Catalan functions and k-schur positivity

Transcription

1 Catalan functions and k-schur positivity Jonah Blasiak Drexel University joint work with Jennifer Morse, Anna Pun, and Dan Summers April 2018

2 Strengthened Macdonald positivity conjecture Theorem (Haiman) The modified Macdonald polynomials are Schur positive: H µ (x; q, t) = λ K λµ (q, t)s λ (x) for K λµ (q, t) N[q, t]. Conjecture (Lapointe-Lascoux-Morse) The atom k-schur functions {A λ (x; t)} λ1 k form a basis for Λ k = span Q(q,t) {H µ (x; q, t)} µ1 k, and are Schur positive; expansion of H µ (x; q, t) Λ k in this basis has coefficients in N[q, t]. Conjecture (Lapointe-Lascoux-Morse) The k + 1-Schur expansion of a k-schur function has coefficients in N[t].

3 Strengthened Macdonald positivity conjecture Theorem (Haiman) The modified Macdonald polynomials are Schur positive: H µ (x; q, t) = λ K λµ (q, t)s λ (x) for K λµ (q, t) N[q, t]. Conjecture (Lapointe-Lascoux-Morse) The atom k-schur functions {A λ (x; t)} λ1 k form a basis for Λ k = span Q(q,t) {H µ (x; q, t)} µ1 k, and are Schur positive; expansion of H µ (x; q, t) Λ k in this basis has coefficients in N[q, t]. Conjecture (Lapointe-Lascoux-Morse) The k + 1-Schur expansion of a k-schur function has coefficients in N[t].

4 Strengthened Macdonald positivity conjecture Theorem (Haiman) The modified Macdonald polynomials are Schur positive: H µ (x; q, t) = λ K λµ (q, t)s λ (x) for K λµ (q, t) N[q, t]. Conjecture (Lapointe-Lascoux-Morse) The atom k-schur functions {A λ (x; t)} λ1 k form a basis for Λ k = span Q(q,t) {H µ (x; q, t)} µ1 k, and are Schur positive; expansion of H µ (x; q, t) Λ k in this basis has coefficients in N[q, t]. Conjecture (Lapointe-Lascoux-Morse) The k + 1-Schur expansion of a k-schur function has coefficients in N[t].

5 Strengthened Macdonald positivity conjecture Example. k = 2 H 1 4 = t 4( s + ts + t 2 s H 211 = t ( s + ts + t 2 s H 22 = ( s + ts + t 2 s ) + ( t 2 + t 3)( s + ts ) + ( 1 + qt 2 )( s + ts ) ( ) + (tq + q) s + ts }{{}}{{} positive sum of q, t-monomials t-positive sum of schur functions ) ( + s + ts ) ( + q s + ts + t 2 s + t 2 s +q 2 (s + ts + t 2 s ) ) )

6 Strengthened Macdonald positivity conjecture Example. k = 2 H 1 4 = t 4( s + ts + t 2 ) ( s + t 2 + t 3)( ) ( ) s + ts + s + ts + t 2 s H 211 = t ( s + ts + t 2 ) ( s qt 2 )( ) ( ) s + ts + q s + ts + t 2 s H 22 = ( s + ts + t 2 ) ( ) ) s + (tq + q) s + ts +q (s 2 + ts + t 2 s }{{}}{{} positive sum of q, t-monomials t-positive sum of schur functions }{{}}{{}}{{} s (2) s (2) s (2) basis for restricted span Λ k of Macdonald polynomials

7 Conjecturally equivalent definitions of k-schurs Schur basis symmetric positive branching [1998:Lapointe,Lascoux,Morse] Tableaux and katabolism [2003:Lapointe,Morse] Jing vertex operators [2006:Lam,Lapointe,Morse,Shimozono] Bruhat order on type-a affine Weyl group / strong tableaux [2010:Chen,Haiman] GL l (C)-equivariant Euler characteristics / Demazure operators [2012:Assaf,Billey] Quasisymmetric functions [2015:Dalal,Morse] Inverting affine Kostka matrix

8 Overview The k-schur functions appear in the study of Macdonald polynomials, the homology of the affine Grassmannian, graded representations of the symmetric group. Prior work on the branching rule: Geometric proof at t = 1 (Lam 2011). Formula for branching at t = 1 as equivalence classes on the k-shape poset (Lam-Lapointe-Morse-Shimozono 2013). Main results: Strong tableaux k-schur functions form a Schur positive basis for Λ k. (Branching rule) positive combinatorial formula for the k + 1-Schur expansion of k-schur functions. Strong tableaux k-schur functions agree with a Catalan function definition of Chen-Haiman.

9 Overview The k-schur functions appear in the study of Macdonald polynomials, the homology of the affine Grassmannian, graded representations of the symmetric group. Prior work on the branching rule: Geometric proof at t = 1 (Lam 2011). Formula for branching at t = 1 as equivalence classes on the k-shape poset (Lam-Lapointe-Morse-Shimozono 2013). Main results: Strong tableaux k-schur functions form a Schur positive basis for Λ k. (Branching rule) positive combinatorial formula for the k + 1-Schur expansion of k-schur functions. Strong tableaux k-schur functions agree with a Catalan function definition of Chen-Haiman.

10 Overview The k-schur functions appear in the study of Macdonald polynomials, the homology of the affine Grassmannian, graded representations of the symmetric group. Prior work on the branching rule: Geometric proof at t = 1 (Lam 2011). Formula for branching at t = 1 as equivalence classes on the k-shape poset (Lam-Lapointe-Morse-Shimozono 2013). Main results: Strong tableaux k-schur functions form a Schur positive basis for Λ k. (Branching rule) positive combinatorial formula for the k + 1-Schur expansion of k-schur functions. Strong tableaux k-schur functions agree with a Catalan function definition of Chen-Haiman.

11 Conjecturally equivalent definitions of k-schurs Schur basis symmetric positive branching [1998:Lapointe,Lascoux,Morse] Tableaux and katabolism [2003:Lapointe,Morse] Jing vertex operators [2006:Lam,Lapointe,Morse,Shimozono] Bruhat order on type-a affine Weyl group / strong tableaux [2010:Chen,Haiman] Catalan functions [2012:Assaf,Billey] Quasisymmetric functions [2015:Dalal,Morse] Inverting affine Kostka matrix [2018:B,Morse,Pun,Summers] Strong tableaux = Catalan functions

12 k-bounded partitions and k + 1-cores Def. A k-bounded partition is a partition with parts of size k. Def. A k + 1-core is a partition whose diagram has no box with hook length k + 1. Proposition. There is a bijection κ p(κ) from k + 1-cores to k-bounded partitions. Example. k = κ p(κ) Def. The k-skew diagram of a k + 1-core κ is the skew shape obtained by removing boxes of hook length > k.

13 k-bounded partitions and k + 1-cores Def. A k-bounded partition is a partition with parts of size k. Def. A k + 1-core is a partition whose diagram has no box with hook length k + 1. Proposition. There is a bijection κ p(κ) from k + 1-cores to k-bounded partitions. Example. k = κ p(κ) Def. The k-skew diagram of a k + 1-core κ is the skew shape obtained by removing boxes of hook length > k.

14 k-bounded partitions and k + 1-cores Def. A k-bounded partition is a partition with parts of size k. Def. A k + 1-core is a partition whose diagram has no box with hook length k + 1. Proposition. There is a bijection κ p(κ) from k + 1-cores to k-bounded partitions. Example. k = k-skew(κ) p(κ) Def. The k-skew diagram of a k + 1-core κ is the skew shape obtained by removing boxes of hook length > k.

15 Strong covers Def. An inclusion τ κ of k + 1-cores is a strong cover, denoted τ κ, if p(τ) + 1 = p(κ). Example. Strong cover with k = 4: corresponding k-skew diagrams: = = p(τ) = p(κ) =

16 Strong covers Def. An inclusion τ κ of k + 1-cores is a strong cover, denoted τ κ, if p(τ) + 1 = p(κ). Example. Strong cover with k = 4: corresponding k-skew diagrams: = = p(τ) = p(κ) =

17 Strong marked covers r Def. A strong marked cover τ == κ is a strong cover τ κ together with a positive integer r which is allowed to be the smallest row index of any connected component of the skew shape κ/τ. Example. The two possible markings of the previous strong cover: τ 6 == κ τ 3 == κ

18 Def. spin ( τ Spin r == κ ) = c (h 1) + N, where c = number of connected components of κ/τ, h = height (number of rows) of each component, N = number of components below the marked one. Example. τ 6 == κ τ 3 == κ spin = 4 spin = 5 spin = c (h 1) + N = 2 (3 1) + 0 = 4 spin = 2 (3 1) + 1 = 5

19 Vertical strong marked tableaux Def. A vertical strong marked tableau T of weight η = (η 1, η 2,... ) is a sequence κ (0) r == 1 κ (1) r == 2 r == m κ (m) such that r vi +1 < r vi +2 < < r vi +η i for all i, where v i := η η i 1. inside(t ) := p(κ (0) ) outside(t ) := p(κ (m) ) Example. For k = 4, a vertical strong marked tableau of weight (5): 5 5 κ (4) 5 == κ (5) 5

20 Vertical strong marked tableaux Def. A vertical strong marked tableau T of weight η = (η 1, η 2,... ) is a sequence κ (0) r == 1 κ (1) r == 2 r == m κ (m) such that r vi +1 < r vi +2 < < r vi +η i for all i, where v i := η η i 1. inside(t ) := p(κ (0) ) outside(t ) := p(κ (m) ) Example. For k = 4, a vertical strong marked tableau of weight (5): 4 4 κ (3) 4 == κ (4)

21 Vertical strong marked tableaux Def. A vertical strong marked tableau T of weight η = (η 1, η 2,... ) is a sequence κ (0) r == 1 κ (1) r == 2 r == m κ (m) such that r vi +1 < r vi +2 < < r vi +η i for all i, where v i := η η i 1. inside(t ) := p(κ (0) ) outside(t ) := p(κ (m) ) Example. For k = 4, a vertical strong marked tableau of weight (5): 3 3 κ (2) 3 == κ (3) 3

22 Vertical strong marked tableaux Def. A vertical strong marked tableau T of weight η = (η 1, η 2,... ) is a sequence κ (0) r == 1 κ (1) r == 2 r == m κ (m) such that r vi +1 < r vi +2 < < r vi +η i for all i, where v i := η η i 1. inside(t ) := p(κ (0) ) outside(t ) := p(κ (m) ) Example. For k = 4, a vertical strong marked tableau of weight (5): κ (1) 2 == κ (2)

23 Vertical strong marked tableaux Def. A vertical strong marked tableau T of weight η = (η 1, η 2,... ) is a sequence κ (0) r == 1 κ (1) r == 2 r == m κ (m) such that r vi +1 < r vi +2 < < r vi +η i for all i, where v i := η η i 1. inside(t ) := p(κ (0) ) outside(t ) := p(κ (m) ) Example. For k = 4, a vertical strong marked tableau of weight (5): 1 κ (0) 1 == κ (1)

24 Vertical strong marked tableaux Def. A vertical strong marked tableau T of weight η = (η 1, η 2,... ) is a sequence κ (0) r == 1 κ (1) r == 2 r == m κ (m) such that r vi +1 < r vi +2 < < r vi +η i for all i, where v i := η η i 1. inside(t ) := p(κ (0) ) outside(t ) := p(κ (m) ) Example. For k = 4, a vertical strong marked tableau of weight (5):

25 Spin k-schur functions We work in the ring of symmetric functions in infinitely many variables x = (x 1, x 2,... ). SMT k η(µ) = set of strong marked tableaux T of weight η with outside(t ) = µ. spin(t ) = sum of the spins of the strong marked covers comprising T. Def. For a k-bounded partition µ, let s µ (k) (x; t) = η Z 0, η = µ T SMT k η(µ) t spin(t ) x η. Their t = 1 specializations agree with another combinatorial definition using weak tableaux (Lam-Lapointe-Morse-Shimozono 2010), are Schubert classes in the homology of the affine Grassmannian Gr SLk+1 of SL k+1 (Lam 2008).

26 Spin k-schur functions We work in the ring of symmetric functions in infinitely many variables x = (x 1, x 2,... ). SMT k η(µ) = set of strong marked tableaux T of weight η with outside(t ) = µ. spin(t ) = sum of the spins of the strong marked covers comprising T. Def. For a k-bounded partition µ, let s µ (k) (x; t) = η Z 0, η = µ T SMT k η(µ) t spin(t ) x η. Their t = 1 specializations agree with another combinatorial definition using weak tableaux (Lam-Lapointe-Morse-Shimozono 2010), are Schubert classes in the homology of the affine Grassmannian Gr SLk+1 of SL k+1 (Lam 2008).

27 Properties of k-schur functions Theorem (B.-Morse-Pun-Summers) The k-schur functions {s (k) µ µ is k-bounded of length l} satisfy (vertical dual Pieri rule) (shift invariance) e d s(k) µ = s (k) µ = e l s(k+1) µ+1 l, T VSMT k (d) (µ) t spin(t ) s (k) inside(t ), (Schur function stability) if k µ, then s (k) µ = s µ. e d End(Λ) is defined by e d (g), h = g, e dh for all g, h Λ. VSMT k η(µ) = set of vertical strong marked tableaux T of weight η with outside(t ) = µ.

28 k-schur branching rule Theorem (B.-Morse-Pun-Summers) For µ a k-bounded partition of length l, the expansion of the k-schur function s (k) µ into k + 1-Schur functions is given by s (k) µ = T VSMT k+1 (l) (µ+1l ) t spin(t ) s (k+1) inside(t ). Proof. The shift invariance property followed by the vertical dual Pieri rule yields s (k) µ = e l s(k+1) µ+1 l = T VSMT k+1 (l) (µ+1l ) t spin(t ) s (k+1) inside(t ).

29 k-schur branching rule Theorem (B.-Morse-Pun-Summers) For µ a k-bounded partition of length l, the expansion of the k-schur function s (k) µ into k + 1-Schur functions is given by s (k) µ = T VSMT k+1 (l) (µ+1l ) t spin(t ) s (k+1) inside(t ). Proof. The shift invariance property followed by the vertical dual Pieri rule yields s (k) µ = e l s(k+1) µ+1 l = T VSMT k+1 (l) (µ+1l ) t spin(t ) s (k+1) inside(t ).

30 k-schur branching rule s (3) = t2 s (4) t 2 s (4) t 2 s (4) s (4) VSMT 4 (5) (33332)

31 k-schur branching rule s (3) = t2 s (4) t 2 s (4) t 2 s (4) s (4) VSMT 4 (5) (33332) T = spin(t ) = = 2 inside(t ) = 3222 outside(t ) = 33332

32 Root ideals Set of positive roots + := { (i, j) 1 i < j l }. Ψ + is an upper order ideal of positive roots. Example. Ψ = {(1, 3), (1, 4), (1, 5), (1, 6), (2, 5), (2, 6), (3, 6)} (1, 3) (1, 4) (1, 5) (1, 6) (2, 5) (2, 6) (3, 6)

33 Catalan functions Def. (Panyushev, Chen-Haiman) Ψ + is an upper order ideal of positive roots, γ Z l. The Catalan function indexed by Ψ and γ: Hγ Ψ (x; t) := (1 tr ij ) 1 s γ (x) (i, j) Ψ where the raising operator R ij acts by R ij (s γ (x)) = s γ+ɛi ɛ j (x). Example. Let µ = (µ 1,..., µ l ) be a partition. Empty root set: H µ (x; t) = s µ (x). Full root set: H + µ (x; t) = H µ (x; t), the modified Hall-Littlewood polynomial.

34 Catalan functions Def. (Panyushev, Chen-Haiman) Ψ + is an upper order ideal of positive roots, γ Z l. The Catalan function indexed by Ψ and γ: Hγ Ψ (x; t) := (1 tr ij ) 1 s γ (x) (i, j) Ψ where the raising operator R ij acts by R ij (s γ (x)) = s γ+ɛi ɛ j (x). Example. Let µ = (µ 1,..., µ l ) be a partition. Empty root set: H µ (x; t) = s µ (x). Full root set: H + µ (x; t) = H µ (x; t), the modified Hall-Littlewood polynomial.

35 k-schur Catalan functions Def. For µ a k-bounded partition of length l, define the root ideal and the Catalan function k (µ) = {(i, j) + k µ i + i < j}, s (k) µ (x; t) := H k (µ) µ = l l i=1 j=k+1 µ i +i ( 1 trij ) 1sµ (x). # nonroots in row i = k µ i

36 Example. k = 4, µ = Examples of Catalan functions 1, 3 1, 4 k (µ) = 2, 4 s (k) µ (x; t) = (i, j) k (µ) (1 tr ij ) 1 s µ (x)

37 Example. k = 4, µ = Examples of Catalan functions 1, 3 1, 4 k (µ) = 2, 4 s (k) µ (x; t) = (i, j) k (µ) (1 tr ij ) 1 s µ (x) = (1 tr 13 ) 1 (1 tr 24 ) 1 (1 tr 14 ) 1 s 3321 (x)

38 Example. k = 4, µ = Examples of Catalan functions 1, 3 1, 4 k (µ) = 2, 4 s (k) µ (x; t) = (i, j) k (µ) (1 tr ij ) 1 s µ (x) = (1 tr 13 ) 1 (1 tr 24 ) 1 (1 tr 14 ) 1 s 3321 (x) = s t(s s s 4320 ) + t 2 (s s s 5310 ) + t 3 (s s s 6300 ) + t 4 (s s )

39 Example. k = 4, µ = Examples of Catalan functions 1, 3 1, 4 k (µ) = 2, 4 s (k) µ (x; t) = (i, j) k (µ) (1 tr ij ) 1 s µ (x) = (1 tr 13 ) 1 (1 tr 24 ) 1 (1 tr 14 ) 1 s 3321 (x) = s t(s s s 4320 ) + t 2 (s s s 5310 ) + t 3 (s s s 6300 ) + t 4 (s s ) = s t(s s 4311 ) + t 2 (s s 5310 ) + t 3 s 5400.

40 Chen-Haiman conjecture Theorem (B.-Morse-Pun-Summers) For any k-bounded partition µ, the k-schur function s (k) µ (x; t) is the Catalan function s (k) µ (x; t).

41 k-schur into Schur Theorem (B.-Morse-Pun-Summers) Let µ be a k-bounded partition of length l and set m = max( µ k, 0). The Schur expansion the k-schur function s µ (k) is given by s (k) µ = T VSMT k+m (l m ) (µ+ml ) t spin(t ) s inside(t ). Proof. Applying the shift invariance property m times followed by the vertical dual Pieri rule, we obtain s (k) µ = (e l )m s (k+m) µ+m l = T VSMT k+m (l m ) (µ+ml ) t spin(t ) s inside(t ). The Schur function stability property ensures this is the Schur function decomposition.

42 k-schur into Schur Theorem (B.-Morse-Pun-Summers) Let µ be a k-bounded partition of length l and set m = max( µ k, 0). The Schur expansion the k-schur function s µ (k) is given by s (k) µ = T VSMT k+m (l m ) (µ+ml ) t spin(t ) s inside(t ). Proof. Applying the shift invariance property m times followed by the vertical dual Pieri rule, we obtain s (k) µ = (e l )m s (k+m) µ+m l = T VSMT k+m (l m ) (µ+ml ) t spin(t ) s inside(t ). The Schur function stability property ensures this is the Schur function decomposition.

43 Schur expansion of s (1) 111 = H t 3 s 3 t 2 s 21 t s 21 s 111 s (1) 111 = t3 s 3 + t 2 s 21 + ts 21 + s 111 The Schur expansion of the 1-Schur function s (1) 111 is obtained by summing t spin(t ) s inside(t ) over the set VSMT 3 (3,3)(3, 3, 3) of vertical strong marked tableaux T given above.

44 Schur function straightening Schur functions may be defined for any γ Z l. The Schur function s γ (x 1, x 2,..., x l ) = s γ (x) is straightened as follows: { sgn(γ + ρ)s sort(γ+ρ) ρ (x) if γ + ρ has distinct nonnegative parts, s γ (x) = 0 otherwise, sort(β) = weakly decreasing sequence obtained by sorting β, sgn(β) = sign of the shortest permutation taking β to sort(β). Example. l = 4, γ = γ + ρ = (3, 1, 2, 5) + (3, 2, 1, 0) = (6, 3, 3, 5) has a repeated part. Hence s 3125 (x) = 0.

45 Schur function straightening Schur functions may be defined for any γ Z l. The Schur function s γ (x 1, x 2,..., x l ) = s γ (x) is straightened as follows: { sgn(γ + ρ)s sort(γ+ρ) ρ (x) if γ + ρ has distinct nonnegative parts, s γ (x) = 0 otherwise, sort(β) = weakly decreasing sequence obtained by sorting β, sgn(β) = sign of the shortest permutation taking β to sort(β). Example. l = 4, γ = γ + ρ = (3, 1, 2, 5) + (3, 2, 1, 0) = (6, 3, 3, 5) has a repeated part. Hence s 3125 (x) = 0.

46 Schur function straightening Schur functions may be defined for any γ Z l. The Schur function s γ (x 1, x 2,..., x l ) = s γ (x) is straightened as follows: { sgn(γ + ρ)s sort(γ+ρ) ρ (x) if γ + ρ has distinct nonnegative parts, s γ (x) = 0 otherwise, sort(β) = weakly decreasing sequence obtained by sorting β, sgn(β) = sign of the shortest permutation taking β to sort(β). Example. l = 4, γ = γ + ρ = (4, 7, 1, 6) + (3, 2, 1, 0) = (7, 9, 2, 6) sort(γ + ρ) = (9, 7, 6, 2) sort(γ + ρ) ρ = (6, 5, 5, 2) Hence s 4716 (x) = s 6552 (x).