DEGENERACY SCHEMES, QUIVER SCHEMES AND SCHUBERT VARIETIES. v. lakshmibai and peter magyar 1

Transcription

1 DEGENERACY SCHEMES, QUIVER SCHEMES AND SCHUBERT VARIETIES v lakshmibai and peter magyar 1 November 1997, revised March 1998 abstract A result of Zelevinsky states that an orbit closure in the space of representations of the equioriented quiver of type A h is in bijection with the opposite cell in a Schubert variety of a partial flag variety SL(n)/P We prove that Zelevinsky s bijection is a scheme-theoretic isomorphism, which shows that the degeneracy schemes of Fulton and Buch are reduced and Cohen-Macaulay in arbitrary characteristic Among all algebraic varieties, the best understood are the flag varieties and their Schubert subvarieties They first appear as interesting examples, but acquire a general importance in the theory of characteristic classes of vector bundles Fulton [8] and Buch-Fulton [6] have recently given a theory of universal degeneracy loci, characteristic classes associated to maps among vector bundles, in which the role of Schubert varieties is taken by certain degeneracy schemes The underlying varieties of these schemes arise in the theory of quivers: they are the orbit-closures in the space of representations of the equioriented quiver A h Many other classical varieties also appear as such quiver varieties, such as determinantal varieties and the variety of complexes (cf 14) The same quiver varieties also arise in Deligne-Langlands theory for the p-adic general linear group [19]: the intersection homology of these varieties gives the p-adic analog of Kazhdan-Lusztig polynomials (which, by Zelevinsky s result below, become identical to ordinary Kazhdan-Lusztig polynomials) It turns out that a separate theory is not necessary to understand these spaces (for this particular quiver) By a remarkable but little-known result of Zelevinsky [20], all the above quiver varieties can be identified set-theoretically with open subsets of Schubert varieties In this paper, we prove a schemetheoretic strengthening of Zelevinksy s identification: the naive determinantal conditions defining each quiver variety generate the same ideal as the Plucker equations defining the corresponding Schubert variety Since the latter ideal is well understood via Standard Monomial Theory, we conclude that the corresponding quiver schemes are reduced and their singularities are identical to those of Schubert varieties In particular, the quiver varieties in arbitrary characteristic are normal, Cohen-Macaulay, etc These properties give a more concrete interpretation to the intersection theory in Fulton and Buch s work Our results extend early work by Hochster-Eagon [10], Kempf [12], and Deconcini-Strickland [7] Musili-Seshadri [16], proved the above scheme-theoretic identification for the variety of complexes Some of the consequences of our identification were known for more general quiver varieties by work of Abeasis, Del 1 Both authors partially supported by the National Science Foundation 1

2 Fra, and Kraft [3],[1]: that the quiver varieties are Cohen-Macaulay with rational singularities over a field of characteristic zero, and that the determinantal conditions generate the reduced ideals of the quiver varieties of codimension one Our methods are similar to those of Gonciulea and Lakshmibai [9] 1 Zelevinsky s bijection In this section we establish the set-theoretic identification between quiver varieties and Schubert varieties In 14, we giveseveralexamples, including Fulton s degeneracy schemes 11 Quiver varieties For the basic results below on quivers, we follow Abeasis-del Fra [2] and Zelevinsky [19] Fix an h-tuple of non-negative integers n =(n 1,,n h ) and a list of vector spaces V 1,,V h over an arbitrary field k with respective dimensions n 1,,n h Define Z, thevariety of quiver representations (of dimension n, of the equioriented quiver of type A h ) to be the affine space of all (h 1)-tuples of linear maps (f 1,,f h 1 ): f 1 f f h 2 f h 1 V 1 V2 2 Vh 1 Vh If we endow each V i with a basis, we get V i = k n i and Z = M(n 2 n 1 ) M(n h n h 1 ), where M(l m) denotes the affine space of matrices over k with l rows and m columns The group acts on Z by G n = GL(n 1 ) GL(n h ) (g 1,g 2,,g h ) (f 1,f 2,,f h 1 )=(g 2 f 1 g 1 1,g 3f 2 g 1 2,,g hf h 1 g 1 h 1 ), corresponding to change of basis in the V i Now, let r =(r ij ) 1 i j h be an array of non-negative integers with r ii = n i, and define r ij = 0 for any indices other than 1 i j h Define the set Z (r) ={(f 1,,f h 1 ) Z i<j, rank(f j 1 f i : V i V j )=r ij } (This set might be empty for a bad choice of r) Proposition The G n -orbits of Z are exactly the sets Z (r) for r =(r ij ) with r ij r i,j+1 r i 1,j + r i 1,j+1 0, 1 i j h 2

3 Proof This is a standard result of algebraic quiver theory [5], [4], first stated in this form by Abeasis-del Fra and Zelevinsky Since this theory is not well known among geometers, we recall it here Consider the abelian category R of quiver representations defined as follows f 1 f h 1 An object of R is a sequence of linear maps (V 1 Vh ), where the V i are any vector spaces of arbitrary dimension A morphism of R from the object f f h 1 (V 1 1 Vh ) to the object (V 1 f 1 f h 1 V h ) is defined to be an h-tuple of linear maps (φ i : V i V i ) such that each of the following squares commutes: V i f i Vi+1 φ i φ i+1 V i f i V i+1 Direct sum of objects is defined componentwise, and it is known by the Krull-Schmidt Theorem [4] that any object R Rcan be written uniquely as a direct sum of indecomposable objects By elementary linear algebra, these indecomposables are seen to be R ij =(0 0 k k 0 0) V i V j for 1 i j h (corresponding to the positive roots of the root system A h ) That is, there are unique multiplicities m ij Z + with R = 1 i j h m ij R ij Our variety Z consists of representations with fixed (V i ) and all possible (f i ) Two points of Z are in the same G n -orbit exactly if they are isomorphic as objects in R So the orbits correspond to arrays (m ij ) 1 i j h with m ij Z + and n i = k i l m kl We can compute the rank numbers r =(r ij ) from the multiplicities m = (m ij ): r ij = m kl, and conversely k i j l m ij = r ij r i,j+1 r i 1,j + r i 1,j+1 Hence the arrays (r ij ) with the stated conditions classify the G n -orbits on Z We define the quiver variety as the algebraic set Z(r) ={(f 1,,f h 1 ) Z i, j, rank(f j 1 f i : V i V j ) r ij } It will follow from Zelevinsky s theorem ( 13) that Z(r) is an irreducible variety and is the Zariski closure of Z (r) (provided the base field k is infinite) 3

4 12 Schubert varieties Given n =(n 1,,n h ), for 1 i h let a i = n 1 + n n i, and n = n n h For positive integers i j, we shall frequently use the notations [i, j] ={i, i +1,,j}, [i] =[1,i], [0] = {} Let k n = V 1 V h have basis e 1,,e n compatible with the V i Consider its general linear group GL(n), the subgroup B of upper-triangular matrices, and the parabolic subgroup P of block upper-triangular matrices P = {(a ij ) GL(n) a ij = 0 whenever j a k <ifor some k} A partial flag of type (a 1 < a 2 < < a h = n) (or simply a flag) isa sequence of subspaces U =(U 1 U 2 U h = k n ) with dim U i = a i Let E i = V 1 V i = e 1,,e ai,ande i = V i+1 V h = e ai+1,,e n,so that E i E i = kn Theflag variety Fl is the set of all flags U as above Fl has a transitive GL(n)-action induced from k n,andp =Stab GL(n) (E), so we may identify Fl = GL(n)/P, g E gp TheSchubert varieties are the closures of B-orbits on Fl Such orbits are usually indexed by certain permutations of [n], but we prefer to use flags of subsets of [n], of the form τ =(τ 1 τ 2 τ h =[n]), #τ i = a i (A permutation w :[n] [n] corresponds to the subset-flag with τ i = w[a i ]= {w(1),w(2),,w(a i )} This gives a one-to-one correspondence between cosets of the symmetric group W = S n modulo the Young subgroup W n = S n1 S nh, and subset-flags) Given such τ, lete i (τ) = e j j τ i be a coordinate subspace of k n,and E(τ) =(E 1 (τ) E 2 (τ) ) Fl Then we may define the Schubert cell X (τ) = B E(τ) { = (U 1 U 2 ) Fl dim U i k j } =#τ i [j] 1 i h, 1 j n and the Schubert variety X(τ) = { X (τ) = (U 1 U 2 ) Fl dim U i k j # τ i [j] 1 i h, 1 j n where k j = e 1,,e j k n We define the opposite cell O Fl to be the set of flags in general position with respect to the spaces E 1 E h 1 : O = {(U 1 U 2 ) Fl U i E i =0} In fact, O = B E, the orbit of the standard flag in Fl under the group B of lower triangular matrices We also define Y (τ) =X(τ) O, an open subset of X(τ) andy (τ) =X (τ) O By abuse of language, we call Y (τ) theopposite cell of X(τ), even though it is not a cell } 4

5 13 The bijection ζ We define a special subset-flag τ max to n =(n 1,,n h ) We want each τi max given the constraints [a j 1 ] τj max recursively by τ max i =(τ1 max τh max =[n]) corresponding to contain numbers as large as possible for all j (here a 0 = 0) Namely, we define τh max =[n]; τi max =[a i 1 ] {largest n i elements of τi+1 max} Furthermore, given r =(r ij ) 1 i j h indexing a quiver variety, define a subsetflag τ r to contain numbers as large as possible given the constraints { # τi r ai r [a j ]= i,j+1 for i j for i>j Namely, a j τ r i = { 1 a i 1 a i a }{{}}{{} i+1 a i+2 n} }{{}}{{}}{{} a i 1 r ii r i,i+1 r i,i+1 r i,i+2 r i,i+2 r i,i+3 r i,h whereweusethevisualnotation a }{{} =[a b+1,a] b Recall that a j = a j 1 + n j and 0 r ij r i,j+1 n j,sothateachτi r is an increasing list of integers Also r ij r i,j+1 r i+1,j r i+1,j+1,sothatτi r τ i+1 r Thus τ r is indeed a subset-flag See 14 for examples Now define the Zelevinsky map where ζ : Z Fl (f 1,,f h 1 ) (U 1 U 2 ) U i = {(v 1,,v h ) V 1 V h = k n j i, v j+1 = f j (v j )} In terms of coordinates, if we identify the linear maps (f 1,,f h 1 )withthe matrices (A 1,,A h 1 ), and identify Fl = GL(n)/P,wehave ζ(a 1,,A h 1 )= where I i is an identity matrix of size n i I A 1 I A 2 A 1 A 2 I 3 0 A 3 A 2 A 1 A 3 A 2 A 3 I 4 mod P 5

6 Theorem (Zelevinsky [20]) (i) ζ is a bijection of Z onto its image Y (τ max ): ζ : Z Y (τ max ) Also, ( ) Y (τ max )={(U 1 U 2 ) i, E i 1 U i, U i E i =0} (ii) ζ restricts to a bijection from Z(r) onto Y (τ r ): ζ : Z(r) Y (τ r ) Also, { ( ) Y (τ r i j, dim U i E j a i r i,j+1, )= (U 1 U 2 ) E i 1 U i, U i E i =0 } Proof Obviously ζ is injective To prove (i), we first show equation ( ) The inclusion is clear For the inclusion, consider a flag U with E i 1 U i for all i SinceactingbyB does not change dim U i k j, we may suppose U = E(µ) for some µ =(µ 1 µ h =[n]) with [a i 1 ] µ i for all i By definition #τ max i [j] is as small as possible given [a i 1 ] τ max i dim U i k j =#µ i [j] #τi max [j], which shows ( ) Now we show that ζ(z) is equal to the right hand side of ( ) The inclusion ζ(z) RHS( ) isclear,soweshowζ(z) RHS( ) Each U i is tansverse to E i,sou i is the graph of a linear map,so (f i,i+1,,f i,h ):E i E i = V i+1 V h Since E i 1 U i, we have f ij (E i 1 ) = 0, and we may consider f ij : V i = E i /E i 1 V j Any element of U j can be written (v 1,,v j,f j,j+1 (v j ),) Let i<j Any (v 1,,v h ) U i is also an element of U j,sov j+1 = f j,j+1 (v j ) Taking f i = f i,i+1, we find U = ζ(f 1,,f h 1 ) The proof of (ii) is similar Equation ( ) follows just as before Now consider a flag U = ζ(f 1 f h 1 ) ζ(z) =Y (τ max ) Then dim U i E j = dime i 1 +dimker(f j f j 1 f i ) = dime i 1 +dimv i rank(f j f j 1 f i ) = a i rank(f j f j 1 f i ) Hence U ζ(z(r)) U RHS( ) =Y (τ r ) Corollary ([2], [19]) For each r, Z (r) isanopendenseg n -orbit in Z(r) Proof Arguing as in the proof of (ii) above, we find that ζ(z (r)) Y (τ r ) ButitisknownthatY (τ r ) is Zariski open and dense in Y (τ r ) Remarks (i) The map ζ is an algebraic isomorphism onto its image, since it is clear from the coordinate definition that ζ is injective on points and on tangent vectors (ii) For each r, X (τ r )isanorbitofp IfweembedG n into P as block-diagonal matrices, then ζ is a G n -equivariant map Now, clearly ζ(z (r)) Y (τ r ) Also the Z (r) areacompletelistofg n -orbits on Z and the Y (τ r )aredisjoint subsets of Y (τ max ) = ζ(z) We conclude that ζ(z (r)) = Y (τ r ) 6

7 14 Examples Example A small generic case Let h =4,n =(2, 3, 2, 2), r = m = where r ij and m ij are written in the usual matrix positions Note that r ij is obtained by summing the entries in m weakly above and to the right of m ij Then we get (a 1,a 2,a 3,a 4 )=(2, 5, 7, 9), n =9,and τ max =( [9]), τ r =( [9]), which correspond to the cosets in W/W n w max = , w r = (The minimal-length representatives of these cosets are the permutations as written; the other elements are obtained by permuting numbers within each block) The partial flag variety is Fl = {U 1 U 2 U 3 k 9 dim U i = a i },and the Schubert varieties are: { } { X(τ max )= U k2 U 2 k 5, X(τ r )= U U 3 U } 1 k 5 U 3, k 2 U 2 dim U 2 k 5 4 The opposite cells Y (τ) are defined by the extra conditions U i E i =0 Example Fulton s universal degeneracy schemes [8] Given m > 0, let Z be the affine space associated to the quiver data h =2m, n =(1, 2,,m,m,,2, 1) For each w S m+1, Fulton defines a degeneracy scheme Ω w = Z(r) as follows (Here Ω w = Z(r) is a variety We will define scheme structures for quiver varieties in 2) Denote i = 2m +1 i, and define r = r(w) = (r ij )and m =(m ij )by: r ij = r ji = i r ij =#[i] w[j] { 1, (i, j) =(w(k), k), k m m ij = 0, otherwise for 1 i, j m The associated Schubert varieties Y (τ r )aregivenbyτ r = (τ r 1 τr 1 )orbycosets w = w 1 w 1 W/W n τi r =[a i 1] {a w 1 (1),a w 1 (2),,a w 1 (i) }, τ r =[a i i 1] {a 1,a 2,,a m} w i =[a i 2 +1,a i 1 ] {a w 1 (i) } w m =[a m 1 +1,a m 1] {a w 1 (m+1) } w j =[a j 2 +1,a j 1 ] for 1 i m, 1 j m 1 Furthermore τ max = τ r(w) and w max = w r(w) for w = e S m+1, the identity permutation 7

8 Example For a given h and n, thevariety of complexes is defined as the union C = r Z(r) overallr =(r ij )withr i,i+2 =0foreachi Thesubvarieties Z(r) correspond to the multiplicity matrices m =(m ij )withm ij =0forall i +2 j, andm ii + m i 1,i + m i,i+1 = n i for all i Musili-Seshadri [16] find the irreducible components (maximal subvarieties) of C, andshowthateach component is isomorphic to the opposite cell of some Schubert variety Example The classical determinantal( variety ) of k l matrices ( of rank ) m (where m k, l) isd = Z(r) forr = l m,andm = l m m Also 0 k 0 k m n = k + l, τ max =([k +1,n] [n]), τ r =([m +1,l] [n m +1,n] [n]) X(τ max )=Fl=Gr(l, k n ), X(τ r ) = {U Gr(l, k n ) dim U k l l m}, D = Z(r) = Y (τ r )={U Gr(l, k n ) dim U k l l m, U E =0}, where E = e l+1,e l+2,,e n 2 Plucker coordinates and determinantal ideals In this section we prove the scheme-theoretic version of Zelevinsky s bijection From now on, we assume our field k is infinite 21 Coordinates on the opposite big cell Consider the opposite cell O GL(n)/P It is easily seen that O consists of those cosets which have a unique representative A of the form I A 21 I A =(a kl )= A 31 A 32 I 3 0 mod P, A h1 A h2 A h3 I h where I i is the identity matrix of size n i,anda ij is an arbitrary matrix of size n i n j That is, O is an affine space with coordinates a kl for those positions (k, l) with1 l a i <k n for some i Its coordinate ring is the polynomial ring k[o] =k[a kl ] For a matrix M M(k l) and subsets λ [k], µ [l], let det M λ µ be the minor with row indices λ and column indices µ Now let σ [n] be a subset of size #σ = a i for some i Define the Plucker coordinate p σ k[o] tobethe a i -minor of our matrix A with row indices σ and column indices the interval [a i ]: p σ = p σ (A) =deta σ [ai] 8

9 Define a partial order on Plucker coordinates by: σ σ σ = {σ(1) <σ(2) < <σ(a i )}, σ = {σ (1) <σ (2) < <σ (a i )}, σ(1) σ (1), σ(2) σ (2),,σ(a i ) σ (a i ) This is a version of the Bruhat order Proposition Let τ =(τ 1 τ h =[n]) be a subset-flag and Y (τ) the intersection of the Schubert variety X(τ) with the opposite cell O Then the (reduced) vanishing ideal I(τ) k[o] of Y (τ) O is generated by those Plucker coordinates p σ which are incomparable with one of the p τi : I(τ) = p σ i, #σ = a i, σ τ i Proof This follows from well-known results of Lakshmibai-Musili-Seshadri in Standard Monomial Theory (see eg [16],[13]) 22 The main theorem Denote a generic element of the quiver space Z = M(n 2 n 1 ) M(n h n h 1 ) by (A 1,,A h 1 ), so that the coordinate ring of Z is the polynomial ring in the entries of all the matrices A i Letr =(r ij ) index the quiver variety Z(r) = {(A 1,,A h 1 ) rank A j 1 A i r ij } Let J (r) k[z] be the ideal generated by the determinantal conditions implied by the definition of Z(r): J (r) = det(a j 1 A j 2 A i ) λ µ j>i, λ [n j ], µ [n i ] #λ =#µ = r ij +1 Clearly J (r) defines Z(r) set-theoretically Theorem J (r) is a prime ideal and is the vanishing ideal of Z(r) Z There are isomorphisms of reduced schemes Z(r) =Spec(k[Z] / J (r)) = Spec(k[O] / I(τ r )) = Y (τ r ) That is, the quiver scheme Z(r) defined by J (r) is isomorphic to the reduced variety Y (τ r ), the opposite cell of a Schubert variety Corollary For a ring R, consider the polynomial ring R[O] =R[a kj ] and the ideal J (r) R R[O] generated by the same determinants as above Define the scheme Z(r) R = Spec(R[O]/J (r) R ) (i) If k is an arbitrary field, then Z(r) k is reduced, irreducible, Cohen-Macaulay, normal, and has rational singularities (ii) If R is a noetherian ring, then Z(r) R is reduced (resp irreducible, Cohen- Macaulay, normal) exactly when Spec(R) is reduced (irreducible, Cohen-Macaulay, 9

10 normal) Proof of Corollary (i) Let k be the algebraic closure of k Then the desired properties of Z(r) k follow from the corresponding properties of Schubert varieties (see eg [11], [15], [17]) But this implies these properties for Z(r) k as well, since Z(r) k Z(r) k is a faithfully flat morphism (See [14], 21E) (ii) By [15], the morphism Z(r) Z Spec(Z) is faithfully flat, so Z(r) R Spec(R) isaswell([14], 3C) Now, the fibers of this latter morphism are reduced, Cohen-Macaulay, and normal by (i), so the corresponding properties hold for the total space Z(r) R exactly when they hold for the base ([14], 21E) Finally, Z(r) R Spec(R) is a closed surjective morphism with irreducible fibers of the same dimension, and it is elementary that the total space is irreducible exactly when the base is irreducible (see [18], I63) Proof of Theorem The map of 13, ζ : Z Y (τ max ) O is an algebraic isomorphism onto its image, so the restriction homomorphism ζ : k[o] k[z] is surjective Now, ζ maps Z(r) isomorphically onto Y (τ r ), so we have Z(r) = Spec(k[Z] / J (r)) = Spec(k[O] / (ζ ) 1 J (r)) = Spec(k[O] / I(τ r )) = Y (τ r ) where J (r) k[z] denotes the (reduced) vanishing ideal of Y (τ r ) We must show J (r) = J (r) Since clearly J (r) J (r), we are left with the other inclusion, which is equivalent to We prove this in the next section (ζ ) 1 J (r) (ζ ) 1 J (r) =I(τ r ) 23 Proof of the main theorem Let A O be the generic matrix of 21 We define ideals I 0, I 1, I 2 k[o] generated by certain minors of A: I 0 =(ζ ) 1 J (r) =(ζ ) 1 det(a j 1 A j 2 A i ) λ µ j>i, λ [n j ], µ [n i ] #λ =#µ = r ij +1 I 1 = I 2 = I(τ r ) = det A λ µ i j, λ [a j +1,n], µ [a i ] #λ =#µ = r ij +1 det A σ [ai] To finish the proof of Theorem 22, we will show I 0 I 1 I 2 1 i h 1, σ [n] #σ = a i, σ τ r i 10

11 Lemma 1 Let X =(x ij ) and Y =(y kl ) be matrices of variables x ij, y kl generating a polynomial ring Let J X (resp J Y ) be the ideal generated by all r+1-minors of X (resp Y ) Then J X and J Y both contain all r+1-minors of the product XY Proof det(xy ) λ µ = det X λ ν det Y ν µ ν Lemma 2 Let (A 1,,A h 1 ) be a generic element of Z, andforj>ilet J ji be the ideal generated by all r +1-minors of the n j n i product matrix A j 1 A i ThenJ ji contains all r+1-minors of the (n a j 1 ) a i matrix A j 1 A 1 A j 1 A 2 A j 1 A i à (ji) A j A 1 A j A 2 A j A i = A h 1 A 1 A h 1 A 2 A h 1 A i Proof Note that we can factor the matrix I j 1 à (ji) A j = A j 1 A i (A i 1 A 1, A i 1 A 2,, A i 1, I i ) A h 1 A j Now apply Lemma 1 twice Lemma 3 I 0 I 1 Proof For generic elements A O and (A 1,,A h 1 ) Z, wehaveby definition ζ (f(a)) = f(ζ(a 1,,A h 1 )) for any polynomial f in the matrix entries Now let λ [a j 1 +1,n], µ [a i ], #λ =#µ = r ij + 1, and consider a generator det A λ µ of I 1 Then ζ (det A λ µ )=detζ(a 1,,A h 1 ) λ µ =detã(ji) λ µ where λ [n a j 1 ] is a translate of λ By Lemma 2, det Ã(ji) λ µ J(r), so I 1 = det A λ µ (ζ ) 1 J (r) =I 0 Lemma 4 (Gonciulea-Lakshmibai) Let A be a generic element of O Let 1 t a i, 1 s n, andσ = {σ(1) <σ(2) < <σ(a i )} [n] with σ(a i t +1) s Thenp σ (A) belongs to the ideal of k[o] generated by t-minors of A with row indices s and column indices a i Proof Choose σ [s, n] σ with #σ = t, andletσ = σ \σ Then the Laplace expansion of p σ (A) with respect to the rows σ, σ,gives p σ (A) =deta σ [ai] = ± det A σ λ det A σ λ, λ λ =[a i] 11

12 where the sum is over all partitions of the interval [a i ] The first factor of each term in the sum is of the form required Lemma 5 I 1 I 2 Proof Let σ [n] with#σ = a i, σ τi r has the largest possible entries such that for some i, 1 i h 1 Now, τ r i τ r i (a i r i,j+1 ) a j, j i, so σ τ r i must violate this condition for some j: Hence by Lemma 4, p σ (A) isini 1 σ(a i r i,j+1 ) a j +1, j i The Main Theorem 22 is therefore proved References [1] S Abeasis, Codimension 1 orbits and semi-invariants for the representations of an oriented graph of type A n, Trans Amer Math Soc 282 (1984), [2] S Abeasis, A Del Fra, Degenerations for the representations of an equioriented quiver of type A m, Boll Univ Math Ital Suppl 2 (1980), [3] S Abeasis, A Del Fra, H Kraft, The geometry of representations of A m, Math Ann 256 (1981), [4] M Auslander, I Reiten, S Smalo, Representation theory of Artin algebras, Cambridge University Press, 1995 [5] J Bernstein, I Gelfand, V Ponomarev, Coxeter functors and Gabriel s theorem, Russ Math Surveys 28 (1973), [6] A Buch, W Fulton, preprint 1997 [7] C DeConcini, E Strickland, On the variety of complexes, Adv in Math 41 (1981), [8] W Fulton, Universal Schubert polynomials, preprint 1997 [9] N Gonciulea, V Lakshmibai, Singular loci of ladder determinantal varieties and Schubert varieties, preprint 1997 [10] M Hochster, JA Eagon, Cohen-Macaulay rings, invariant theory, and the generic perfection of determinantal loci, Amer J Math 93 (1971), [11] JC Jantzen, Representations of Algebraic Groups, Academic Press, 1987 [12] G Kempf, Images of homogeneous vector bundles and the variety of complexes, Bull AMS 81 (1975), [13] V Lakshmibai, CS Seshadri, Standard Monomial Theory, in Proc Hyderabad Conf on Alg Groups, S Ramanan (ed), Manoj Prakashan (1991), [14] H Matsumura, Commutative Algebra, 2nd ed, Benjamin/Cummings, Reading, MA (1980) 12

13 [15] VB Mehta, A Ramanathan, Frobenius splitting and cohomology vanishing for Schubert varieties, Annals of Math 122 (1985), [16] C Musili, CS Seshadri, Schubert varieties and the variety of complexes, in Arithmetic and Geometry, Vol II, Progress in Math 36, Birkhauser (1983), [17] A Ramanathan, Schubert varieties are arithmetically Cohen-Macaulay, Invent Math 80 (1985), [18] I Shafarevitch, Basic Algebraic Geometry, vol I, Springer-Verlag (1988) [19] A Zelevinsky, A p-adic analogue of the Kazhdan-Lusztig conjecture Funct Anal App 15 (1981), 9-21 [20] A Zelevinsky, Two remarks on graded nilpotent classes, Uspekhi Math Nauk 40 (1985), no 1 (241),