Schubert Varieties. P. Littelmann. May 21, 2012

Transcription

1 Schubert Varieties P. Littelmann May 21, 2012

2 Contents Preface 1 1 SMT for Graßmann varieties The Plücker embedding Monomials and tableaux Straightening relation Schubert Varieties in Gr d,n SMT for Schubert varieties in the Graßmannian Consequences SMT for flag varieties SMT for Schubert varieties in the flag variety A basis for H 0 (X, L) and vanishing of higher cohomology groups Ideal theory of Schubert varieties Lexicographic shellability and Cohen-Macaulayness References 55

3 2 Contents

4 1 Standard Monomial Theory for Graßmann varieties Standard Monomial theory (abbreviated as SMT) is the central theme of this book. We think that the classical case of Schubert varieties in the Graßmannian deserves special attention because the philosophy and the strategy are the same as in the general case, but the technical requirements used in the constructions and in the proofs are elementary. We shall see how one can develop SMT in this classical case by using just the Plücker embedding and the associated Plücker coordinates. It should be supportive to have this elementary case ready as a guideline for the later chapters. SMT consists in constructing explicit bases for the homogeneous coordinate rings of the Graßmann variety and its Schubert varieties. After introducing the Graßmannian and its Schubert varieties, we present the Standard Monomial Theory for Schubert varieties in the Graßmannian in Section 1.5. As an application we present a proof of the vanishing theorems for these Schubert varieties, where we also deduce their normality. Let us fix some notation that we shall use throughout this chapter. Let k be an algebraically closed field of arbitrary characteristic and set k = k \ {0}. Let V = k n and let e 1,..., e n be the standard basis of V. The group SL n (k) acts naturally on V. Let T be the subgroup of of SL n (k) consisting of diagonal matrices, and B the subgroup consisting of upper triangular matrices. We have B = T U = UT where U is the subgroup of B consisting of upper triangular unipotent matrices. Given a positive integer d, we denote by S d the symmetric group on d letters.

5 4 1. SMT for Graßmann varieties 1.1 The Plücker embedding Graßmann variety. Let us start with the most simple example of a Graßmann variety, the projective space P n 1. Recall that the projective space P n 1 is defined as the set of all lines in V. Another way to formulate the definition is to say that the projective space is the quotient (V \{0})/, where the equivalence relation is defined by: v v if there exists an element t k such that tv = v. The definition of a Graßmann variety is a straight forward generalization of the above, only one has to replace lines, i.e. 1-dimensional subspaces, by d-dimensional subspaces. Definition Let 1 d < n. The Graßmann variety Gr d,n is defined as the set of all d-dimensional subspaces in V. In particular, Gr 1,n = P n 1. To get a description of Gr d,n as a quotient similar to the description of the projective space above, let U Gr d,n be a d-dimensional subspace of k n. Fix a basis {v 1,..., v d } of U, then we can associate to U an n d matrix A = (a i,j ) of rank d such that the j-th column consists of the coefficients of v j with respect to the standard basis {e 1,..., e n } of V, i.e. v j = n i=1 a ije i. Vice versa, to an n d matrix A M n,d (k) of rank d one associates naturally the d-dimensional subspace U of V obtained as the span of the column vectors. In this language we can give a description of Gr d,n similar to that of the projective space above: let Z be the set of n d matrices of rank strictly less than d, then Gr d,n = (M n,d (k) \ Z)/, where the equivalence relation is defined by: A A if the column vectors span the same subspace of V. Above we defined the relation on V \ {0} in terms of the group action of k on V. Here we can do the same by using the fact that GL d (k) acts transitively on the set of bases of a d-dimensional subspace: Gr d,n = (M n,d (k) \ Z)/, where A A A = AC for some C GL d (k) For d = 1, this is exactly the description of the projective space P n 1 = Gr 1,n given above Gr d,n as homogeneous space. Another very useful description of the Graßmann variety is that of Gr d,n as a homogeneous space. If U V is a d-dimensional subspace and g SL n (k), then gu = {gu u U} is again a d-dimensional subspace. In fact, given U, U Gr d,n, there exists always a g SL n (k) such that gu = U. Denote by F j V the j-dimensional subspace F j = e 1, e 2..., e j spanned by the first j elements of the standard basis. Then we can identify Gr d,n with the coset space SL n (k)/p d, where P d is the isotropy group of

6 1.1 The Plücker embedding 5 the d-dimensional subspace F d. Now g SL n (k) is an element of P d if and only if ge j F d for 1 j d, and hence: Gr d,n = SL n (k)/p d, where P d = { A SL n (k) Note that the isotropy group P d contains B. ( A = 0 (n d) d )} Plücker coordinates. To endow the Graßmann variety with the structure of an algebraic variety, we will identify Gr d,n with a subset of the projective space P(Λ d V ). A first step in this direction is the introduction of Plücker coordinates, which can be viewed as linear functions on Λ d V as well as multilinear alternating functions on M n,d (k). The d-fold wedge product is alternating, the ordered products of elements in the canonical basis of V : e i1 e id, 1 i 1 < < i d n, form a basis of Λ d V, called the canonical basis of Λ d V. Definition Let I d,n := {i = (i 1,..., i d ) 1 i 1 < < i d n} be the set of all strictly increasing sequences of length d between 1 and n. For i = (i 1,..., i d ) I d,n we write e i = e i1 e id. We define a partial order on I d,n as follows: i j i t j t for all t = 1,..., d. So the canonical basis of Λ d V can be written as {e i i I d,n }. Denote by {p i i I d,n } the dual basis of (Λ d V ), i.e., p i (e j ) = δ i,j. Definition The linear functions p i, i I d,n, on Λ d V are called Plücker coordinates. By the definition of the d-fold wedge product the space of linear functions on Λ d V can be naturally identified with the space of multilinear alternating functions on d-copies of V, i.e., on M d,n (k) = V... V. }{{} d times Remark We use the same name Plücker coordinates and the same symbol p i for the linear functions on Λ d V as well as the corresponding multilinear alternating function on the space M n,d (k). To make this relationship more explicit, recall that we have a natural map, the exterior product map: π d : M n,d (k) Λ d V A = (v 1,..., v d ) v 1 v d. (1.1) Here v 1,..., v d are the column vectors of the matrix A. If we express the product v 1 v d as a linear combination of the elements of the canonical basis, then, by the definition of the dual basis, we have v 1 v d = i I d,n p i (π d (A))e i.

7 6 1. SMT for Graßmann varieties The alternating multilinear function on M n,d (k) associated to p i is just the i-th coordinate of the linear combination above, i.e., it is the composition p i π d. So by abuse of notation we write just p i (A) instead of p i (π d (A)). Denote by A i the d d submatrix of A consisting of the i 1 -th, i 2 -th,... and the i d -th row of A. It follows that: Lemma p i (A) is the determinant det A i of the submatrix A i of A Plücker embedding. Our next step is to identify the Graßmann variety with a subset of the projective space P(Λ d V ). For A M n,d (k) of rang d let v 1,..., v d k n be the column vectors, let U V be the span of these column vectors and let u 1,..., u d U. Denote by C = (c i,j ) the d d-matrix expressing the u j as linear combinations of the v i. i.e., u j = d i=1 c i,jv i. The exterior product is alternating, so we get v 1... v d = (det C)u 1... u d. As a consequence we see that the exterior product map induces a well defined map: π : Gr d,n = ((M n,d (k) \ Z)/ ) P(Λ d V ) called the Plücker embedding. We have a left action of SL n (k) on M n,d (k) defined by g(v 1,..., v d ) = (gv 1,..., gv d ), and we have a natural action of SL n (k) on Λ d V given by g(v 1 v d ) = (gv 1 ) (gv d ). It follows that the exterior product map π d : M n,d (k) Λ d V is equivariant with respect to these SL n (k)-actions, and hence so is the Plücker embedding. The term embedding is justified because: Proposition The Plücker map π : Gr d,n P(Λ d V ) is injective. Proof. Let F d be the d-dimensional subspace of V spanned by e 1,..., e d. By the homogeneity of the SL n (k)-action on Gr d,n, it is sufficient to show if π(u) = π(f d ), then U = F d. So suppose π(u) = π(f d ) and let {v 1,..., v d } be a basis of U. Denote by A M n,d (k) the corresponding matrix. Since [π d (A)] = [e 1... e d ], we can choose the basis such that π d (A) = e 1... e d. It follows that the submatrix A 1,...,d consisting of the first d-rows of A has determinant one, so by replacing A by A A 1 1,...,d if necessary we can (and will) assume that the submatrix of A consisting of the first d rows is the d d identity matrix. Now all d d minors except p 1,2,...,d (A) vanish. In particular, for i > d we have ±a i,j = p 1,...,j 1,j+1,...,d,i (A) = 0 and hence U = F d Again Plücker coordinates. In section we introduced the name Plücker coordinate for the dual basis p i of the canonical basis of Λ d V. To simplify the notation we use p i in the following for arbitrary d-tuples and not only for elements i I d,n.

8 1.1 The Plücker embedding 7 We give a description of the functions as alternating multilinear functions on the columns of M n,d (k) (instead of describing them as linear functions on Λ d V ). For 1 i 1,..., i d n (not necessarily distinct nor in increasing order) set i = (i 1,..., i d ). For an n d matrix A let A i be the d d matrix having as first row the i 1 -th row of A, as second row the i 2 -th row of A and so on. We set p i (A) = det A i. Clearly, p i = 0 if the i j s are not distinct, and if they are all distinct, then p i1,...,i d = sgn(σ)p σ(i1),...,σ(i d ) (1.2) where σ S d is such that (σ(i 1 ),..., σ(i d )) I d,n Alternating functions. In view of Proposition , we can identify Gr d,n with Im π. In general the image will not be all of P(Λ d V ), so the Plücker coordinates restricted to Gr d,n must satisfy some relations. By definition, the Plücker coordinates are i) linear functions on Λ d V as well as ii) multilinear alternating functions on the columns of M n,d (k) (the latter being identified with d-copies of V ). These functions are defined as determinants of maximal submatrices, so they have a third property: iii) the Plücker coordinate p i is a multilinear and alternating function in the i 1 -th, i 2 -th etc. row of M n,d (k). Suppose now i j =, then the product p i p j is a quadratic function on Λ d V which is definitely not anymore multilinear in the columns of M n,d (k). But this function is still multilinear in the i 1 -th, i 2 -th,..., j 1 -th, j 2 -th etc. row of M n,d (k), and alternating separately in the i k and j l. So if we alternate this function so that it becomes alternating in the rows say i 1,..., i d, j 1, then we have an alternating function on d + 1-copies of a d-dimensional vector space (the space of row vectors of M n,d (k)). Hence this function is zero on M n,d (k). Or, in other words, viewed as a quadratic function on Λ d V, we have a function such that the restriction to Im π d vanishes. Example Before starting with the formal approach consider the example Gr 2,4 and the product of Plücker coordinates p 1,2 p 3,4 k[λ 2 k 4 ]. The composition with π 2 : M 4,2 Λ 2 k 4 gives a function which is of course not anymore multilinear in the columns of M 4,2 (k), but which is still multilinear in the rows of this space of matrices. We will formally alternate this function. For example (we will see below why this is the alternated function) p 1,2 p 3,4 + p 2,3 p 1,4 p 2,4 p 1,3 (1.3) is a quadratic polynomial on Λ 2 k 4. The restriction to Im π 2 is a multilinear function on M 4,2 (k) which is alternating in the first, the third and the fourth row of M 4,2 (k). The only function with this property (i.e. being alternating on 3 copies of a 2-dimensional space) is the zero function, so the function above vanishes identically on Im π 2. But this means that the

9 8 1. SMT for Graßmann varieties restriction of the quadratic polynomial in (1.3) to Gr 2,4 is identically zero, and hence the Plücker coordinates satisfy on Gr 2,4 a quadratic relation. To formalize this idea, let us start with some generalities. We work inside the ring k[x i,j ] of polynomial functions on M n,d (k) and we write just x 1,..., x n for the vector variables corresponding to the rows of M n,d (k). Let f(x 1,..., x n ) be a multilinear function, then we can alternate it by setting: Alt(f) := σ S n sgn(σ)( σ f)(x 1,..., x n ), where σ f(x 1,..., x n ) = f(x σ 1 (1),..., x σ 1 (n)). Suppose n d+1. Instead of assuming that the function is multilinear in all vector variables, fix a subset M = {k 1,..., k d+1 }, 1 k 1... k d+1 n, of pairwise different indices, and assume the function is multilinear in the rows corresponding to the indices k 1,..., k d+1. The function Alt M (f) := sgn(σ)f(..., x kσ 1 (1),..., x kσ 1 (d+1),...) σ S d+1 (i.e. all vector variables different from x k1,..., x kd+1 are not changed) is alternating and multilinear in x k1,..., x kd+1. For 1 t < d + 1 let M = M 1 M 2 be a disjoint decomposition such that M 1 = t. If f is alternating separately in the variables {x k k M 1 } and {x l l M 2 }, then sgn(σ)f(..., x kσ 1 (1),..., x kσ 1 (d+1),...) = sgn(σ )f(..., x kσ 1 (1),..., x kσ 1 (d+1),...) whenever σ and σ are in the same coset in S d+1 /S t S d+1 t. Here we identify the subgroup S t S d+1 t with the subgroup of permutations in S d+1 which separately permute only the elements in M 1 and M 2 among themselves. So to get an alternating function one has to take the sum Alt M1,M 2 (f) := sgn(σ)f(..., x kσ 1 (1),..., x kσ 1 (d+1),...) σ S d+1 /S t S d+1 t only over a system of representatives of the cosets. Example Suppose n = 4 and d = 2. Let f(x 1, x 2, x 3, x 4 ) = p 1,2 p 3,4 be the product of these two Plücker coordinates, then f is a multilinear function on M 4,2 (k), alternating separately in the 1st and 2nd and the 3rd and 4th row. Set M 1 = {1}, M 2 = {3, 4} and M = M 1 M 2, and denote by S M respectively S Mi the permutation groups of the sets. Then ( ) 134 id =, σ = ( ), σ 2 = ( ),

10 1.1 The Plücker embedding 9 is a set of representatives of S M /S M1 S M2 (see section for a procedure to get the representatives) and Alt M1,M 2 (f) = f + sgn(σ 1 )( σ1 f) + sgn(σ 2 )( σ2 f) = f(x 1, x 2, x 3, x 4 ) f(x 3, x 2, x 1, x 4 ) + f(x 4, x 2, x 1, x 3 ) = p 1,2 p 3,4 + p 2,3 p 1,4 p 2,4 p 1,3 is the function on M 4,2 (k) in equation 1.3, which is alternating in the 1st, 3rd and 4th row Quadratic relations. A product f = p i p j of Plücker coordinates is a quadratic polynomial on Λ d V. Suppose now all indices i k, j l are different. The product is a function on M n,d (k) which is multilinear with respect to the rows of this space of matrices. Fix 1 t < d, then f is, by construction, alternating separately in the (row) vector variables x i1,..., x it and x jt,..., x jd. Given σ S d+1, note that σ shuffles the indicees i 1,..., i t and j t,..., j d. Denote by i σ and j σ the d-tuples (σ 1 (i 1 ),..., σ 1 (i t ), i t+1,..., i d ) and (j 1,..., j t 1, σ 1 (j t ),..., σ 1 (j d )). Recall that the function sgn(σ)( σ f), σ S d+1 /S t S d+1 t, is independent of the choice of a representative for σ. The function we get by alternating f = p i p j is: Alt {i1,...,i t},{j t,...,j d }(p i p j ) = sgn(σ)p i σp j σ. σ S d+1 /S t S d+1 t Lemma Suppose n 2d. Let i, j be two d-tuples, 1 i k, j l n, such that the entries are all distinct. Fix 1 t < d, the homogeneous polynomial Alt {i1,...,i t},{j t,...,j d }(p i p j ) k[λ d V ] vanishes on Gr d,n P[Λ d V ]. Proof. By composing the function with the exterior product map, we see that the quadratic polynomial vanishes on Gr d,n if and only if, viewed as a sum of products of minors, the function vanishes on M n,d (k). But this function is multilinear and alternating in the d + 1 row vector variables x i1,..., x it, x jt,..., x jd. The space of the row vectors is of dimension d, so this function vanishes on M n,d (k). To weaken the condition that all indices have to be different, consider two arbitrary d-tuples i and j, 1 i k, j l n. We will now define a new pair i, j such that all entries are different. Set i k = i k + mn where m = {l l < k, i k = i l } j k = j k + mn where m = {l j k = i l } + {l l < k, j k = j l } For example, suppose i 1, i 2, j 1, j 2 are pairwise different, then this procedure applied to the pair i = (i 1, i 2, i 1, i 1, i 2 ) j = (j 1, i 2, i 1, j 1, j 2 ) i = (i 1, i 2, i 1 + n, i 1 + 2n, i 2 + n) j = (j 1, i 2 + 2n, i 1 + 3n, j 1 + n, j 2 )

11 10 1. SMT for Graßmann varieties provides a new pair (i, j ) such that all entries are different. So we can formally define the quadratic polynomial (in a larger ring with more vector variables) Alt {i 1,...,i t },{j t,...,j d } (p i p j ). (1.4) We define the polynomial (which is either zero or a quadratic polynomial) Alt (i1,...,i t),(j t,...,j d )(p i p j ) now as the function obtained from (1.4) by replacing in the Plücker coordinates all indices i k, j l by the original indices, i.e. all indices i k, j l > n are replaced by i k (mod n) respectively j l (mod n). Example Suppose n = 5, d = 3 and i = (2, 1, 5) and j = (1, 3, 4). Then i = i and j = (6, 3, 4). For t = 1 we have M 1 = {2}, M 2 = {6, 3, 4} and M = M 1 M 2. For the permutation groups we have S M1 S 1, S M2 S 3, S M S 4 and S M /(S M1 S M2 ) S 4 /(S 1 S 3 ). Denote by s 1, s 2, s 3 the simple transpositions of S 4. The elements id, s 1, s 2 s 1, s 3 s 2 s 1 form a system of representatives for the cosets in S 4 /S 1 S 3 and we get Alt {2},{6,3,4} (p i p j ) = p 2,1,5 p 6,3,4 p 6,1,5 p 2,3,4 + p 3,1,5 p 2,6,4 p 4,1,5 p 2,6,3 After specializing (i.e. replacing 6 back by 1) we get Alt (2),(1,3,4) (p i p j ) = p 2,1,5 p 1,3,4 p 1,1,5 p 2,3,4 + p 3,1,5 p 2,1,4 p 4,1,5 p 2,1,3 = p 1,2,5 p 1,3,4 + p 1,3,5 p 1,2,4 p 1,4,5 p 1,2,3 Theorem Let i and j, 1 i k, j l n, be two arbitrary d-tuples. For all 1 t < d, the polynomial Alt (i1,...,i t),(j t...,j d )(p i p j ) vanishes on the Graßmann variety Gr d,n. Proof. Suppose the polynomial is different from zero. As above, by composing the function with the exterior product map, one sees that this quadratic polynomial vanishes on Gr d,n if and only if, viewed as a sum of products of minors, the function vanishes on M n,d (k). If the entries in i and j are all different, then this is Lemma Otherwise consider first the multilinear function Alt (i 1,...,i t ),(j t...,j d ) (p i p j ) defined in (1.4), this function is defined on the space M 2dn,d (k) of 2dn d matrices, and vanishes identically since it is multilinear and alternating in d + 1 of the vector variables. The original space M n,d (k) can be seen as a subspace of M 2dn,d (k) by identifying a n d-matrix A with the 2dn d-matrix obtained by putting 2d copies of A on the top of each other. By construction we have then Alt (i1,...,i t),(j t...,j d )(p i p j ) = Alt (i 1,...,i t ),(j t...,j d ) (p i p j ) Mn,d (k) 0.

12 1.1 The Plücker embedding 11 Definition If Alt (i1,...,i t),(j t...,j d )(p i p j ) is not the zero polynomial in k[λ d V ], then this quadratic polynomial is called a shuffle relation or a Plücker relation Shuffles. We will describe how to obtain shuffles or coset representatives. Fix 1 t d, we want to describe a special set of coset representatives of S d+1 /S t S d+1 t. Let S d+1 act on the set {1,..., d + 1}. Then a coset σ S d+1 /S t S d+1 t is identified by the relative position of the first t and the second d + 1 t elements. Expressed in a pictorial way: suppose we are given a configuration of t-balls and d + 1 t triangles:... If we fill the balls with any permutation of {1,..., t} and the triangles with any permutation of {t + 1,..., d + 1}, we always get a permutation which is an element of the same coset. A canonical representative of such a coset is hence obtained by putting 1, 2,..., t in order in the balls and t + 1,..., d + 1 in order in the triangles. Such a representative is called a t-shuffle. Example To determine the set of all 2-shuffles in S 4 consider first the set of all configuration of 2-balls and 2 triangles:,,,,,. The 2-shuffles and the decomposition of the inverse are given by: ( ) ( ) ( ) ( ) ( ) ( ) σ =,,,,, σ 1 = id, s 2, s 2 s 1, s 2 s 3, s 2 s 1 s 3, s 2 s 1 s 3 s 2 Example Let n = 5, d = 3, i = (2, 3, 4), j = (1, 4, 5), t = 2. By the example above we have Alt (2,3)(4,5) p i p j = p 2,3,4 p 1,4,5 p 2,4,4 p 1,3,5 + p 3,4,4 p 1,2,5 +p 2,5,4 p 1,3,4 p 3,5,4 p 1,2,4 + p 4,5,4 p 1,2,3 = p 2,3,4 p 1,4,5 p 2,4,5 p 1,3,4 + p 3,4,5 p 1,2, Closed embedding. Next we will see that one can identify Gr d,n with is a closed subset of P(Λ d V ), i.e. the Graßmann variety is naturally endowed with the structure of a projective variety. Theorem The Graßmann variety Gr d,n P(Λ d V ) is the zero set of the homogeneous ideal generated by the following polynomials: d+1 ( 1) l p i1,...,î l,...,i d+1 p j1,...,j d 1,i l, (1.5) l=1 where i 1,..., i d+1 and j 1,..., j d 1 are any numbers between 1 and n.

13 12 1. SMT for Graßmann varieties Proof. The relation in (1.5) is a special case of the shuffle relations (see Theorem , t = d), so Gr d,n is contained in the zero set of the homogeneous ideal generated by these equations. Conversely, let y = [ i I d,n y i e i ] satisfy the equations in (1.5). Suppose y l1,...,l d 0 for some l = (l 1,..., l d ) I d,n, without loss of generality we may (and will) assume y l1,...,l d = 1. For 1 i n, 1 j d, set a ij = y l1,...,l j 1,i,l j+1,...,l d. We apply the usual rules as in (1.2): y l1,...,l j 1,i,l j+1,...,l d is zero if two indices are equal etc. Let A be the n d matrix A = (a ij ). By construction A l1,...,l d = I d because a lj,j = y l1,...,l d = 1 for j = 1,..., d and for i j we have a lj,i = y l1,...,l i 1,l j,l i+1,...,l d = 0. Clearly rank A = d, let U be the d-dimensional subspace spanned by the columns of A. We have to show that π(u) = [ i I d,n p i (A)e i ] = [ i I d,n y i e i ] = y and hence y Gr d,n. For two d-tuples k, k denote by {k k } the number of common entries. We will show p j (A) = y j by decreasing induction on {l j}. We know already that p l (A) = 1 = y l. For j = (l 1,..., l j 1, i, l j+1,..., l d ) we have p j (A) = a i,j = y j by the definition of A, so this proves the claim if {l j} d 1. Let j be arbitrary such that {l j} < d 1. There exists an entry in j which is not an entry in l. Without loss of generality (i.e., after permuting the entries if necessary) we assume that j d has this property. Now y satisfies all the relations in (1.5), so the coordinates y l and y j satisfy a relation of the form above: y l y j + ±y l y j = 0, where l differs from l in just one place. Further, if y l y j 0, then {j l} > {j l} since j d has been replaced by an element in l. Thus we know by induction y l = p l (A), y j = p j (A). By Theorem , the d-minors of A satisfy the relations in (1.5), so p l (A)p j (A) + ±p l (A)p j (A) = 0. Now p l (A) = y l = 1, so p j (A) = ±p l (A)p j (A) = ±y l y j = y j. 1.2 Monomials and tableaux Let I(Gr d,n ) = {f k[λ d V ] f Grd,n 0} be the homogeneous vanishing ideal of the Grassmann variety and denote by k[gr d,n ] = k[λ d V ]/I(Gr d,n ) the homogeneous coordinate ring of this projective variety. The homogeneous coordinate ring of the projective space P(Λ d V ) is the polynomial ring k[λ d V ] = k[p i i I d,n ]. This ring has a k-basis the monomials in the Plücker coordinates. Our aim is to find a subset of special monomials,

14 1.2 Monomials and tableaux 13 the standard monomials, such that the classes of these monomials form a k-basis of k[gr d,n ]. We prepare in this section the necessary combinatorial background. To have a normal form for the monomials in k[λ d V ], we fix a total order on the set I d,n : i j if and only if 1 t d : i 1 = j 1,..., i t 1 = j t 1, i t > j t. It follows that the ordered or weakly standard monomials p i p j p k such that i j... k form a k-basis for the ring k[λ d V ]. Note that i j (see Definition 1.1.5) implies i j. Definition Let p i p k be a weakly standard monomial (i.e. i... k) in the Plücker coordinates. The monomial is called a standard monomial if i... k. It is often convenient to use the language of Young tableaux for tuples of elements in I d,n. Expressed in a pictorial way, a tableau is a filling of a Young diagram. Definition A Young diagram of shape m d = (m,..., m) is a sequence of d left adjusted rows of boxes, each row of length m, and a Young tableau is a filling of the boxes with the numbers {1,..., n}. Such a filling is called column standard if the entries in the columns are strictly increasing (top to bottom). Unless stated otherwise, all tableaux we consider are column standard. So by abuse of notation we write in the following just Young tableau for a column standard Young tableau. Let T be a Young tableau of shape m d. The entries in each column, read from top to the bottom, define an element in I d,n. So we can view a tableau T of shape m d also as an m-tuple of elements in I d,n. Since the world of combinatorics and the world of commutative algebra do not always live in harmony, note a slight slip in the notation. Tuples are to be read from the left to the right, filling of diagrams are to be read from the right to the left. So the tuple associated to the Young tableau of shape 6 2 : T = (1.6) is ((4, 5), (3, 5), (3, 4), (2, 4), (1, 5), (1, 2)). The monomial p T associated to a tableau is the product of the Plücker coordinates corresponding to the columns of the tableau. For the example above we have p T = p 4,5 p 3,5 p 3,4 p 2,4 p 1,5 p 1,2.

15 14 1. SMT for Graßmann varieties Definition Let T be a Young tableau of shape m d and let (i, j,..., k) be the corresponding m-tuple of elements in I d,n. The tableau is called weakly standard if i j... k, and the tableau is called semi-standard if i j... k. In terms of the entries of the diagram, this means the tableau is semi-standard if and only if the entries in the boxes are strictly increasing in the columns and weakly increasing (left to right) in the rows. In this language we see that {p T T Young tableau, shape m d, weakly standard} is the set of all weakly standard monomials of degree m, and {p T T Young tableau, shape m d, semi-standard} is the set of all semi-standard monomials of degree m. Remark The odd looking correspondence of names semi-standard tableaux standard monomials comes from the fact that the name standard tableau is already reserved for a class of Young tableau occuring in the representation theory of the symmetric group. We denote by l the induced homogeneous lexicographic ordering on the weakly standard tableaux, i.e. if T = (i 1, i 2,..., i r ) and T = (j 1, j 2,..., j s ) are weakly standard tableaux, then we say T l T if r > s or if r = s, and i k j k for the first index k such that i k j k. We define an induced ordering on the weakly standard monomials by p T l p T if T l T. Note that l defines a monomial ordering on k[λ d k n ], i.e. if p T l p T, then we have for weakly standard tableaux T, T, T : p T p T l p T p T l p T. 1.3 Straightening relation Our aim is to show that the standard monomials form a basis of k[gr d,n ] = k[λ d V ]/I(Gr d,n ). As a first step we show that the Plücker relations imply straightening relations, i.e. provide an algorithm to straighten out nonstandard monomials as a sum of standard monomials. Proposition The images of the standard monomials in k[gr d,n ] span the homogeneous coordinate ring. The proof uses the Plücker relations described in Theorem We start with some preliminaries. Let i, j I d,n be such that i j but i j, then there exists 1 t d such that i r j r, 1 r t 1, and i t < j t. Consider the shuffle relation Alt (i1,...,i t),(j t...,j d )(p i p j ) = sgn(σ)p i σp j σ. (1.7) σ S d+1 /S t S d+1 t

16 1.3 Straightening relation 15 Denote by i σ the tuple obtained from i σ by writing the entries in ascending order. Lemma If σ id, then i σ i, j j σ. Proof. If σ id, then the shuffle replaces some elements in {i 1,..., i t } by some elements in {j t,..., j d }. Now by the choice of t we know j t,..., j d > i 1,..., i t. So after reordering the elements we get for i σ = (i 1,..., i d ): i 1 i 1,..., i t i t, and for at least one 1 l t we have i l > i l. It follows: i σ i j. The proof for j σ is similar. Proposition (Straightening relation). If p i p j is weakly standard but not standard, then p i p j i >j i i,j j a j,i p i p j mod I(Gr d,n) (1.8) In other words: the image of p i p j in k[gr d,n ] is a linear combination of the images of standard monomials. A relation as in (1.8) is called a straightening relation. Proof. If the product is weakly standard but not standard, then let 1 t d be as above. The polynomial Alt (i1,...,i t),(j t...,j d )(p i p j ) vanishes on Gr d,n, so, modulo I(Gr d,n ), by (1.7) we can write p i p j as a linear combination of products p i σ p j σ, σ id. These products are again weakly standard by Lemma 1.3.2, and T = (i σ, j σ ) l T = (i, j). If one of the T is not semi-standard, then we can repeat the straightening procedure and express p T (modulo I(Gr d,n )) as a linear combination of p T, where T l T l T. The number of weakly standard tableaux of shape 2 d is finite, so this process has to end after a finite number of steps. As a consequence we see that we can express p T = p i p j as linear combination of standard monomials p T (modulo I(Gr d,n )) such that i i, j j for T = (i, j ). Proof of Proposition The homogeneous coordinate ring is spanned by the classes of the weakly standard monomials. If a monomial p T is weakly standard but not standard, then we can find a pair of columns in the corresponding tableau satisfying the conditions of Proposition Since we have a monomial order, it follows that, modulo I(Gr d,n ), we can rewrite the monomial as a linear combination of monomials p T where T and T are of the same shape and T T. If one of the T is not standard, then we can repeat the procedure. But the number of weakly standard Young tableaux of a fixed shape is finite, so the algorithm has to end and expresses p T, modulo I(Gr d,n ), as a linear combination of standard monomials, which finishes the proof.

17 16 1. SMT for Graßmann varieties 1.4 Schubert Varieties in Gr d,n For 1 t n, let F t be the standard t-dimensional subspace of V spanned by {e 1,..., e t }. We denote by F the complete flag of subspaces F : F 0 = 0 F 1 F 2... F n 1 F n = V. For a d-dimensional subspace W Gr d,n of V consider the intersections of the subspace with the flag: 0 (F 1 W ) (F 2 W )... (F n 1 W ) W = W F n. The tuple (i 1,..., i d ) given by the indices where the dimension jumps (i.e. dim(f ij W ) = j = dim(f ij 1 W ) + 1) is an element in I d,n. For i I d,n let C i be defined by C i = { W Gr d,n 1 t d : } dim(w F it ) = t and. dim(w F l ) < t for all l < i t Definition The subset C i Gr d,n is called the Schubert cell C i associated to i. The Graßmann variety is the disjoint union of the Schubert cells. The Schubert variety associated to i is defined to be X i = {W Gr d,n dim(w F it ) t, 1 t d}. We have obviously C i X i. As in the case of the Graßmannian, there are other descriptions of Schubert varieties and Schubert cells. Let M i M n,d (k) be the set of matrices A = (a i,i ) such that a i,j = 0 for i > i j and a ik,l = δ k,l (so the submatrix A i is the d d identity matrix). Example For n = 8, d = 4 and i = (2, 3, 5, 7) the matrices in M i M n,d (k) are of the form Lemma The exterior product map π d : (v 1,..., v d ) [v 1... v d ] induces a bijection π d : M i C i.

18 1.4 Schubert Varieties in Gr d,n 17 Proof. Let W C i. By choosing a basis {v 1 } W F i1, {v 1, v 2 } W F i2 etc., we can fix a basis {v 1,..., v d } of W such that v k = e ik + l<i k a l,k e l, denote A = (a p,q) the corresponding matrix. The submatrix A i is upper triangular and unipotent. By replacing A by A = A (A i ) 1 we get a matrix in M i such that the columns span W. On the other hand, a subspace spanned by the column vectors of a matrix in M i satisfies the intersection conditions defining C i, so the map π d : M i C i, (v 1,..., v d ) [v 1... v d ], is well defined and surjective. The coefficients a i,j of the matrix A can be obtained (see also proof of Theorem ) back from W = [v 1... v d ]: a i,j = p i (W )/p i (W ), where i = (i 1,..., i j 1, i, i j+1,..., i d ), so the map is bijective. The subspaces F t are stable under the subgroups B and U, and hence X i as well as C i are stable under the induced action of U, T and B = T U = UT. The points [ e i ] can be characterized as follows: the T -fixed points in P(Λ d V ) correspond to the T -stable lines in Λ d V. Now as T -module we have Λ d V i I d,n ke i, and every one-dimensional T -submodule is of the form ke i for some i. So the only T -fixed points in P(Λ d V ) are the [e i ] Gr d,n. Lemma C i = B [e i ] = U [ ei ], and Grd,n = i I d,n B [e i ]. Proof. For W C i let A M i be the corresponding matrix. By standard linear algebra arguments, we can find an upper triangular unipotent matrix u U such that ua = E i = (e p,q ) is the matrix with entries e ik,k = 1 for k = 1,..., d, and all other entries equal to zero. Since the exterior product map is equivariant with respect to the SL n (k)-action, this means u.w = [ e i ]. It follows that Ci = U [e i ], and, since [ ei ] is a T -fixed point, C i = B [e i ]. The rest is immediate. A Schubert variety is in fact a projective variety, for example it can be seen as the intersection of Gr d,n with a finite number of hyperplanes in P(Λ d V ): Lemma X i = {W Gr d,n p j (W ) = 0 j I d,n such that j i}, C j X i if and only if j i, and C j X i = if and only if j i. Proof. Let W C j, by definition dim(w F jt ) = t and dim(w F l ) < t for l < j t. By the definition of X i it follows that W X i if and only if j 1 i 1, j 2 i 2, etc., or, in other words, i j. Hence: C j X i if and only if j i, and the intersection C j X i is empty otherwise. Suppose now W C j X i, let A M j be the corresponding matrix and let v 1,..., v d be the column vectors. Since v k = e jk + l<j k a l,k e l, we have in P(Λ d V ) W = [v 1... v d ] = [e j + j <j a j e j ]

19 18 1. SMT for Graßmann varieties In particular, p k (W ) = 0 for all k j. Next suppose W Gr d,n is such that p k (W ) = 0 for all k i. Now W C j for some j I d,n. Since p j (e j ) 0 by the above, we have j i and hence W C j X i. Proposition The Schubert variety X i is the Zariski closure of the Schubert cell C i, and X j X i if and only if j i. Proof. The second part follows from the first and Lemma Suppose first that j < i is such that j k = i k 1 for some k and all other entries are equal. Let E i,j be the matrix having 1 as entry in the i-th row and j-th column, and having entry zero everywhere else. Let I be the identity matrix, then (I + te ik 1,i k ).[e i ] = [e i + te j ] = [t 1 e i + e j ] C i for all t k. It follows: lim t [t 1 e i + e j ] = [e j ] and hence [e j ] C i. The cell is B-stable and hence so is the Zariski closure. It follows C j C i. Let now i > j be arbitrary and let k be minimal such that i k > j k. Let i I d,n be obtained from i by replacing the entry i k by i k 1, then i > i j. Since C i C i, to prove C j C i it suffices to prove C j C i. But this follows now by decreasing induction (repeating the procedure above). It follows j i C j C i X i, and hence C i = X i by Lemma Proposition C i is an affine cell of dimension d(i) = 1 t d i t t and X i is an irreducible projective variety of dimension d(i). Proof. The Schubert variety is the Zariski closure of the Schubert cell, so the first part implies the second. Let P(Λ d V ) pi be the affine open set P(Λ d V ) pi = {[w] P(Λ d V ) p i ([w]) 0}, then C i is the affine variety X i P(Λ d V ) pi. The map π d : M i C i and its inverse defined in the proof of Lemma are morphisms of affine varieties, and hence define an isomorphism between C i and the affine space M i of dimension d(i) = 1 t d i t t. Lemma Let X i, X j be two Schubert varieties in Gr d,n. Then X i X j is irreducible, i.e. X i X j is a Schubert variety (set-theoretically). Proof. The intersection is closed and B-stable, so it has to be a union of Schubert varieties. Set k l = min{i l, j l }, 1 l d, and k = (k 1,..., k d ). Then k I d,n, and if X k X i X j, then k i, j and hence k k. It follows that X i X j = X k. Lemma Let X i, X j be two distinct Schubert divisors in a Schubert variety X k in Gr d,n, i.e. X i, X j are Schubert varieties of codimension 1 in X k. Then the intersection is a Schubert variety of codimension 2 in X k.

20 1.5 SMT for Schubert varieties in the Graßmannian 19 Proof. By Lemma we know that the intersection is the Schubert variety X k, where k l = min{i l, j l }. Since k > i, by the dimension fomula in Proposition we know that k and i are the same but for one place where the entry in i is one less than the corresponding entry in k, and the same holds for j. It follows that k and k are the same but for two places where the entries in k are one less than the corresponding entries in k, and hence X k is of codimension 2 in X k. 1.5 SMT for Schubert varieties in the Graßmannian Let R = k[gr d,n ] = k[λ d V ]/I(Gr d,n ) be the homogeneous coordinate ring of Gr d,n for the Plücker embedding with the induced grading R = m N R m. More generally, for i I d,n, let R(i) = k[x i ] be the homogeneous coordinate ring of the Schubert variety X i in Gr d,n. In this section, we present a Standard Monomial Theory for X i and as a consequence, we obtain a basis for the graded components R(i) m, m Z +. Recall that Gr d,n = X [n d+1,...,n], so this includes the case of the Graßmann variety. Definition A monomial p T = p i 1 p i m is said to be standard on a Schubert variety X i, if the monomial is standard (or, equivalently, the tableau T is semi-standard) and, in addition, i i 1 (so i i 1... i m ). Remark By Lemma we know that p j vanishes identically on X i if and only if j i. So for a standard monomial p T (on Gr d,n ) we conclude p T is standard on X i p T Xi is not identically zero Theorem The standard monomials on X i form a basis of the homogeneous coordinate ring R(i). Proof. By Proposition 1.3.1, we know that the standard monomials span R. Since the standard monomials, not standard on X i, vanish identically on X i, it follows that the standard monomials, standard on X i, span the homogeneous coordinate ring R(i). It remains to prove the linear independence. We proceed by induction on dim X i, the case i = (1, 2,..., d) (i.e. dim X i = 0) being obvious because k[x i ] = k[p (1,2,...,d) ] is isomorphic to the polynomial ring in one variable. Suppose now dim X i > 0 and let r c Tl p Tl 0, c Tl k, (1.9) l=1 be a linear relation of standard monomials p Tl of degree m. If m = 1, then let T 1 = (j 1 ),..., T r = (j r ). Note that j s i for all s = 1,..., r. Without

21 20 1. SMT for Graßmann varieties loss of generality let j 1 be a minimal element such that c j 1 0. But then p j t([e j 1]) = 0 for all t 2 and hence the sum in (1.9) does not vanish in [e j 1], which is a contradiction. Suppose now m > 1 and assume there exists an l such that T l = (j 1,..., j d ) is standard and j 1 i. Now note: a) the restriction of the standard monomials p Tk occuring in this sum which are not standard on X j1, they vanish identically on X j1, and b) at least p Tl is standard on X j1. So the restriction of (1.9) to X j1 is a non-trivial linear dependence relation between standard monomials, standard on X j1. By induction on the dimension, this is impossible. So for all l = 1,..., r, p Tl is of the form p i p T l, where p T l is a standard monomial, standard on X i. Now X i is an irreducible variety, so r ( r ) c Tl p Tl 0 p i c Tl p T l 0 l=1 l=1 r l=1 c Tl p T l 0 Now the latter is a homogeneous linear dependence relation of lesser degree, and hence by induction c Tl = 0 for all l. As a consequence of Theorem 1.5.3, we have a qualitative description of a typical quadratic relation on a Schubert variety X i, which is a stronger version of Proposition 1.3.3: Proposition Let k, i, j I d,n, k > i, j. We assume that i j but i j, so p i p j is a monomial of degree 2 on X k but not standard. Let p i p j = a,b I d,n k a b c a,b p a p b, c a,b k (1.10) be the expression for p i p j as a sum of standard monomials on X k. Then for every a, b on the right hand side, we have a > i, j > b. Proof. Among all the a s occuring in (1.10) choose a minimal one, call it a 0. Restricting (1.10) to X a0, the restriction of the right hand side is a non-zero sum of standard monomials on X a0. Note that the restriction of p a p b to X a0 is (by the minimality of a 0 ) non-zero if and only if a = a 0 ; and there is at least one term, namely p a0 p b whose restriction to X a0 is nonzero. Hence in view of the linear independence of standard monomials on X a0, we obtain that the restriction of the left hand side to X a0 is non-zero. From this we conclude that a 0 both i and j. In fact, a 0 > both i and j for if a 0 equals one of them, say a 0 = i, then p i p j would vanish on X a0 because j and i = a 0 are not comparable. But this is not possible. Among all the b s choose a maximal one, call it b 0. For c = (c 1,..., c d ) I d,n set c = (n + 1 c d,..., n + 1 c 1 ), then c I d,n and induces an

22 1.5 SMT for Schubert varieties in the Graßmannian 21 order reversing involution on I d,n. Let w 0 be the permutation exchanging i and n + 1 i for all i = 1,..., n. Denote by n w0 the corresponding permutation matrix, then n w0 (p c ) = cp c for some c k. So by applying n w0 to (1.10), we conclude from the above that b 0 > both i and j, i.e. b 0 < both i and j Equations defining Schubert varieties in the Graßmannian. As a first consequence of standard monomial theory we give a description of the vanishing ideals of the Graßmann variety and its Schubert varieties. Proposition The homogeneous vanishing ideal I(Gr d,n ) k[λ d (V )] of the embedded Graßmann variety Gr d,n P(Λ d V ) is the homogeneous ideal generated by the Plücker relations (see Definition ). 2. For i I d,n, the vanishing ideal I i k[gr d,n ] of the Schubert variety X i Gr d,n is the ideal generated by the Plücker coordinates p j such that j i. 3. The standard monomials p T, T = (i 1,..., i m ), such that i 1 i, form a basis of the kernel of the restriction map k[gr d,n ] k[x i ]. Proof. Let I be the ideal generated by the Plücker relations. By Proposition 1.3.3, we can write a homogeneous function f as a linear combination of standard monomials of the same degree plus an element of I. The linear independence of the standard monomials (Theorem 1.5.3) implies: f vanishes on Gr d,n if and only f I. For i I d,n let B i the set of standard monomials, standard on X i and let K i be the set of standard monomials, not standard on X i. Let f k[gr d,n ], then we can write f as a linear combination of standard monomials. We can break up this sum into two parts f = k i +b i, where k i is a linear combination of the elements in K i and b i is a linear combination of elements in B i. By Theorem 1.5.3, f is in the kernel of the restriction map k[gr d,n ] k[x i ] if and only if b i = 0, which proves part 3). Part 2) of the proposition is an immediate consequence of this. Remark Part 2) of the proposition above implies that X i is schemetheoretically (even at the cone level) the intersection of Gr d,n with all hyperplanes in P(Λ d V ) containing X i. Further, as a closed subvariety of Gr d,n, X i is defined scheme-theoretically by the vanishing of the p j s, j i. We now extend the results above to unions of Schubert varieties. Let i 1,..., i s I d,n and let X = l=1,...,s X i be a union of Schubert varieties. l Definition A monomial p T in the Plücker coordinates is called standard on the union X of Schubert varieties if it is standard on at least one of the Schubert varieties X i 1,..., X i s.

23 22 1. SMT for Graßmann varieties Theorem Let R X be the homogeneous coordinate ring of the union of Schubert varieties X Gr d,n. Then the (images of the) set of standard monomials, standard on X, forms a basis for R X. Proof. Let B X be the set of standard monomials, standard on X and let K X be the set of standard monomials, not standard on X. Let f k[gr d,n ], then we can write f as a linear combination of standard monomials. We can break up this sum into two parts f = k X + b X, where k X is a linear combination of the elements in K X and b X is a linear combination of elements in B X. The elements in K X vanish identically on X, so we see that R X is spanned by the (images of the) elements in B X. To prove the linear independence, suppose one has a linear dependence relation between the standard monomials, standard on X. The restriction of this relation to one of the Schubert varieties X i l gives a linear dependence relation for those summands in this linear dependence relation which are standard on X i l. But by Theorem 1.5.3, this is impossible, so non of the summands is standard on X i l. This holds for all l = 1,..., s, so the standard monomials, standard on X, are linearly independent. 1.6 Consequences Theorem Let X 1, X 2 be unions of Schubert varieties in Gr d,n. Then the scheme-theoretic union X 1 X 2 and the scheme-theoretic intersection X 1 X 2 are reduced. Proof. For a closed subscheme Y in Gr d,n, let I(Y ) denote the ideal defining Y in Gr d,n. We have I(X 1 X 2 ) = I(X 1 ) I(X 2 ), and hence I(X 1 X 2 ) is reduced since I(X 1 ), I(X 2 ) are reduced. Let X be the set theoretic intersection of X 1 and X 2, then X is a union of Schubert varieties. By Theorem we know that the vanishing ideal I k[gr d,n ] of X has as basis all standard monomials not standard on X. Similar, let I j, j = 1, 2 be the vanishing ideal of X j, then I j has as basis all standard monomials not standard on X j. But then I 1 + I 2 has as basis all standard monomials which are not standard on either X 1 or X 2, in other words, which are not standard on X. But this implies I = I 1 + I 2, and hence X is also the scheme theoretic intersection of the two. By a Schubert divisor in X i we mean a Schubert variety X j X i of codimension one. The union X i = r l=1 X jl of all the Schubert divisors in X i is called the border of X i. Theorem (Pieri s formula) We have the following scheme-theoretic equality X i {p i = 0} = X i.

24 1.6 Consequences 23 Proof. Let R be the homogeneous coordinate ring of X i and denote by I k[x i ] the vanishing ideal of X i, so R = k[x i ]/I. Clearly, the principle ideal (p i ) R Xi generated by p i is contained in I. Let f I, then we can write f as f = a k p Tk + b j p T j where a k, b j k, and in the first sum we have standard monomials p Tk in k[x i ] starting with p i, while in the second sum we have standard monomials p T j in k[x i ] starting with p j, where j < i. Obviously, the first sum is an element in (p i ) I, and hence also the second sum b j p T j vanishes identically on X i. But the p T j are all standard on X i by construction, so b j = 0 for all j and f (p i ) Vanishing theorem and basis for cohomology. Denote by L the line bundle on Gr d,n obtained as the restriction of O P(Λ d V ) to Gr d,n P(Λ d V ). Denote by S(Y, m) the set of standard monomials of degree m, standard on a union of Schubert varieties Y, and let s(y, m) denote the cardinality of this set. The restrictions of the standard monomials to Y form a linearly independent subset of the space of global sections H 0 (Y, L m ) of the line bundle L m restricted to Y. The aim of this section is to show that SMT actually provides a basis for H 0 (Y, L m ) and implies the vanishing of higher cohomology. Theorem (Vanishing Theorems) Let Y be a union of Schubert varieties in Gr d,n. Then (a) H i (Y, L m ) = 0 for i 1, m 0. (b) The set S(Y, m) is a basis for H 0 (Y, L m ). (c) H i (X i, L m ) = 0 for 0 i < dim X i, m < 0 for all i I d,n. We start with some preliminaries. In a first step we will reduce the proof of the theorem to case where Y is just a Schubert variety. If Y = X i for some i, then S(Y, m) and s(y, m) will also be denoted by S(i, m), respectively s(i, m). Lemma Let Y = Y 1 Y 2 where Y 1, Y 2 are unions of Schubert varieties in Gr d,n. Then s(y, m) = s(y 1, m) + s(y 2, m) s(y 1 Y 2, m). 2. s(i, m) = s(i, m 1) + s( X i, m) Proof. Both (1) and (2) are easy consequences of Theorem 1.5.9, Theorem and Theorem

25 24 1. SMT for Graßmann varieties Proposition Let r be an integer d(n d). Suppose that all Schubert varieties X in Gr d,n of dimension at most r satisfy the following two properties: 1. H i (X, L m ) = 0, for i 1, m The set S(X, m) is a basis for H 0 (X, L m ), m 0. Then all unions and intersections of Schubert varieties of dimension at most r satisfy (1) and (2). Proof. We will prove the result by induction on r and induction on the number of irreducible components. Let S r denote the set of Schubert varieties X in Gr d,n of dimension at most r. Let Y be a union of Schubert j is a decomposition of Y as a union of its irre- S r. Set Y 1 = t 1 j=1 X i j and set Y 2 = X i t. varieties, say Y = t j=1 X i ducible components, where X i j Consider the exact sequence 0 O Y O Y1 O Y2 O Y1 Y 2 0. Tensoring with L m, we obtain the long exact sequence, H i 1 (Y 1 Y 2, L m ) H i (Y, L m ) H i (Y 1, L m ) H i (Y 2, L m ) H i (Y 1 Y 2, L m ). Now by Theorem 1.6.1, Y 1 Y 2 is reduced and Y 1 Y 2 is a union of Schubert varieties in S r 1. Hence by the induction hypothesis, (1) and (2) hold for Y 1 Y 2. In particular, if m 0, then (2) implies that the map H 0 (Y 1, L m ) H 0 (Y 2, L m ) H 0 (Y 1 Y 2, L m ) is surjective. Hence we obtain that the sequence 0 H 0 (Y, L m ) H 0 (Y 1, L m ) H 0 (Y 2, L m ) H 0 (Y 1 Y 2, L m ) 0 is exact. This implies that H 0 (Y 1 Y 2, L m ) H 1 (Y, L m ) is the zero map. Also, H 1 (Y, L m ) H 1 (Y 1, L m ) H 1 (Y 2, L m ) is the zero map since by induction H 1 (Y 1, L m ) = 0 = H 1 (Y 2, L m ). Hence we obtain H 1 (Y, L m ) = 0, for m 0. For i 2, the assertion that H i (Y, L m ) = 0, for m 0 follows from the long exact cohomology sequence above and the induction hypothesis. This proves assertion (1) for Y. To prove assertion (2) for Y, we observe that dim H 0 (Y, L m ) = dim H 0 (Y 1, L m ) + dim H 0 (Y 2, L m ) dim H 0 (Y 1 Y 2, L m ) = s(y 1, m) + s(y 2, m) s(y 1 Y 2, m). Hence Lemma and the induction hypothesis imply that dim H 0 (Y, L m ) = s(y, m).