F. Martin, Representations of the Symmetric Group, the Specialization Order, Systems and Grassmann Manifolds, L'Enseignement Math., t, 19(1983), 53-87. [Hul] J. E. Humphreys, Linear Algebraic Groups, Springer-Verlag, Berlin-New York, (1975). [Hu2] J. E. Humphreys, Reflection Groups and Coxeter Group, Cambridge University Press (Cambridge Studies in Advanced Mathematics: 29), (1989). 35 Bibliography 36 [Hu3] J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, New York, (1970). [KL] S. Kleiman and D. Laksov, Schubert Calculus, Am. Math Monthly 79(1972), 1061-1082. [S] R. Steinberg, Conjugacy classes in algebraic groups, Lecture Notes in Math. 366. Springer-Verlag, Berlin, Heidelberg, New York (1974). [SR] R. Stanley, Weyl Groups, the Hard Lefschetz Theorem and the Spemer Property, SIAM J, Algebraic Discrete Methods 1(1980), 168-184. [ST] D, Stanton, Unimodality and Young's Lattice, Journal of Combinatorial Theory, Series A54(1990), 41-53. 

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