JOURNAL OF ALGEBRA 195, 130᎐140Ž. 1997 ARTICLE NO. JA977092
Straightening for Standard Monomials on Schubert Varieties
V. Reiner
Uni¨ersity of Minnesota, Minneapolis, Minnesota
and
M. Shimozono*
Massachusetts Institute of Technology, Cambridge, Massachusetts
Communicated by D. A. Buchsbaum
Received December 10, 1996
We give an elementary proof of the following known fact: any multihomogenous component in the homogeneous coordinate ring of a Schubert variety inside GLnrB has basis given by the standard monomialsŽ V. Lakshmibai et al., 1986, J. Algebra 100, 462᎐557. . ᮊ 1997 Academic Press
INTRODUCTION
Hodge and Pedoewx 5 constructed bases of the homogeneous coordinate rings of the Schubert varieties in the Grassmannian, consisting of certain products of Plucker¨ coordinates called standard monomials. Lakshmibai, Musili, and Seshadriwx 6 have generalized this theory extensively, giving standard monomial bases of the spaces of global sections of certain line bundles over unions of Schubert varieties in GrB where G is a classical group and B is a Borel subgroup. The purpose of this article is to give an elementary proof that the standard monomials yield a basis of the coordi- nate ring of a single Schubert subvariety of the flag varietyŽ that is, G is of type A.. The part of the original proof that is not elementary is the
*The second author was supported by NSF Postdoctoral Research Fellowship DMS- 9407639.
130
0021-8693r97 $25.00 Copyright ᮊ 1997 by Academic Press All rights of reproduction in any form reserved. STANDARD MONOMIALS OF SCHUBERT VARIETIES 131 argument that the standard monomials are a spanning set. This is accom- plished here using an explicit straightening algorithm based on a classical determinantal identity. As a byproduct of the proof, the Plucker¨ coordi- nates that vanish on a given Schubert variety are shown to generate its prime ideal. Our methods are valid over the integers.
2. SCHUBERT VARIETIES
This section establishes notation and reviews standard facts regarding the Schubert varieties in the flag varietyŽ seewx 3, 8. . For convenience of exposition the base field k is assumed to be algebraically closed. Let E be an n-dimensional vector space over k with the fixed ordered Ä4Ž. Ž. basis ein:1FiFn,GsGL E s GL k the general linear group, B the Borel subgroup of lower triangular matrices in G, and ᒄ and ᑿ the Lie algebras of G and B, respectively. Let F be the set of complete flags in E, Ž. whose typical element has the form F.s F012; F ; F ; иии ; Fn, where Fi is a subspace of E of dimension i. Composing the well-known Plucker¨ and Segre maps
Plucker¨ 6 Ł nnސ r Segre 6 ސ r F Z [ rs1Ž.H Ž.E Ž.mrs1H Ž.E gives an embedding of F into the projective space of one-dimensional n rŽ. subspaces of mrs1 H E . It can be shown that the image of this map is irreducible and closed, giving F the structure of a projective variety. There is an isomorphism given by the right G-equivariant map B _ G ª F such that Bg ¬ F., where Fr is the span of the first r rows of the matrix g. The multihomogeneous coordinates of the embedding of F in Z can be described as follows. The vectors e e иии e form a basis of uus n n ur rŽ. 1 HE, where u s uu12...urruns over the strictly increasing words of length r with letters in the set wxn s Ä41, 2, . . . , n . Projective coordinates on rŽ. Ä4 rŽ. Ä4 HEare given by the basis xuuof H E * dual to e , so that the multihomogeneous coordinate ring kZwxof Z is isomorphic to the polyno- n mial ring kxwxuu. The ring kx wxis ގ -graded; every variable x uindexed by a word of length r has degree given by the rth standard basis vector of ގn.
Let kzwxijbe the coordinate ring of ᒄ, where z ij is the Ž.i, j th matrix entry function. For an arbitrary word u s uu12...uriwith u g wxn and 0 F r F n, let puijbe the minor of the generic matrix Ž.z with row indices 1, 2, . . . , r and column indices u12, u ,...,ur. By direct calculation, the multihomoge- Ž. neous coordinates of the image of F in Z are given by Bg ¬ pgu , where uis a strictly increasing word with letters in wxn . Since G is Zariski dense in ᒄ, the prime ideal of the image of F in Z is given by the kernel I of the map : kxwxuijuuªkz wxgiven by x ¬ p . 132 REINER AND SHIMOZONO
The coordinate ring kwxF has the following well-known presentation. A proof is given inwx 5 , where it is attributed to Mertens.
THEOREM 1. The prime ideal I of the image F in Z is generated by the quadratic relationsŽ. see Section 5.
Let w be a permutation in the symmetric group Sn on the set wxn s Ä41, 2, . . . , n . We also denote by w the permutation matrix in G whose Ž. o i,jth entry is ␦wŽi., jw. The Schubert cell is defined by X s B _ BwB. Every right B-orbit in B _ G is a Schubert cell. The Schubert ¨ariety Xw is o defined to be the Zariski closure of the Schubert cell Xw in the projective variety B _ G. To describe the right B-orbit decomposition of Xw , recall theŽ. strong Bruhat order on Sn. The minimal length coset representatives Ž. in Snrr S = S nyrcan be identified with the strictly increasing words of length r in the alphabet wxn .If uand uЈ are two such words, write u F uЈ X Ž. if uiiF u for all i. Let rnnrn: S ª S r S = S yrbe the natural projection. Under the above identification, rŽ.w is the word obtained by sorting the Ž. Ž. Ž. Ž . values w 1,w2 ,...,wr into increasing order. Then let ¨ F w if r ¨ Ž. Fr wfor all 1 F r F n. The Schubert variety has the well-known orbit decomposition
o Xws " X¨.2.1Ž. ¨Fw
Consider the homomorphism of coordinate rings w: kwxᒄ ª kw wᑿ x given by a restriction of functions on ᒄ to the subvariety wᑿ. The following result appears as Exercise 10.5.11 inwx 3 . Ž. LEMMA 2. The radical ideal I Xww of X is prime and is gi¨en by the kernel of the composite map wu( : kxwxªkw wᑿ x.
Proof. Since kwwxᑿ is a polynomial ring, the kernel of w ( is a prime ideal. Let f g kxwxu . Then the assertion of the lemma is deduced by the Ž. following chain of equivalences: f g IXw if and only if f vanishes on o Xw , if and only if Ž.f vanishes on BwB, if and only if Ž.f vanishes on wB, if and only if Ž.f vanishes on wᑿ. The first and second equivalences are trivial, and the fourth follows since wᑿ is the closure of wB in ᒄ.Soit only remains to show the third equivalence. Without loss of generality assume that f g kxwxu is homogeneous of multidegree m s Ž. m12,m,...,mnr. Let be the partition such that s m12q m q иии qmr for 1 F r F n. A direct calculation shows that Ž.Ž.fbgs Ž.Ž.Ž.Łn r rs1 brr fgfor all b g B and g g G. Ž. Let Jwube the ideal in kxwxgenerated by I the ideal of F ; Z and the Ž. variables xursuch that u g w , where r is the length of the word u. STANDARD MONOMIALS OF SCHUBERT VARIETIES 133
LEMMA 3. The ideal Jww defines X set-theoretically. Ž. Proof. Suppose u s uu12...urris a word such that u g w . Let m be an index such that um is greater than the mth largest letter in wŽ.1w Ž.2...wr Ž.. Consider wu Žp ., which is the Plucker¨ coordinate p u with the variables zij set to zero if j ) wiŽ.. It vanishes, being the determinant of an r = r matrix that has a rectangular Ž.r q 1 y m = m Ž. submatrix of zeros. By Lemma 2, we have Jww: Ix , so the zero set Y of Jwwcontains X . On the other hand, the ideal J wis B-stable under the action Ž.Ž.bf x s fxb Ž.. It follows that Y is closed and right B-stable, hence a union of Schubert varieties. If ¨ g w, then there is an index l Ž. Ž . Ž. Ž. such that ll¨ g w . Setting u s l¨ , we have x uwug J and p ¨ / o Ž. 0, so X¨ Y. In light of 2.1 we have Y s Xw . We will obtain an elementary proof of the following result.
THEOREM 4. The ideal Jww is the prime ideal of the Schubert ¨ariety X .
3. STANDARD MONOMIALS
Recall that the homogeneous coordiante ring kwxF of the flag variety is isomorphic to the subring kpwxuij:kz wxgenerated by the Plucker¨ coordi- nates puum. Let kpwxdenote the mth graded component kp wx u, where Ž.n msm12,m,...,mnrgގ . Let be the partition defined by s m1q m2 qиии qmrumfor 1 F r F n. In characteristic zero, kpwxaffords the irreducible representation of G of highest weight . This module was first constructed by Deruytswx 2 . A tableau of shape is a map from the Ferrers Ž. ÄŽ. 4 diagram D s i, j :1FjFi of to the set wxn . It is clear that kpwxumis spanned by the elements p Ts pp u12 u иии , where T is a tableau of shape whose ith column ui is strictly increasing from top to bottom. A tableau T is said to be standard if its rows are weakly increasing from left to right and its columns are strictly increasing from top to bottom. By abuse of language the monomial pTuin the Plucker¨ coordinates p is said to be standard if the tableau T is standard. The following basis was known to Youngwx 10 .
THEOREM 5. The standard monomials form a basis of the homogeneous coordinate ring of F.
The original definition of a standard monomial on a Schubert ¨ariety wx6 can be translated into the following. Let T be a standard tableau of shape .A defining chain for T is a Bruhat-increasing chain of permutations
1 2 w F w F иии F w 1 134 REINER AND SHIMOZONO
k X such that X Žw . is the kth column of T, where is the number of cells k k in the kth column of the Ferrer diagram DŽ. of . Say that T is standard
on Xw Žand by abuse of language that the monomial pT is standard on i Xw . if there is a defining chain Äw 4 for T whose last permutation is less than or equal to w, that is, w 1 F w. EXAMPLE 6. A standard tableau T and defining chain for T are given
below. This chain shows that T is standard on Xw for any w g S9 such that w G 596281347: 125 369 Ts58 7 9 1 ws135792468 2 w s 268391457 3 w s 596281347. The following lemma about defining chains was proven by Deodhar.
LEMMA 7wx 8, Chap. 4, Sect. 6 . For any standard tableau T, there is a 12 j minimal defining chain w F w F иии F w1 for T in the sense that if Ĩ 4is j j another defining chain for T, then ¨ G w for all j. In the previous example the sequence Žw123, w , w .is the minimal defining chain for T.
As a consequence of this lemma, checking standardness of pTwon X is 1 2 easy if one knows the minimal defining chain w F w F иии F w 1 for T, since one only needs to check if w 1 F w. There is a nice way to compute this minimal defining chain, involving the notion of the right key of a Ž. tableauwx 7 . Given a standard tableau T of shape , the right key Kq T of j, j j, j 1 j, T is the tableau of shape given as follows. Let u F u q F иии F u 1 X be the unique chain of strictly increasing wordsŽ each of length j . defined by the following properties:
j, j ⅷ u is the jth column of T, and j, kj,kj,k1 ⅷ uis minimal among words with u G u ywhich contain the kth column of T as a subwordŽ. see paragraph below Lemma 8 .
Ž. j,1 Then the jth column of KTq is defined to the last word u . Define the canonical lift wŽ. T of T to be the shortest permutation such that XŽŽ..wT equals the jth column of KT Ž.. We then have the j q following relationship. STANDARD MONOMIALS OF SCHUBERT VARIETIES 135
LEMMA 8.wx 7 For any standard tableau T, the minimal defining chain 1 2 j w F w F иии F w 1 for T has w equal to the canonical lift of the tableau consisting of the first j columns of T. Ž. Therefore, T is standard on Xw if and only if wT Fw. The process involved in passing from u j, ky1 to u j, k is equivalent to a jeu de taquin on a two-column tableau or a greedy partial matching. Here j, k 1 is an explicit algorithm which starts with the word u s u y and pro- j, k duces the word ¨ s u . By definition the letters TkŽ.,1 -Tk Ž.,2 - иии Ž.Ž X. -Tk,q for q s k must occur as a subword of ¨, say in positions X 1 F r12- r - иии - rqjF , that are determined as follows. Let r qbe maximal such that u TkŽ.,q. Given r , let r be maximal such that rmqF q1m r-rand u TkŽ.,m. It follows from the standardness of T that mmq1 rmF at each stage the index rm exists. Define ¨ to be the word such that
TkŽ.,m if r s rm ¨rs ½ur if r f Ä4r12, r ,...,rq.
XAMPLE Ј Ž. E 9. Three tableaux, T, T , KTq , are depicted below. The Ј j, k Ž. kth column of the tableau T is the word u in the definition of KTq , where j s 1. A letter in the kth column of TЈ is starred if and only if it does not appear in the kth column of T. The last column of TЈ yields the Ž. Ž. first column since j s 1 of the right key KTq of T. Performing similar Ž. calculations for j ) 1, one obtains KTq , also shown below. From this we see that the canonical lift of T is wTŽ.s596281347:
125 12 2* 255 369 33*5 569 Ј Ts58 Ts56 6* KTqŽ.s69 7788*8 999*99
p i Let Y s D is1 Xw be a union of Schubert varieties. Say that a tableau T is standard on Y if it is standard on Xw i for some i, and again by abuse of language say that pT is a standard monomial on Y. Our main goal is to give an elementary proof of the following result in the special case that Y is a single Schubert variety.
THEOREM 10.wx 6, 8 The standard monomials on Y form a basis of the homogeneous coordinate ring kwx Y . 136 REINER AND SHIMOZONO
This result has the following well-known consequence for Demazure
modules wx1 . Let be a partition with at most n parts and mi the number of columns of of length i. The partition defines a B-equivariant line n mm i bundle L [ mis1 Liion Y, where L is the restriction from F s B _ G to Y of the pullback of OŽ.1 under the composite morphism
Plucker¨ 6 i B _ G ª Pi_ G ސŽ.n Ž.E
and Pi is the maximal parabolic subgroup generated by the subgroups
GLin= GL yinand B of GL . The Demazure module is defined to be the 0 space of global sections HYŽ.,L and can be shown to be isomorphic to Ž. the m s m1,...,mn th multihomogeneous component of kYwx. There- fore Theorem 10 implies that the Demazure module has a basis given by
the elements pT where T is a tableau of shape that is standard on Y. In the case Y s Xw , Theorems 4 and 10 follow immediately from Lemma 3 and the following two results.
THEOREM 11. The monomials that are standard on a union Y of Schubert ¨arieties are linearly independent on Y.
THEOREM 12. The monomials that are standard on Xwuw span kwx x rJ . Theorems 11 and 12 are proved in Sections 4 and 5, respectively.
4. INDEPENDENCE
For the sake of completeness we repeat the elementary proof of Theo- rem 11 furnished by standard monomial theorywx 8 . We work in the ring kpwxu. Suppose there is a linear dependence relation among a collection T of
standard monomials pT on Y. Without loss of generality we may assume that all the tableaux in T have the same shape , and that has a minimal number of columns. Let
ÝcpTTYNs04.1Ž. TgT
be such a relation with as few terms as possible, so that 0 / cT g k, where f N Y denotes the restriction of the function f to the subvariety Y. Among all the strictly increasing words that appear as the last column of a tableau Ž. TgT, let u be any minimal element. Let Tsu resp. T/ u be the set of TgTwhose last column is equalŽ. resp. not equal to u. Let YЈ s X Ž ., which is contained in Y since each T T is assumed to be DT g Tsu w T g standard on Y. STANDARD MONOMIALS OF SCHUBERT VARIETIES 137
Suppose first that T/ u is empty, that is, every tableau T g T has last column u. For T g T, let Tˆ be the tableau obtained by removing the last column of T. The relationŽ. 4.1 still holds when restricted further to the subvariety YЈ : Y, and factors into the relation
pcPuTTYÝ ˆNsЈ0.Ž. 4.2 TgT Let T g T. By Lemma 8 and the definitions, wTŽ.contains u as an initial o subword. Thus, pu does not vanish at any point of the Schubert cell XwŽT .. Ž. o It follows that the sum in 4.2 vanishes on DT g T XwŽT . and hence on its closure YЈ. This gives a linear dependence relation among monomials that are standard on YЈ, which correspond to tableaux with one less column than , which is a contradiction.
Otherwise suppose the set T/ u is nonempty. Restricting the relation Ž. 4.1 to YЈ : Y, one has pTYNsЈ 0 for T g T/u, since the Plucker¨ coordinate corresponding to the last column of such a tableau T vanishes on YЈ. This means that there is a linear dependency among the monomials
pT for T g F=u, which are standard on YЈ. This new relation has fewer turns, which is a contradiction. Remark 13. Notice that the inductive nature of this proof naturally leads one to consider spaces Y which are unions of Schubert varieties, rather than just single Schubert varieties Xw .
5. SPANNING
Two kinds of straightening relations are required here, the famous quadratic relations Ž.see Theorem 1 and another relation that is also given by a classical determinantal identity. Both straightening relations use the following total order on tableaux. Given a tableau T of shape , let < PROPOSITION 14. Let T be a tableau of shape that is not standard on Xw . Then there is an expression of the form pTSSw' ÝcpŽ.mod J ,5.1 Ž. S-T where cS g ޚ. 138 REINER AND SHIMOZONO It may be assumed that the columns of T are strictly increasing from top to bottom. There are two cases: 1. T is not standard, that is, one of its rows is not weakly increasing from left to right. 2. T is standard but not standard on Xw . In the first case a quadratic relation may be employed to produce a relation of the formŽ. 5.1 . For the sake of completeness we describe the quadratic relations and repeat the classical argument. Consider two adjacent columns j and j q 1 of a tableau T of shape , X and let 1 F i F jq1. Then one has the relation pTSs Ýp ,5.2Ž. S where S runs over all the tableaux obtained from T by exchanging any i of the letters in the jth column with the first i letters of the Ž.j q 1 st. So if T is not standard, TiŽ.,j)Ti Ž,jq1 . for some cell Ž.i, j . Note that the first i letters of the Ž.j q 1 st column of T are all strictly smaller X than the last j y i letters in the jth column: X T Ž.j, j ) иии ) TiŽ.Ž.Ž.q1, j ) Ti,j)Ti,jq1 )иии ) T Ž.Ž.2, j q 1 ) T 1, j q 1. It is not hard to verify that on the right hand side ofŽ. 5.2 , each tableau S satisfies S - T, for in passing from T to S, at least one of the largest X j y i letters in the jth column of T is replaced by a strictly smaller letter. This yields a relation of the formŽ. 5.1 . For the second case, a little more notation is required. Let SA be the symmetric group on the set A and ColŽ. the Young subgroup of SDŽ . that stabilizes each column of DŽ. . The group SDŽ. has a natural right action on tableaux of shape by Ž.Ž.T i, j s T ŽŽ.. i, j . We now describe the second kind of straightening relation. Let A be a subset of DŽ. that has at most one cell in each row, j the minimum index of a column that contains a cell of A, and AЈ be the portion of A that is strictly to the right of the jth column. Let f be any injection from AЈ into the jth column of DŽ. whose image CЈ is disjoint from A. Finally, let C consist of the cells in the jth column of DŽ. that are not in A nor in CЈ. STANDARD MONOMIALS OF SCHUBERT VARIETIES 139 Let  be the involution in SDŽ . that exchanges each cell in AЈ with its image under f. Then for any tableau T of shape , the Plucker¨ coordinates satisfy the relation Ž.1 p Ž.1p.5.3 Ž. ÝÝy TsyT Ž Ž .. Ž Ž .. gSAAr S lCol gS AЈjCArSЈlClCol This follows from the determinantal identityw 9, III.11Ž. Ix . So assume that T is standard but not standard on Xw . This means that Ž. Ž. wT gw, or in other words, for some j, the jth column u of KTq satisfies u XŽ.w . Let j be maximal with this property, so that all g j columns to the right of the jth are strictly shorter than the jth. Let r be an index such that u exceeds the rth letter in XŽ.w . r j We now determine a set A as inŽ. 5.3 by selecting a suitable strictly decreasing subword of wordŽ.T . This is most easily done by examining the tableau TЈ, whose columns consist of the words u j, j, u j, jq1,...,uj,1 that occur in the calculation ofŽ. the jth column of the right key of T. Let us index the columns of TЈ from left to right by j through 1. By the definition of right key, the rightmost column of TЈ is equal to u. X Let ur be the letter in the rth row and last column of TЈ. It follows from X X the right key algorithm that given ursthrough u 1, there is an unstarred X y X letter us in the sth row of TЈ that is in a weakly earlier column that us 1 X X y with uss) u y1. Repeating this process, one eventually has that the letter X uXmust be given by the bottom letter of the jth column of TЈ. Since the j unstarred letters in the pth column of TЈ correspond to letters in the pth X X X column of T, the word uЈ u X ...u u may be regarded as a strictly s j rq1 r decreasing subword of wordŽ.T . Let A be the set of cells in D Ž. correspond to the letters of uЈ. It is enough to show that the desired relationŽ. 5.1 is obtained by applying Ž. 5.3 to the tableau T and the set of cells A. The reader is urged to keep the following example in mind while reading the remainder of the proof. EXAMPLE 15. Let T be as in the running example, w s 64281357, and Ž. js1, so that TЈ is as before. Here u s 25689 and 5 w s 24689, so r s 2. One choice for uЈ is given in bold in TЈ and in T: 12 2* 125 33*5 369 TЈs56 6* T s 58 788* 7 99* 9 9 Let f be any injection from AЈ into the jth row of DŽ. whose image C is disjoint from A. Consider any tableau T appearing on the right hand side ofŽ. 5.3 . Its jth column Ž call it x .contains the letters of uЈ. Since uЈ 140 REINER AND SHIMOZONO X X has jrjy r q 1 letters, the smallest of which is u , there at least y r q 1 X letters in x that are greater than or equal to urj. Since x has letters, its rth largest letter must be greater than or equal to ur . It follows that x Ž.w, so that p and p vanish on X . Consider the left hand side grxTw ofŽ. 5.3 . Since uЈ appears as a strictly decreasing subword of word Ž.T ,it follows that pT appears once and the rest of the summands have the form "pS where S - T. Thus one has a relation of the formŽ. 5.1 . PROBLEM 16. Devise a straightening algorithm that proves spanning for standard monomials on unions of Schubert varieties. REFERENCES 1. M. Demazure, Desingularisation´´´´´´ des varietes de Schubert generalises, Ann. Sci. Ecole Norm. Sup. 7 Ž.1974 , 53᎐88. 2. J. 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Turnbull, ‘‘The Theory of Determinants, Matrices, and Invariants,’’ 3rd ed., Dover, New York, 1960. 10. A. Young, On quantitative substitutional analysis, II, Proc. London Math. Soc. Ž.1 34 Ž.1902 , 362᎐397.