Kra´skiewicz-Pragacz modules and some positivity properties of Schubert polynomials

Kra´skiewicz-Pragacz modules and some positivity properties of Schubert polynomials

Masaki Watanabe

Graduate School of Mathematical Sciences, the University of Tokyo

December 9, 2015

Schubert Polynomials

w: permutation S_w ∈Z[x₁,x₂, . . .]

(S_wsi =∂iS_w = ^S_x^w−siSw

i−x_i+1 (`(wsi)< `(w))

S_w0=x₁ⁿ⁻¹x₂ⁿ⁻²· · ·xn−1 for the longest elementw₀∈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 x1, . . . ,xi).

Examples for w ∈S3:

S₁₂₃ = 1 S₁₃₂ =x₁+x₂ S₂₁₃ =x₁ S₂₃₁ =x1x2

Kra´skiewicz-Pragacz modules

w: permutation S_w ∈Z[x1,x2, . . .]

(S_wsi =∂_iS_w = ^S_x^w−siSw

i−x_i+1 (`(ws<sub>i</sub>)< `(w))

S_w0=x₁ⁿ⁻¹x₂ⁿ⁻²· · ·xn−1 for the longest elementw₀∈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 x1, . . . ,x_i).

Schur polynomials: characters of irreducible representations ofgl_n(C).

Schubert polynomials: characters ofKra´skiewicz-Pragacz modules.

Kra´skiewicz-Pragacz modules: Definition

K: field with characteristic 0

b=b_n= Lie algebra ofn×n upper triangular matrices Kⁿ=L

1≤i≤nKui: vector representation of b

D(w): Rothe diagramof a permutation w S_w: KP module Example: w = 3142

3 1 4 2

Kra´skiewicz-Pragacz modules: Definition

K: field with characteristic 0

b=b_n= Lie algebra ofn×n upper triangular matrices Kⁿ=L

1≤i≤nKu_i: vector representation of b

D(w): Rothe diagramof a permutation w S_w: KP module Example: w = 3142

3 1 4 2 D(w) ={(i,w(j)) :i <j,w(i)>w(j)}

={(1,1),(1,2),(3,2)}

Kra´skiewicz-Pragacz modules: Definition

K: field with characteristic 0

b=b_n= Lie algebra ofn×n upper triangular matrices Kⁿ=L

1≤i≤nKu_i: vector representation of b

D(w): Rothe diagramof a permutation w S_w: KP module Example: w = 3142

3 1 4 2

Kra´skiewicz-Pragacz modules: Property

For λ∈Zⁿ,M_λ ={m∈M :hm=P

iλ_ih_i (∀h =diag(h₁, . . . ,h_n)∈b)}.

ch(M) =P

λdimM_λx^λ: characterof 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 (Kra´skiewicz-Pragacz) ch(S_w) =S_w.

eg.

S₃₁₄₂ =hu₁⊗(u₁∧u3),u1⊗(u₁∧u₂)i ch(S₃₁₄₂) =x₁²x3+x₁²x2=S₃₁₄₂.

Kra´skiewicz-Pragacz modules: Examples

w =si: simple transposition

usi =ui

S_si =Ku₁⊕ · · · ⊕Ku_i ch(S_si) =x₁+· · ·+x_i =S_si

Kra´skiewicz-Pragacz modules: Examples

w: grassmannian i.e. w(1)<· · ·<w(i),w(i + 1)<w(i+ 2)<· · · eg. w = 13524

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 inx1, . . . ,xi) =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-avoidingw).

Filtrations by Kra´skiewicz-Pragacz modules

A KP filtrationof a (weight) b-module M is a filtration

M =M_r ⊃Mr−1 ⊃ · · · ⊃M₀ = 0 such that each M_i/M_i−1 is isomorphic to KP modules.

Question

What kind ofb-modules M have KP filtrations?

Question

Why such question?

Motivation: Schubert positivity.

Filtrations by Kra´skiewicz-Pragacz modules

A KP filtrationof a (weight) b-module M is a filtration

M =M_r ⊃Mr−1 ⊃ · · · ⊃M₀ = 0 such that each M_i/M_i−1 is isomorphic to KP modules.

Question

What kind ofb-modules M have KP filtrations?

Question

Why such question?

Motivation: Schubert positivity.

Filtrations by Kra´skiewicz-Pragacz modules

A KP filtrationof a (weight) b-module M is a filtration

M =M_r ⊃Mr−1 ⊃ · · · ⊃M₀ = 0 such that each M_i/M_i−1 is isomorphic to KP modules.

Question

What kind ofb-modules M have KP filtrations?

Question

Why such question?

Motivation: Schubert positivity.

Schubert positivity

The productS_wS_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_wS_v =P

uc_wv^u S_u as the number of intersections of certain subvarieties in a flag variety.

Combinatorists seek for some combinatorial objects which “counts” the coefficientsc_wv^u (like eg. Littlewood-Richardson tableaux for products of Schur functions). Or at least combinatorial proofs for the positivity.

Representation-theoretic way?

Theplethysm of a symmetric functionf with a polynomial (or a power series)g =x^α+x^β+· · · is defined asf[g] =f(x^α,x^β, . . .).

Classical fact: the plethysms_λ[s_µ] of Schur functions is Schur-positive. sλ[Sw]: Schubert-positive?

Schubert positivity

The productS_wS_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_wS_v =P

uc_wv^u S_u as the number of intersections of certain subvarieties in a flag variety.

Combinatorists seek for some combinatorial objects which “counts” the coefficientsc_wv^u (like eg. Littlewood-Richardson tableaux for products of Schur functions). Or at least combinatorial proofs for the positivity.

Representation-theoretic way?

Theplethysm of a symmetric functionf with a polynomial (or a power series)g =x^α+x^β+· · · is defined asf[g] =f(x^α,x^β, . . .).

Classical fact: the plethysms_λ[s_µ] of Schur functions is Schur-positive.

s [S ]: Schubert-positive?

Schubert positivity and KP modules

IfM has a KP filtration M =Mr ⊃Mr−1 ⊃ · · · ⊃M0 = 0 with M_i/Mi−1 ∼=S_wi thench(M) =Pch(S_wi) =P

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 anyλand w?

Question

What kind ofb-modules M have KP filtrations?

Schubert positivity and KP modules

IfM has a KP filtration M =Mr ⊃Mr−1 ⊃ · · · ⊃M0 = 0 with M_i/Mi−1 ∼=S_wi thench(M) =Pch(S_wi) =P

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 anyλand w?

Theorem (W.)

A weight b-module M with weights⊂Zⁿ≥0 has a KP filtration if and only if Extⁱ(M,S_w^∗ ⊗K(k,k−1,...,k−n+1)) = 0 for anyw, any k ∈Zand any i ≥1 (K_λ denotes the one-dimensional b-module with weightλ).

Highest weight category structure onb-modules such that

standard objects = KP modules (cf. Polo, van der Kallen, . . . for Demazure modules).

Highest weight structure needs an ordering on the set of weights (roughly, we want an order≤such thatS_w becomes a projective cover of K_code(w) in the category of b-modules with weights≤code(w)): Example of our ordering:

(1,3,0,1,0)

|| code(25143)

≤(0,3,2,0,0)

|| code(15423)

since 25143⁻¹≥

lex

15423⁻¹.

Theorem (W.)

A weight b-module M with weights⊂Zⁿ≥0 has a KP filtration if and only if Extⁱ(M,S_w^∗ ⊗K(k,k−1,...,k−n+1)) = 0 for anyw, any k ∈Zand any i ≥1 (K_λ denotes the one-dimensional b-module with weightλ).

Highest weight category structure onb-modules such that

standard objects = KP modules (cf. Polo, van der Kallen, . . . for Demazure modules).

Highest weight structure needs an ordering on the set of weights (roughly, we want an order≤such thatS_w becomes a projective cover of K_code(w) in the category of b-modules with weights≤code(w)):

Example of our ordering: (1,3,0,1,0)

|| code(25143)

≤(0,3,2,0,0)

|| code(15423)

since 25143⁻¹≥

lex

15423⁻¹.

Theorem (W.)

A weight b-module M with weights⊂Zⁿ≥0 has a KP filtration if and only if Extⁱ(M,S_w^∗ ⊗K(k,k−1,...,k−n+1)) = 0 for anyw, any k ∈Zand any i ≥1 (K_λ denotes the one-dimensional b-module with weightλ).

Highest weight category structure onb-modules such that

standard objects = KP modules (cf. Polo, van der Kallen, . . . for Demazure modules).

Highest weight structure needs an ordering on the set of weights (roughly, we want an order≤such thatS_w becomes a projective cover of K_code(w) in the category of b-modules with weights≤code(w)):

Example of our ordering:

(1,3,0,1,0)

||

code(25143)

≤(0,3,2,0,0)

||

code(15423)

since 25143⁻¹≥

lex

15423⁻¹.

Theorem (W.)

A weight b-module M with weights⊂Zⁿ≥0 has a KP filtration if and only if Extⁱ(M,S_w^∗ ⊗K(k,k−1,...,k−n+1)) = 0 (∀w,k,∀i ≥1).

Corollary

IfM has a KP filtration then so does its direct summands.

If 0→L→M →N →0 is exact andM,N have KP filtrations then so doesL.

So:

Plethysm question can be reduced to Tensor product question since s_λ(S_w) is a direct summand ofS_w^⊗|λ|.

Tensor product question can also be reduced to simplest caseS_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 ofS_wS_v

sλ(S_w) has a KP filtration for anyw, λ.

S-positivity ofsλ[Sw]

Theorem (W.)

A weight b-module M with weights⊂Zⁿ≥0 has a KP filtration if and only if Extⁱ(M,S_w^∗ ⊗K(k,k−1,...,k−n+1)) = 0 (∀w,k,∀i ≥1).

Corollary

IfM has a KP filtration then so does its direct summands.

If 0→L→M →N →0 is exact andM,N have KP filtrations then so doesL.

So:

Plethysm question can be reduced to Tensor product question since s_λ(S_w) is a direct summand ofS_w^⊗|λ|.

Tensor product question can also be reduced to simplest caseS_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 ofS_wS_v

sλ(S_w) has a KP filtration for anyw, λ.

S-positivity ofsλ[Sw]

Theorem (W.)

A weight b-module M with weights⊂Zⁿ≥0 has a KP filtration if and only if Extⁱ(M,S_w^∗ ⊗K(k,k−1,...,k−n+1)) = 0 (∀w,k,∀i ≥1).

Corollary

IfM has a KP filtration then so does its direct summands.

If 0→L→M →N →0 is exact andM,N have KP filtrations then so doesL.

So:

Plethysm question can be reduced to Tensor product question since s_λ(S_w) is a direct summand ofS_w^⊗|λ|.

Tensor product question can also be reduced to simplest caseS_si⊗ S_w (corresponding to Monk’s formula) which can be checked directly.

Theorem (W.)

S-positivity of S_wS_v S-positivity ofsλ[Sw]

Theorem (W.)

A weight b-module M with weights⊂Zⁿ≥0 has a KP filtration if and only if Extⁱ(M,S_w^∗ ⊗K(k,k−1,...,k−n+1)) = 0 (∀w,k,∀i ≥1).

Corollary

IfM has a KP filtration then so does its direct summands.

If 0→L→M →N →0 is exact andM,N have KP filtrations then so doesL.

So:

Plethysm question can be reduced to Tensor product question since s_λ(S_w) is a direct summand ofS_w^⊗|λ|.

Tensor product question can also be reduced to simplest caseS_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_wS_v s_λ(S_w) has a KP filtration for anyw, λ. S-positivity ofs_λ[Sw]

Sketch of S

⊗ S

case = ⇒ general case

Corollary

(1) IfM has a KP filtration then so does its direct summands.

(2) If 0→L→M →N →0 is exact andM,N have KP filtrations then so doesL.

Repeated use of Monk case S_si⊗ S_sj⊗ · · · ⊗ S_sk has a KP filtration.

Then so does its direct summandN

i

Vli(S_si) =:T by (1).

Character calculation (T :S_w) = 1 and every other constituentS_u satisfyu⁻¹ >

lexw⁻¹ (for suitable li).

Ext¹(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. byS_u (u⁻¹ >

lexw⁻¹))→0. 0→ S_w⊗ S_v →T ⊗ S_v →(mod. with filtr. byS_u⊗ S_v)→0. induction proceeds by (2).

Sketch of S

⊗ S

case = ⇒ general case

Corollary

(1) IfM has a KP filtration then so does its direct summands.

(2) If 0→L→M →N →0 is exact andM,N have KP filtrations then so doesL.

Repeated use of Monk case S_si⊗ S_sj⊗ · · · ⊗ S_sk has a KP filtration.

Then so does its direct summandN

i

Vli(S_si) =:T by (1).

Character calculation (T :S_w) = 1 and every other constituentS_u satisfyu⁻¹ >

lexw⁻¹ (for suitable li).

Ext¹(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. byS_u (u⁻¹ >

lexw⁻¹))→0. 0→ S_w⊗ S_v →T ⊗ S_v →(mod. with filtr. byS_u⊗ S_v)→0. induction proceeds by (2).

Sketch of S

⊗ S

case = ⇒ general case

Corollary

(1) IfM has a KP filtration then so does its direct summands.

(2) If 0→L→M →N →0 is exact andM,N have KP filtrations then so doesL.

Repeated use of Monk case S_si⊗ S_sj⊗ · · · ⊗ S_sk has a KP filtration.

Then so does its direct summandN

i

Vli(S_si) =:T by (1).

Character calculation (T :S_w) = 1 and every other constituentS_u satisfyu⁻¹ >

lexw⁻¹ (for suitable li).

Ext¹(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. byS_u (u⁻¹ >w⁻¹))→0.

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(Kⁱ) (character expansion known as Pieri rule) Basic strategy: if ch(M) =S_w1+· · ·+S_wr, find v₁, . . . ,v_r ∈M and suitable b-homs φ₁, . . . , φ_r so thatφ_i(v_i) =u_wi,φ_i(v_j) = 0 (j <i) hv₁, . . . ,vii/hv₁, . . . ,vi−1iS_wi simple dimension-counting argument shows that 0⊂ hv₁i ⊂ hv₁,v2i ⊂ · · · gives a desired filration.

Main difficulty: we only knoww₁, . . . ,w_r up to their indexing. Highest weight structure: Ext¹(S_w,S_u) = 0 ifu⁻¹ >

lexw⁻¹ so we can always take w₁⁻¹ ≤

lex

w₂⁻¹≤

lex

· · · (if we takev andφ correctly). Theorem (W.)

Explicit construction of KP filtrations for S_w⊗S^d(Kⁱ) andS_w⊗Vd(Kⁱ). (remark: Pieri rule for KP modules another proof for the h.w. structure)

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(Kⁱ) (character expansion known as Pieri rule) Basic strategy: if ch(M) =S_w1+· · ·+S_wr, find v₁, . . . ,v_r ∈M and suitable b-homs φ₁, . . . , φ_r so thatφ_i(v_i) =u_wi,φ_i(v_j) = 0 (j <i) hv₁, . . . ,vii/hv₁, . . . ,vi−1iS_wi simple dimension-counting argument shows that 0⊂ hv₁i ⊂ hv₁,v2i ⊂ · · · gives a desired filration.

Main difficulty: we only knoww₁, . . . ,w_r up to their indexing.

Highest weight structure: Ext¹(S_w,S_u) = 0 ifu⁻¹ >

lexw⁻¹ so we can always take w₁⁻¹ ≤

lex

w₂⁻¹≤

lex

· · · (if we takev andφ correctly). Theorem (W.)

Explicit construction of KP filtrations for S_w⊗S^d(Kⁱ) andS_w⊗Vd(Kⁱ). (remark: Pieri rule for KP modules another proof for the h.w. structure)

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(Kⁱ) (character expansion known as Pieri rule) Basic strategy: if ch(M) =S_w1+· · ·+S_wr, find v₁, . . . ,v_r ∈M and suitable b-homs φ₁, . . . , φ_r so thatφ_i(v_i) =u_wi,φ_i(v_j) = 0 (j <i) hv₁, . . . ,vii/hv₁, . . . ,vi−1iS_wi simple dimension-counting argument shows that 0⊂ hv₁i ⊂ hv₁,v2i ⊂ · · · gives a desired filration.

Main difficulty: we only knoww₁, . . . ,w_r up to their indexing.

Highest weight structure: Ext¹(S_w,S_u) = 0 ifu⁻¹ >

lexw⁻¹ so we can always take w₁⁻¹ ≤

lex

w₂⁻¹≤

lex

· · · (if we takev andφ correctly).

Theorem (W.)

Explicit construction of KP filtrations for S_w⊗S^d(Kⁱ) andS_w⊗Vd(Kⁱ). (remark: Pieri rule for KP modules another proof for the h.w. structure)

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(Kⁱ) (character expansion known as Pieri rule) Basic strategy: if ch(M) =S_w1+· · ·+S_wr, find v₁, . . . ,v_r ∈M and suitable b-homs φ₁, . . . , φ_r so thatφ_i(v_i) =u_wi,φ_i(v_j) = 0 (j <i) hv₁, . . . ,vii/hv₁, . . . ,vi−1iS_wi simple dimension-counting argument shows that 0⊂ hv₁i ⊂ hv₁,v2i ⊂ · · · gives a desired filration.

Main difficulty: we only knoww₁, . . . ,w_r up to their indexing.

Highest weight structure: Ext¹(S_w,S_u) = 0 ifu⁻¹ >

lexw⁻¹ so we can always take w₁⁻¹ ≤

lex

w₂⁻¹≤

lex

· · · (if we takev andφ correctly).

Theorem (W.)