On certain family of B-modules

Transcription

1 On certain family of B-modules Piotr Pragacz (IM PAN, Warszawa) joint with Witold Kraśkiewicz with results of Masaki Watanabe

2 Issai Schur s dissertation (Berlin, 1901): classification of irreducible polynomial representations of GL n : Homomorphisms GL n GL N sending X to a matrix [P ij (X )], where P ij is a polynomial in the entries of X. Two actions on E n (E vector space over a field K of char. 0). of the symmetric group S n via permutations of the factors, the diagonal action of GL(E). Irreductible representations S λ of the symmetric group S n are labeled by partitions of n.

3 Partition of n: λ = (λ 1 λ k 0) s.t. λ λ k = n. Graphical presentation for 8742: Schur module: V λ (E) := Hom Z[Sn](S λ, E n ) V λ ( ) is a functor: if E, F are R-modules, f : E F is an R-homomorphism, then f induces an R-homomorphism V λ (E) V λ (F ). In this way, we get all irreducible polynomial representations of GL n.

4 Let us label the boxes of the diagram with 1,..., n P:= sum of the permutations preserving the rows (P Z[S n ]), N:= sum of the permutations with their signs, preserving the columns. e(λ) := N P the Young idempotent ; V λ (E) = e(λ)e n. Example: V (n) (E) = S n (E), V (1 n )(E) = n (E).

5 Let T be the subgroup of diagonal matrices in GL n : x 1 x x 3 Consider the action of T on V λ (E) induced from the action of GL n via restriction. Main result of Schur s Thesis: The trace of the action of T on V λ (E) is equal to the Schur function:... s λ (x 1,..., x n ) = det ( s λp p+q(x 1,..., x n ) ) 1 p,q k, where s i (x 1,..., x n ) is the ith complete symmetric function.

6 Permutation: bijection N N, which is the identity off a finite set. A := Z[x 1, x 2,...]. We define i : A A i (f ) := f (x 1,..., x i, x i+1,...) f (x 1,..., x i+1, x i,...) x i x i+1. For a simple reflection s i = 1,..., i 1, i + 1, i, i + 2,..., we put si := i. Let w = s 1 s k = t 1 t k be two reduced words of w. Then s1 sk = t1 tk.

7 Thus for any permutation w, we can define w as s1 sk independently of a reduced word of w. Let n be a natural number such that w(k) = k for k > n. Schubert polynomial (Lascoux-Schützenberger 1982): S w := w 1 w 0 (x n 1 1 x n 2 2 x 1 n 1x 0 n) where w 0 is the permutation (n, n 1,..., 2, 1), n + 1, n + 2,....

8 We define the kth inversion set of w: I k (w) := {l : l > k, w(k) > w(l)} k = 1, 2,... Code of w (c(w)): sequence i k = I k (w), k = 1, 2,.... c(5, 2, 1, 6, 4, 3, 7, 8,...) is equal to (4, 1, 0, 2, 1, 0,...). Schubert polynomial S w is symmetric in x k i x k+1 if and only if w(k) < w(k + 1) (or equivalently if i k i k+1 ). If w(1) < w(2) < < w(k) > w(k + 1) < w(k + 2) < (or i 1 i 2 i k, 0 = i k+1 = i k+2 = ), then S w is equal to s ik,...,i 2,i 1 (x 1,..., x k ). If i 1 i 2, then S w = x i 1 1 x i 2 2 is a monomial.

9 If the sets I k (w) form a chain (w.r.t. inclusion), then w is called a vexillary permutation. Theorem (Lascoux-Schützenberger, Wachs) If w is a vexillary permutation with code (i 1, i 2,..., i n > 0, 0...), then S w = s (i1,...,i n) (min I 1(w) 1,..., min I n (w) 1). Flag Schur function: For two sequences of natural numbers i 1 i k and 0 < b 1 b k, s i1,...,i k (b 1,..., b k ) := det ( s ip p+q(x 1,..., x bp ) ) 1 p,q k

10 R commutative Q-algebra, E. : E 1 E 2 a flag of R-modules. Suppose that I = [i k,l ], k, l = 1, 2,..., is a matrix of 0 s and 1 s s.t. i k,l = 0 for k l; l i k,l is finite for any k; I has a finite number of nonzero rows. Such a matrix I is called a shape: =

11 Shape of permutation w is the matrix: I w = [i k,l ] := [χ k (l)], k, l = 1, 2,... where χ k is the characteristic function of I k (w). For w = 5, 2, 1, 6, 4, 3, 7, 8,..., the shape I w is equal to We define a module S w (E.), associated with a permutation w and a flag E. as S Iw (E.); this leads to a functor S w ( ).

12 From now on, let E. be a flag of K-vector spaces with dim E i = i. Let B be the Borel group of linear endomorphisms of E := E i, which preserve E.. The modules used in the definition of S w (E.) are Z[B]-modules, and maps are homomorphisms of Z[B]-modules. Let {u i : i = 1, 2,...} be a basis of E such that u 1, u 2,..., u k span E k. Then S w (E) as a cyclic Z[B]-submodule in l ĩl E l, generated by the element u w := l u k1,l u k2,l u kil,l where k 1,l < k 2,l < < k il,l are precisely those indices for which i kr,l,l = 1. E.g. S 5,2,1,6,4,3,7,... (E.) is generated over Z[B] by u 1 u 1 u 2 u 1 u 4 u 1 u 4 u 5.

13 Theorem (K-P) The trace of the action of a maximal torus T B on S w (E.) is equal to the Schubert polynomial S w. About the proof: we study multiplicative properties of S w (E.). t p,q (... w(p)... w(q)...) = (... w(q)... w(p)...) Chevalley-Monk formula for multiplication by S sk : S w (x x k ) = S w tp,q, the sum over p, q s.t. p k, q > k and l(w t p,q ) = l(w) + 1. For example S (x 1 +x 2 ) = S S S

14 Transition formula: Let (j, s) be a pair of positive integers s.t. j < s and w(j) > w(s), for any i ]j, s[, w(i) / [w(s), w(j)], for any r > j, if w(s) < w(r) then there exists i ]j, r[ s.t. w(i) [w(s), w(r)]. Then S w = S v x j + m p=1 S v p, where v = w t j,s, v p = w t j,s t kp,j, the sum over k p s.t. k p < j and w(k p ) < w(s), if i ]k p, j[ then w(i) / [w(k p ), w(s)].

15 Such a pair (j, s) always exists for a nontrivial permutation: it suffices to take the maximal pair in the lexicographical order s.t. w(j) > w(s). E.g. S = = S x 5 + S S maximal transition = S x 4 + S S S = S x 2.

16 We prove that for the maximal transition, there exists a filtration of Z[B]-modules 0 = F 0 F 1 F k F = S w (E.) and isomorphisms F/F k S v (E.) E j /E j 1 and F p /F p 1 S vp (E.) for p = 1,..., m. There exist flag S λ ( ), for which we have Theorem (K-P) If w is a vexillary permutation with code (i 1, i 2,..., i n > 0, 0...), then S w (E.) = S (i1,...,i n) ( Emin I1 (w) 1,..., E min In(w) 1).

17 Let b be the Lie algebra of n n upper matrices, t that of diagonal matrices, and U(b) the enveloping algebra of b. M a U(b)-module, λ = (λ 1,..., λ n ) Z n, M λ = {m M : hm =< λ, h > m} weight space of λ, < λ, h >= λ i h i If M is a direct sum of its weight spaces and each weight space has finite dimension, then M is called a weight module ch(m) := λ dim M λx λ, where x λ = x λ 1 1 x n λn

18 Let e ij be the matrix with 1 at the (i, j)-position and 0 elsewhere. Let K λ be a 1-dim l U(b)-module, where h acts by < λ, h > and the matrices e ij, where i < j, acts by zero. Any finite dim l weight module admits a filtration by these 1 dim l modules. w S (n) := {w : w(n + 1) < w(n + 2) <...}. E = 1 i n Ku i. For each j N, let {i < j : w(i) > w(j)} = {i 1 <... < i lj } u (j) w = u i1 u ilj Λ l j E u w = u 1 w u 2 w S w = U(b)u w The weight of u w is c(w).

19 Thm (K-P) For any w S (n), S w is a weight module and ch(s w ) = S w. What is the annihilator of u w? 1 i < j n m ij (w) = #{k > j : w(i) < w(k) < w(j)} e m ij +1 ij annihilates u w. Let I w U(b) be the ideal generated by h < c(w), h >, h t and e m ij (w)+1 ij, i < j. There exists U(b)/I w S w s.t. 1 mod I w u w. Theorem (W) This surjection is an isomorphism.

20 For λ Z n 0 we set S λ := S w where c(w) = λ. For λ Z n take k s.t. λ + k1 Z n 0 (1 = (1,..., 1) n times), and set S λ = K k1 S λ+k1. Similarly for S λ. QUESTIONS: 1. When a weighted module admits a filtration with subquotients isomorphic to some S λ s? 2. Does S λ S µ have such a filtration? ρ = (n 1, n 2,..., 2, 1, 0), K ρ dualizing module C category of all weight modules, for Λ Z n, C Λ is the full subcategory of C consisting of all weight modules whose weights are in Λ. Λ < Λ = {ρ λ : λ Λ} C Λ = C op Λ M M K ρ Lemma For any Λ Z n, C Λ has enough projectives.

21 Orders: w, v S w lex v if w = v or there exists i > 0 s.t. w(j) = v(j) for j < i and w(i) < v(i). For λ Z n, define λ = λ i. If λ = c(w), µ = c(v), we write λ µ if λ = µ and w 1 lex v 1. For general λ, µ Z n take k s.t. λ + k1, µ + k1 Z n 0, and define λ µ iff λ + k1 µ + k1. For λ Z n, set C λ := C {ν:ν λ}. All Ext s over U(b), in C λ. Prop. For λ Z n the modules S λ and S ρ λ K ρ are in C λ. Moreover S λ is projective and S ρ λ K ρ is injective. Theorem (W) For µ, ν λ, Ext i (S µ, S ρ ν K ρ ) = 0, i 1.

22 Theorem (W) Let M C λ. If Ext 1 (M, S ρ µ K ρ ) = 0 for all µ λ, then M has a filtration s.t. each of its subquotients is isomorphic to some S ν (ν λ). Cor. (1) If M = M 1... M r, then M has such a filtration iff each M i has. (2) If 0 L M N 0 is exact and M, N have such filtrations, then L also has. Proof (1) Ext 1 (M, N) = Ext 1 (M i, N) for any N. (2) Ext 1 (M, A) Ext 1 (L, A) Ext 2 (N, A) exact for any A.

23 Prop. w S (n), 1 k n 1. Then S w S sk filtration. (KP for k = 1, W in general) Theorem (W) S w S v has such a filtration for w, v S (n). Consider a B-module T w = 2 i n (Λ l i (w) K i 1 ). has such a T w is a direct sum component of 2 i n S si 1 S si 1, l i (w) times Prop. w S n. Then there is an exact sequence 0 S w T w N 0, where N has a filtration whose subquotients are S u with u 1 > lex w 1. from Cauchy formula i+j n (x i + y j ) = w S w(x)s ww0 (y)

24 Proof of the thm Exact sequence: 0 S w S v T w S v N S v 0 filtr. by Cor.(2) filtr. by Prop. filtr. by ind. on lex(w) Theorem (W) Let λ Z n and M C λ. Then we have ch(m) ν λ dim K(Hom b (M, S ρ ν K ρ ))S ν The equality holds if and only if M has a filtration with all subquotients isomorphic to S µ, where µ λ.

25 As a corollary, we get a formula for the coefficient of S w in S u S v : Cor. This coefficient is equal to the dimension of Hom b (S u S v, S w0 w K ρ ) = Hom b (S u S v S wow, K ρ ). Proof We use ch: S u S v = ch(s u S v ) = w (S u S v, S ρ λ K ρ)s w.

26 Some plethysm Let s σ denote the Schur functor associated to a partition σ Prop. s σ (S λ ) has a filtration with its subquotients isomorphic to some S ν. Proof (S λ ) k has such a filtration for any λ and any k. Hence Ext 1 ((S λ ) k, S ν K ρ ) = 0 for any ν. s σ (S λ ) is a direct sum factor of (S λ ) σ. Hence Ext 1 (s σ (S λ ), S ν K ρ ) = 0 for any ν, and s σ (S λ ) has the desired filtration.

27 Cor. If S w is a sum of monomials x α + x β +, then s σ (x α, x β,...) is a positive sum of. The End

28 References: W. Kraśkiewicz, P. Pragacz, Foncteurs de Schubert, C. R. Acad. Sci. Paris, 304(9) (1987), W. Kraśkiewicz, P. Pragacz, Schubert functors and Schubert polynomials, Eur. J. Comb., 25(8) (2004), M. Watanabe, Filtrations of b-modules with successive quotients isomorphic to Kraśkiewicz and Pragacz s modules realizing as their characters, arxiv: v3. M. Watanabe, Tensor product of Kraśkiewicz and Pragacz s modules, arxiv: