Kraśkiewicz-Pragacz modules and some positivity properties of Schubert polynomials

Transcription

1 Kraśkiewicz-Pragacz modules and some positivity properties of Schubert polynomials Masaki Watanabe Graduate School of Mathematical Sciences, the University of Tokyo December 9, 2015 Masaki Watanabe KP modules and Schubert positivity December 9, / 16

2 Schubert Polynomials { w: permutation S w Z[x 1, x 2,...] Swsi = i S w = Sw s i S w x i x i+1 (l(ws i ) < l(w)) S w0 = x1 n 1 x2 n 2 x n 1 for the longest element w 0 S n {S w } Schubert classes in the cohomology rings of flag varieties. w: grassmannian i.e. i, w(1) < < w(i), w(i + 1) < w(i + 2) < = S w = (a Schur polynomial in x 1,..., x i ). Examples for w S 3 : S 123 = 1 S 132 = x 1 + x 2 S 213 = x 1 S 231 = x 1 x 2 S 312 = x 2 1 S 321 = x 2 1 x 2 Masaki Watanabe KP modules and Schubert positivity December 9, / 16

3 Kraśkiewicz-Pragacz modules { w: permutation S w Z[x 1, x 2,...] Swsi = i S w = Sw s i S w x i x i+1 (l(ws i ) < l(w)) S w0 = x1 n 1 x2 n 2 x n 1 for the longest element w 0 S n {S w } Schubert classes in the cohomology rings of flag varieties. w: grassmannian i.e. i, w(1) < < w(i), w(i + 1) < w(i + 2) < = S w = (a Schur polynomial in x 1,..., x i ). Schur polynomials: characters of irreducible representations of gl n (C). Schubert polynomials: characters of Kraśkiewicz-Pragacz modules. Masaki Watanabe KP modules and Schubert positivity December 9, / 16

4 Kraśkiewicz-Pragacz modules: Definition K: field with characteristic 0 b = b n = Lie algebra of n n upper triangular matrices K n = 1 i n Ku i: vector representation of b D(w): Rothe diagram of a permutation w S w : KP module Example: w = Masaki Watanabe KP modules and Schubert positivity December 9, / 16

5 Kraśkiewicz-Pragacz modules: Definition K: field with characteristic 0 b = b n = Lie algebra of n n upper triangular matrices K n = 1 i n Ku i: vector representation of b D(w): Rothe diagram of a permutation w S w : KP module Example: w = D(w) = {(i, w(j)) : i < j, w(i) > w(j)} = {(1, 1), (1, 2), (3, 2)} Masaki Watanabe KP modules and Schubert positivity December 9, / 16

6 Kraśkiewicz-Pragacz modules: Definition K: field with characteristic 0 b = b n = Lie algebra of n n upper triangular matrices K n = 1 i n Ku i: vector representation of b D(w): Rothe diagram of a permutation w S w : KP module Example: w = u w = u 1 (u 1 u 3 ) K n 2 (K n ) S w = U(b)u w = u 1 (u 1 u 3 ), u 1 (u 1 u 2 ) Masaki Watanabe KP modules and Schubert positivity December 9, / 16

7 Kraśkiewicz-Pragacz modules: Property For λ Z n, M λ = {m M : hm = i λ ih i ( h = diag(h 1,..., h n ) b)}. ch(m) = λ dim M λx λ : character of M (we only consider weight b-modules: i.e. finite dimensional modules having weight-space decompositions with respect to the subalgebra of diagonal matrices). Theorem (Kraśkiewicz-Pragacz) ch(s w ) = S w. eg. S 3142 = u 1 (u 1 u 3 ), u 1 (u 1 u 2 ) ch(s 3142 ) = x 2 1 x 3 +x 2 1 x 2 = S Masaki Watanabe KP modules and Schubert positivity December 9, / 16

8 Kraśkiewicz-Pragacz modules: Examples w = s i : simple transposition S si u si = u i = Ku 1 Ku i ch(s si ) = x x i = S si Masaki Watanabe KP modules and Schubert positivity December 9, / 16

9 Kraśkiewicz-Pragacz modules: Examples w: grassmannian i.e. w(1) < < w(i), w(i + 1) < w(i + 2) < eg. w = u w = a lowest vector in an irreducible representation of gl i (C) S w = an irreducible representation of gl i (C) ch(s w ) = (a Schur polynomial in x 1,..., x i ) = S w Remark: the examples we have seen are all special cases of Demazure modules, but in general they are different (equal only for 2143-avoiding w). Masaki Watanabe KP modules and Schubert positivity December 9, / 16

10 Filtrations by Kraśkiewicz-Pragacz modules A KP filtration of a (weight) b-module M is a filtration M = M r M r 1 M 0 = 0 such that each M i /M i 1 is isomorphic to KP modules. Question What kind of b-modules M have KP filtrations? Masaki Watanabe KP modules and Schubert positivity December 9, / 16

11 Filtrations by Kraśkiewicz-Pragacz modules A KP filtration of a (weight) b-module M is a filtration M = M r M r 1 M 0 = 0 such that each M i /M i 1 is isomorphic to KP modules. Question What kind of b-modules M have KP filtrations? Question Why such question? Masaki Watanabe KP modules and Schubert positivity December 9, / 16

12 Filtrations by Kraśkiewicz-Pragacz modules A KP filtration of a (weight) b-module M is a filtration M = M r M r 1 M 0 = 0 such that each M i /M i 1 is isomorphic to KP modules. Question What kind of b-modules M have KP filtrations? Question Why such question? Motivation: Schubert positivity. Masaki Watanabe KP modules and Schubert positivity December 9, / 16

13 Schubert positivity The product S w S v of Schubert polynomials is always Schubert-positive, i.e. a positive sum of Schubert polynomials. Classically proved geometrically: by interpreting the coefficients in S w S v = u cu wv S u as the number of intersections of certain subvarieties in a flag variety. Combinatorists seek for some combinatorial objects which counts the coefficients c u wv (like eg. Littlewood-Richardson tableaux for products of Schur functions). Or at least combinatorial proofs for the positivity. Representation-theoretic way? Masaki Watanabe KP modules and Schubert positivity December 9, / 16

14 Schubert positivity The product S w S v of Schubert polynomials is always Schubert-positive, i.e. a positive sum of Schubert polynomials. Classically proved geometrically: by interpreting the coefficients in S w S v = u cu wv S u as the number of intersections of certain subvarieties in a flag variety. Combinatorists seek for some combinatorial objects which counts the coefficients c u wv (like eg. Littlewood-Richardson tableaux for products of Schur functions). Or at least combinatorial proofs for the positivity. Representation-theoretic way? The plethysm of a symmetric function f with a polynomial (or a power series) g = x α + x β + is defined as f [g] = f (x α, x β,...). Classical fact: the plethysm s λ [s µ ] of Schur functions is Schur-positive. s λ [S w ]: Schubert-positive? Masaki Watanabe KP modules and Schubert positivity December 9, / 16

15 Schubert positivity and KP modules If M has a KP filtration M = M r M r 1 M 0 = 0 with M i /M i 1 = Swi then ch(m) = ch(s wi ) = S wi is Schubert-positive. So: Question Does the tensor product module S w S v have a KP filtration for any w and v? Question Does the Schur-functor image s λ (S w ) of a KP module have a KP filtration for any λ and w? Masaki Watanabe KP modules and Schubert positivity December 9, / 16

16 Schubert positivity and KP modules If M has a KP filtration M = M r M r 1 M 0 = 0 with M i /M i 1 = Swi then ch(m) = ch(s wi ) = S wi is Schubert-positive. So: Question Does the tensor product module S w S v have a KP filtration for any w and v? Question Does the Schur-functor image s λ (S w ) of a KP module have a KP filtration for any λ and w? Question What kind of b-modules M have KP filtrations? Masaki Watanabe KP modules and Schubert positivity December 9, / 16

17 Theorem (W.) A weight b-module M with weights Z n 0 has a KP filtration if and only if Ext i (M, Sw K (k,k 1,...,k n+1) ) = 0 for any w, any k Z and any i 1 (K λ denotes the one-dimensional b-module with weight λ). Highest weight category structure on b-modules such that standard objects = KP modules (cf. Polo, van der Kallen,... Demazure modules). for Masaki Watanabe KP modules and Schubert positivity December 9, / 16

18 Theorem (W.) A weight b-module M with weights Z n 0 has a KP filtration if and only if Ext i (M, Sw K (k,k 1,...,k n+1) ) = 0 for any w, any k Z and any i 1 (K λ denotes the one-dimensional b-module with weight λ). Highest weight category structure on b-modules such that standard objects = KP modules (cf. Polo, van der Kallen,... Demazure modules). Highest weight structure needs an ordering on the set of weights (roughly, we want an order such that S w becomes a projective cover of K code(w) in the category of b-modules with weights code(w)): for Masaki Watanabe KP modules and Schubert positivity December 9, / 16

19 Theorem (W.) A weight b-module M with weights Z n 0 has a KP filtration if and only if Ext i (M, Sw K (k,k 1,...,k n+1) ) = 0 for any w, any k Z and any i 1 (K λ denotes the one-dimensional b-module with weight λ). Highest weight category structure on b-modules such that standard objects = KP modules (cf. Polo, van der Kallen,... Demazure modules). Highest weight structure needs an ordering on the set of weights (roughly, we want an order such that S w becomes a projective cover of K code(w) in the category of b-modules with weights code(w)): Example of our ordering: for (1, 3, 0, 1, 0) code(25143) (0, 3, 2, 0, 0) code(15423) since lex Masaki Watanabe KP modules and Schubert positivity December 9, / 16

20 Theorem (W.) A weight b-module M with weights Z n 0 has a KP filtration if and only if Ext i (M, Sw K (k,k 1,...,k n+1) ) = 0 ( w, k, i 1). Corollary If M has a KP filtration then so does its direct summands. If 0 L M N 0 is exact and M, N have KP filtrations then so does L. Masaki Watanabe KP modules and Schubert positivity December 9, / 16

21 Theorem (W.) A weight b-module M with weights Z n 0 has a KP filtration if and only if Ext i (M, Sw K (k,k 1,...,k n+1) ) = 0 ( w, k, i 1). Corollary So: If M has a KP filtration then so does its direct summands. If 0 L M N 0 is exact and M, N have KP filtrations then so does L. Plethysm question can be reduced to Tensor product question since s λ (S w ) is a direct summand of Sw λ. Tensor product question can also be reduced to simplest case S si S w (corresponding to Monk s formula) which can be checked directly. Masaki Watanabe KP modules and Schubert positivity December 9, / 16

22 Theorem (W.) A weight b-module M with weights Z n 0 has a KP filtration if and only if Ext i (M, Sw K (k,k 1,...,k n+1) ) = 0 ( w, k, i 1). Corollary So: If M has a KP filtration then so does its direct summands. If 0 L M N 0 is exact and M, N have KP filtrations then so does L. Plethysm question can be reduced to Tensor product question since s λ (S w ) is a direct summand of Sw λ. Tensor product question can also be reduced to simplest case S si S w (corresponding to Monk s formula) which can be checked directly. Theorem (W.) S w S v has a KP filtration for any w, v. s λ (S w ) has a KP filtration for any w, λ. Masaki Watanabe KP modules and Schubert positivity December 9, / 16

23 Theorem (W.) A weight b-module M with weights Z n 0 has a KP filtration if and only if Ext i (M, Sw K (k,k 1,...,k n+1) ) = 0 ( w, k, i 1). Corollary So: If M has a KP filtration then so does its direct summands. If 0 L M N 0 is exact and M, N have KP filtrations then so does L. Plethysm question can be reduced to Tensor product question since s λ (S w ) is a direct summand of Sw λ. Tensor product question can also be reduced to simplest case S si S w (corresponding to Monk s formula) which can be checked directly. Theorem (W.) S w S v has a KP filtration for any w, v. S-positivity of S w S v s λ (S w ) has a KP filtration for any w, λ. S-positivity of s λ [S w ] Masaki Watanabe KP modules and Schubert positivity December 9, / 16

24 Sketch of S si S w case = general case Corollary (1) If M has a KP filtration then so does its direct summands. (2) If 0 L M N 0 is exact and M, N have KP filtrations then so does L. Repeated use of Monk case S si S sj S sk has a KP filtration. Masaki Watanabe KP modules and Schubert positivity December 9, / 16

25 Sketch of S si S w case = general case Corollary (1) If M has a KP filtration then so does its direct summands. (2) If 0 L M N 0 is exact and M, N have KP filtrations then so does L. Repeated use of Monk case S si S sj S sk Then so does its direct summand i li (S si ) =: T by (1). has a KP filtration. Masaki Watanabe KP modules and Schubert positivity December 9, / 16

26 Sketch of S si S w case = general case Corollary (1) If M has a KP filtration then so does its direct summands. (2) If 0 L M N 0 is exact and M, N have KP filtrations then so does L. Repeated use of Monk case S si S sj S sk has a KP filtration. Then so does its direct summand i li (S si ) =: T by (1). Character calculation (T : S w ) = 1 and every other constituent S u satisfy u 1 > lex w 1 (for suitable l i ). Ext 1 (S w, S u ) = 0 for such u (a consequence of highest weight structure, in fact shown along the proof of highest weight structure) 0 S w T (mod. with filtr. by S u (u 1 > lex w 1 )) 0. 0 S w S v T S v (mod. with filtr. by S u S v ) 0. induction proceeds by (2). Masaki Watanabe KP modules and Schubert positivity December 9, / 16

27 Highest weight structure & explicit filtrations Highest weight stucture for KP modules also turns out to be useful in constructing explicit KP filtrations of some modules: eg. M = S w S d (K i ) (character expansion known as Pieri rule) Basic strategy: if ch(m) = S w1 + + S wr, find v 1,..., v r M and suitable b-homs φ 1,..., φ r so that φ i (v i ) = u wi, φ i (v j ) = 0 (j < i) v 1,..., v i / v 1,..., v i 1 S wi simple dimension-counting argument shows that 0 v 1 v 1, v 2 gives a desired filration. Masaki Watanabe KP modules and Schubert positivity December 9, / 16

28 Highest weight structure & explicit filtrations Highest weight stucture for KP modules also turns out to be useful in constructing explicit KP filtrations of some modules: eg. M = S w S d (K i ) (character expansion known as Pieri rule) Basic strategy: if ch(m) = S w1 + + S wr, find v 1,..., v r M and suitable b-homs φ 1,..., φ r so that φ i (v i ) = u wi, φ i (v j ) = 0 (j < i) v 1,..., v i / v 1,..., v i 1 S wi simple dimension-counting argument shows that 0 v 1 v 1, v 2 gives a desired filration. Main difficulty: we only know w 1,..., w r up to their indexing. Masaki Watanabe KP modules and Schubert positivity December 9, / 16

29 Highest weight structure & explicit filtrations Highest weight stucture for KP modules also turns out to be useful in constructing explicit KP filtrations of some modules: eg. M = S w S d (K i ) (character expansion known as Pieri rule) Basic strategy: if ch(m) = S w1 + + S wr, find v 1,..., v r M and suitable b-homs φ 1,..., φ r so that φ i (v i ) = u wi, φ i (v j ) = 0 (j < i) v 1,..., v i / v 1,..., v i 1 S wi simple dimension-counting argument shows that 0 v 1 v 1, v 2 gives a desired filration. Main difficulty: we only know w 1,..., w r up to their indexing. Highest weight structure: Ext 1 (S w, S u ) = 0 if u 1 > lex w 1 so we can always take w1 1 w2 1 (if we take v and φ correctly). lex lex Masaki Watanabe KP modules and Schubert positivity December 9, / 16

30 Highest weight structure & explicit filtrations Highest weight stucture for KP modules also turns out to be useful in constructing explicit KP filtrations of some modules: eg. M = S w S d (K i ) (character expansion known as Pieri rule) Basic strategy: if ch(m) = S w1 + + S wr, find v 1,..., v r M and suitable b-homs φ 1,..., φ r so that φ i (v i ) = u wi, φ i (v j ) = 0 (j < i) v 1,..., v i / v 1,..., v i 1 S wi simple dimension-counting argument shows that 0 v 1 v 1, v 2 gives a desired filration. Main difficulty: we only know w 1,..., w r up to their indexing. Highest weight structure: Ext 1 (S w, S u ) = 0 if u 1 > lex w 1 so we can always take w1 1 w2 1 (if we take v and φ correctly). lex lex Theorem (W.) Explicit construction of KP filtrations for S w S d (K i ) and S w d (K i ). (remark: Pieri rule for KP modules another proof for the h.w. structure) Masaki Watanabe KP modules and Schubert positivity December 9, / 16