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 November 2018

2 Theorem (Haiman) Macdonald positivity conjecture 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]. H 22 (x; q, t) = t 2 s +(t+qt+qt 2 )s +(1+q 2 t 2 )s +(q+qt+q 2 t)s +q 2 s H 22 (x; 0, t) = t 2 s + ts + s Theorem (Lascoux-Schützenberger) The modified Hall-Littlewood polynomials have the Schur expansion H µ (x; t) = H µ (x; 0, t) = T t charge(t ) s shape(t ) (x), the sum over semistandard Young tableaux T of content µ.

3 Theorem (Haiman) Macdonald positivity conjecture 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]. H 22 (x; q, t) = t 2 s +(t+qt+qt 2 )s +(1+q 2 t 2 )s +(q+qt+q 2 t)s +q 2 s H 22 (x; 0, t) = t 2 s + ts + s Theorem (Lascoux-Schützenberger) The modified Hall-Littlewood polynomials have the Schur expansion H µ (x; t) = H µ (x; 0, t) = T t charge(t ) s shape(t ) (x), the sum over semistandard Young tableaux T of content µ.

4 Strengthened Macdonald positivity conjecture Conjecture (Lapointe-Lascoux-Morse) The k-schur functions {A (k) λ (x; t)} λ 1 k form a basis for Λ k = span Q(q,t) {H µ (x; q, t)} µ1 k, are Schur positive, expansion of H µ (x; q, t) Λ k in this basis has coefficients in N[q, t]. 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 + (q + qt) s + ts +q (s 2 + ts + t 2 s }{{}}{{} positive sum of q, t-monomials t-positive sum of Schur functions }{{} A (2)

5 Strengthened Macdonald positivity conjecture Conjecture (Lapointe-Lascoux-Morse) The k-schur functions {A (k) λ (x; t)} λ 1 k form a basis for Λ k = span Q(q,t) {H µ (x; q, t)} µ1 k, are Schur positive, expansion of H µ (x; q, t) Λ k in this basis has coefficients in N[q, t]. 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 + (q + qt) s + ts +q (s 2 + ts + t 2 s }{{}}{{} positive sum of q, t-monomials t-positive sum of Schur functions }{{} A (2)

6 Strengthened Macdonald positivity conjecture Conjecture (Lapointe-Lascoux-Morse) The k-schur functions {A (k) λ (x; t)} λ 1 k form a basis for Λ k = span Q(q,t) {H µ (x; q, t)} µ1 k, are Schur positive, expansion of H µ (x; q, t) Λ k in this basis has coefficients in N[q, t]. 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 + (q + qt) s + ts +q (s 2 + ts + t 2 s }{{}}{{} positive sum of q, t-monomials t-positive sum of Schur functions }{{} A (2)

7 Strengthened Macdonald positivity conjecture Conjecture (Lapointe-Lascoux-Morse) The k-schur functions {A (k) λ (x; t)} λ 1 k form a basis for Λ k = span Q(q,t) {H µ (x; q, t)} µ1 k, are Schur positive, expansion of H µ (x; q, t) Λ k in this basis has coefficients in N[q, t]. 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 + (q + qt) s + ts +q (s 2 + ts + t 2 s }{{}}{{} positive sum of q, t-monomials t-positive sum of Schur functions }{{}}{{}}{{} A (2) A (2) A (2)

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

9 Conjecturally equivalent definitions of k-schurs Schur H µ (x; q, t) are basis positive branching k-schur positive [1998:Lapointe,Lascoux,Morse] Young tableaux and katabolism [2001:Lapointe,Morse] (q = 0) Jing operators / k-split polynomials [2006:Lam,Lapointe,Morse,Shimozono] (q = 0) Bruhat order on type A affine Weyl group / strong tableaux [2008:Chen,Haiman] (q = 0) Catalan functions [2014:Dalal,Morse and Lapointe,Pinto] (q = 0) Inverting affine Kostka matrix [2004:Lapointe,Morse] (q = 0) Weak tableaux (t = 1) [2005:Lam] (q = 0) Schubert classes in H (Gr) (t = 1)

10 Overview Part I: introduction to the k-schur and Catalan functions. Part II: properties of k-schur functions including branching, Schur positivity, and shift invariance. Part III: k-schur expansions of several families of symmetric functions and relation to Gromov-Witten invariants.

11 Conjecturally equivalent definitions of k-schurs Schur H µ (x; q, t) are basis positive branching k-schur positive [1998:Lapointe,Lascoux,Morse] Young tableaux and katabolism [2001:Lapointe,Morse] (q = 0) Jing operators / k-split polynomials [2006:Lam,Lapointe,Morse,Shimozono] (q = 0) Bruhat order on type A affine Weyl group / strong tableaux [2008:Chen,Haiman] (q = 0) Catalan functions [2014:Dalal,Morse and Lapointe,Pinto] (q = 0) Inverting affine Kostka matrix [2004:Lapointe,Morse] (q = 0) Weak tableaux (t = 1) [2005:Lam] (q = 0) Schubert classes in H (Gr) (t = 1)

12 Catalan functions A family of symmetric functions studied by Panyushev and Chen-Haiman Contain the modified Hall-Littlewood polynomials H µ (x; t) and their parabolic generalizations. Can be defined in terms of Demazure operators or raising operators on Schur functions. Are equal to GL l -equivariant Euler characteristics of vector bundles on the flag variety.

13 Schur function straightening We work in the ring Λ = Q(q, t)[h 1, h 2,... ] of symmetric functions in infinitely many variables x = (x 1, x 2,... ). Schur functions may be defined for any γ Z l by s γ = s γ (x) = det(h γi +j i(x)) 1 i,j l Λ. Proposition (Schur function straightening) { 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.

14 Schur function straightening We work in the ring Λ = Q(q, t)[h 1, h 2,... ] of symmetric functions in infinitely many variables x = (x 1, x 2,... ). Schur functions may be defined for any γ Z l by s γ = s γ (x) = det(h γi +j i(x)) 1 i,j l Λ. Proposition (Schur function straightening) { 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.

15 Schur function straightening We work in the ring Λ = Q(q, t)[h 1, h 2,... ] of symmetric functions in infinitely many variables x = (x 1, x 2,... ). Schur functions may be defined for any γ Z l by s γ = s γ (x) = det(h γi +j i(x)) 1 i,j l Λ. Proposition (Schur function straightening) { 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).

16 Root ideals Set of positive roots + := { (i, j) 1 i < j l }. A root ideal 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)

17 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.

18 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.

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

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

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

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

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

24 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.

25 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.

26 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.

27 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) =

28 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) =

29 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

30 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

31 Strong marked tableaux Def. For a word w = w 1 w m Z m 1, a strong tableau marked by w is a sequence of strong marked covers of the form κ (0) w m ==== κ (1) w m 1 ===== w 1 === κ (m). Example. For k = 4, a strong tableau marked by 54321: 5 5 κ (4) 5 == κ (5) 5 For a strong tableau T, inside(t ) := p(κ (0) ) and outside(t ) := p(κ (m) ). spin(t ) = sum of spins of strong covers comprising T. SMT k (w ; µ) = set of strong tableaux T marked by w with outside(t ) = µ.

32 Strong marked tableaux Def. For a word w = w 1 w m Z m 1, a strong tableau marked by w is a sequence of strong marked covers of the form κ (0) w m ==== κ (1) w m 1 ===== w 1 === κ (m). Example. For k = 4, a strong tableau marked by 54321: 4 4 κ (3) 4 == κ (4) For a strong tableau T, inside(t ) := p(κ (0) ) and outside(t ) := p(κ (m) ). spin(t ) = sum of spins of strong covers comprising T. SMT k (w ; µ) = set of strong tableaux T marked by w with outside(t ) = µ.

33 Strong marked tableaux Def. For a word w = w 1 w m Z m 1, a strong tableau marked by w is a sequence of strong marked covers of the form κ (0) w m ==== κ (1) w m 1 ===== w 1 === κ (m). Example. For k = 4, a strong tableau marked by 54321: 3 3 κ (2) 3 == κ (3) 3 For a strong tableau T, inside(t ) := p(κ (0) ) and outside(t ) := p(κ (m) ). spin(t ) = sum of spins of strong covers comprising T. SMT k (w ; µ) = set of strong tableaux T marked by w with outside(t ) = µ.

34 Strong marked tableaux Def. For a word w = w 1 w m Z m 1, a strong tableau marked by w is a sequence of strong marked covers of the form κ (0) w m ==== κ (1) w m 1 ===== w 1 === κ (m). Example. For k = 4, a strong tableau marked by 54321: κ (1) 2 == κ (2) For a strong tableau T, inside(t ) := p(κ (0) ) and outside(t ) := p(κ (m) ). spin(t ) = sum of spins of strong covers comprising T. SMT k (w ; µ) = set of strong tableaux T marked by w with outside(t ) = µ.

35 Strong marked tableaux Def. For a word w = w 1 w m Z m 1, a strong tableau marked by w is a sequence of strong marked covers of the form κ (0) w m ==== κ (1) w m 1 ===== w 1 === κ (m). Example. For k = 4, a strong tableau marked by 54321: 1 κ (0) 1 == κ (1) For a strong tableau T, inside(t ) := p(κ (0) ) and outside(t ) := p(κ (m) ). spin(t ) = sum of spins of strong covers comprising T. SMT k (w ; µ) = set of strong tableaux T marked by w with outside(t ) = µ.

36 Strong marked tableaux Def. For a word w = w 1 w m Z m 1, a strong tableau marked by w is a sequence of strong marked covers of the form κ (0) w m ==== κ (1) w m 1 ===== w 1 === κ (m). Example. For k = 4, a strong tableau marked by 54321: For a strong tableau T, inside(t ) := p(κ (0) ) and outside(t ) := p(κ (m) ). spin(t ) = sum of spins of strong covers comprising T. SMT k (w ; µ) = set of strong tableaux T marked by w with outside(t ) = µ.

37 Strong marked tableaux Def. For a word w = w 1 w m Z m 1, a strong tableau marked by w is a sequence of strong marked covers of the form κ (0) w m ==== κ (1) w m 1 ===== w 1 === κ (m). Example. For k = 4, a strong tableau marked by 54321: inside(t ) = 3222 outside(t ) = spin(t ) = = 2 For a strong tableau T, inside(t ) := p(κ (0) ) and outside(t ) := p(κ (m) ). spin(t ) = sum of spins of strong covers comprising T. SMT k (w ; µ) = set of strong tableaux T marked by w with outside(t ) = µ.

38 Strong Pieri operators There is a natural t-analog of Q[h 1,..., h k ] Q[h 1, h 2,... ] compatible with k-schur functions. Proposition Λ k := span Q(q,t) { Hµ (x; q, t) µ 1 k } = span Q(q,t) { Hµ (x; t) µ 1 k } = span Q(q,t) { s (k) µ (x; t) µ 1 k }. Def. The strong Pieri operators u 1, u 2, End(Λ k ) are defined by s (k) µ u p = t spin(t ) s (k) inside(t ). Hence for a word w = w 1 w m, T SMT k (p ;µ) s (k) µ u w = s (k) µ u w1 u wm = T SMT k (w ;µ) t spin(t ) s (k) inside(t ).

39 Strong Pieri operators There is a natural t-analog of Q[h 1,..., h k ] Q[h 1, h 2,... ] compatible with k-schur functions. Proposition Λ k := span Q(q,t) { Hµ (x; q, t) µ 1 k } = span Q(q,t) { Hµ (x; t) µ 1 k } = span Q(q,t) { s (k) µ (x; t) µ 1 k }. Def. The strong Pieri operators u 1, u 2, End(Λ k ) are defined by s (k) µ u p = t spin(t ) s (k) inside(t ). Hence for a word w = w 1 w m, T SMT k (p ;µ) s (k) µ u w = s (k) µ u w1 u wm = T SMT k (w ;µ) t spin(t ) s (k) inside(t ).

40 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 ( ) (dual Pieri rule) ed s(k) µ = s (k) µ u i1 u id, i 1 > >i d (shift invariance) s (k) µ = e l s(k+1) µ+1 l, (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. u i = operator for removing a strong cover marked in row i.

41 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 = s (k+1) µ+1 l u l u 1 = T SMT k+1 (l 2 1 ; µ+1 l ) t spin(t ) s (k+1) inside(t ). Proof. The shift invariance property followed by the dual Pieri rule yields s (k) µ = e l s(k+1) µ+1 l = s (k+1) µ+1 l u l u 1.

42 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 = s (k+1) µ+1 l u l u 1 = T SMT k+1 (l 2 1 ; µ+1 l ) t spin(t ) s (k+1) inside(t ). Proof. The shift invariance property followed by the dual Pieri rule yields s (k) µ = e l s(k+1) µ+1 l = s (k+1) µ+1 l u l u 1.

43 k-schur branching rule s (3) = t3 s (4) t 2 s (4) t 2 s (4) s (4) SMT 4 (54321; 33332)

44 k-schur branching rule s (3) = t3 s (4) t 2 s (4) t 2 s (4) s (4) SMT 4 (54321; 33332) T = spin(t ) = = 2 inside(t ) = 3222 outside(t ) = 33332

45 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 (k) µ is given by s (k) µ = T SMT k+m ( (l 1) m ; µ+m l ) t spin(t ) s inside(t ). Proof. Applying the shift invariance property m times followed by the dual Pieri rule, we obtain s µ (k) = (el )m s (k+m) = s (k+m) (u µ+m l µ+m l l u 1 ) m = T SMT k+m ((l 1) m ;µ+m l ) t spin(t ) s inside(t ). The Schur function stability property ensures this is the Schur function decomposition.

46 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 (k) µ is given by s (k) µ = T SMT k+m ( (l 1) m ; µ+m l ) t spin(t ) s inside(t ). Proof. Applying the shift invariance property m times followed by the dual Pieri rule, we obtain s µ (k) = (el )m s (k+m) = s (k+m) (u µ+m l µ+m l l u 1 ) m = T SMT k+m ((l 1) m ;µ+m l ) t spin(t ) s inside(t ). The Schur function stability property ensures this is the Schur function decomposition.

47 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 SMT 3 (321321; 333) of strong tableaux T above.

48 Unifying the definitions of k-schur functions s (k) µ (x; t) defined as a sum of monomials over strong tableau. Equivalent to the symmetric functions satisfying the dual Pieri rule. Ã (k) µ (x; t) defined recursively using Jing vertex operators. Combining our results with those of Lam and Lam-Lapointe-Morse-Shimozono: Theorem The k-schur functions defined from Jing vertex operators, k-schur Catalan functions, and strong tableau k-schur functions coincide: Ã (k) µ (x; t) = s (k) µ (x; t) = s (k) µ (x; t) for all k-bounded µ. Moreover, their t = 1 specializations {s (k) µ (x; 1)} match a definition using weak tableaux, and represent Schubert classes in the homology of the affine Grassmannian Gr G of G = SL k+1.

49 Shift invariance!?!! s (k) µ = e l s(k+1) µ+1 l What is the geometric meaning of shift invariance?

50 k-schur positive expansions Symmetric functions known or conjectured to be k-schur positive: modified Macdonald polynomials H µ (x; q, t) for k-bounded µ, LLT polynomials of bandwidth k, products of k-schur functions at t = 1, Catalan functions with partition weight and band k.

51 (k-)schur positivity conjectures Conjecture (Chen-Haiman) The Catalan function Hµ Ψ is Schur positive for any root ideal Ψ and partition µ. Strengthens earlier conjectures of Broer and Shimozono-Weyman.

52 (k-)schur positivity conjectures Conjecture (Chen-Haiman) The Catalan function Hµ Ψ is Schur positive for any root ideal Ψ and partition µ. Strengthens earlier conjectures of Broer and Shimozono-Weyman. Conjecture (B.-Morse-Pun-Summers) The Catalan function Hµ Ψ is k-schur positive whenever µ is a partition and max(band(ψ, µ)) k, where band(ψ, µ) i := µ i + # of roots in row i of + \ Ψ. Example band This Catalan function is 6-Schur positive.

53 k-schur positive expansions We have obtained formulas for the k-schur expansions of modified Hall-Littlewood polynomials proving the q = 0 case of the strengthened Macdonald positivity conjecture, the product of a Schur function and a k-schur function when the indexing partitions concatenate to a partition, describing a class of Gromov-Witten invariants for the quantum cohomology of complete flag varieties, k-split polynomials, solving a substantial special case of a problem of Broer and Shimozono-Weyman on parabolic Hall-Littlewood polynomials.

54 k-schur positive expansions We have obtained formulas for the k-schur expansions of modified Hall-Littlewood polynomials proving the q = 0 case of the strengthened Macdonald positivity conjecture, the product of a Schur function and a k-schur function when the indexing partitions concatenate to a partition, describing a class of Gromov-Witten invariants for the quantum cohomology of complete flag varieties, k-split polynomials, solving a substantial special case of a problem of Broer and Shimozono-Weyman on parabolic Hall-Littlewood polynomials. Is k-schur positivity easier to prove than Schur positivity?

55 k-schur positive expansions We have obtained formulas for the k-schur expansions of modified Hall-Littlewood polynomials proving the q = 0 case of the strengthened Macdonald positivity conjecture, the product of a Schur function and a k-schur function when the indexing partitions concatenate to a partition, describing a class of Gromov-Witten invariants for the quantum cohomology of complete flag varieties, k-split polynomials, solving a substantial special case of a problem of Broer and Shimozono-Weyman on parabolic Hall-Littlewood polynomials. Is k-schur positivity easier to prove than Schur positivity? Yes?

56 Strengthened Macdonald positivity Conjecture (Lapointe-Lascoux-Morse) H µ (x; q, t) is k-schur positive whenever µ 1 k. Interesting even when q = 0. H µ (x; t) = H µ (x; 0, t) = H + µ (x; t) is the modified Hall-Littlewood polynomial. Theorem (Lascoux-Schützenberger) H µ = T t charge(t ) s shape(t ), the sum over semistandard Young tableaux T of content µ.

57 Strengthened Macdonald positivity Conjecture (Lapointe-Lascoux-Morse) H µ (x; q, t) is k-schur positive whenever µ 1 k. Interesting even when q = 0. H µ (x; t) = H µ (x; 0, t) = H + µ (x; t) is the modified Hall-Littlewood polynomial. Theorem (Lascoux-Schützenberger) H µ = T t charge(t ) s shape(t ), the sum over semistandard Young tableaux T of content µ.

58 Strengthened Macdonald positivity Z θ = superstandard tableau of shape θ. colword(t ) is the word obtained by concatenating the columns of T, reading each from bottom to top, starting with the leftmost. Theorem (B.-Morse-Pun-Summers) Let µ be a k-bounded partition of length l. Set w = colword(z k l /µ). H µ = s (k) u k l w = t spin(t ) s (k) inside(t ). Example. k = 3, µ = Z (3333)/(2211) = H 2211 = s (3) 3333 u 4u 3 u 4 u 3 u 2 u 1 = T SMT k (w ; k l ) and colword(z (3333)/(2211) ) = SMT 3 ( ; 3333) t spin(t ) s (3) inside(t )

59 Strengthened Macdonald positivity Z θ = superstandard tableau of shape θ. colword(t ) is the word obtained by concatenating the columns of T, reading each from bottom to top, starting with the leftmost. Theorem (B.-Morse-Pun-Summers) Let µ be a k-bounded partition of length l. Set w = colword(z k l /µ). H µ = s (k) u k l w = t spin(t ) s (k) inside(t ). Example. k = 3, µ = Z (3333)/(2211) = H 2211 = s (3) 3333 u 4u 3 u 4 u 3 u 2 u 1 = T SMT k (w ; k l ) and colword(z (3333)/(2211) ) = SMT 3 ( ; 3333) t spin(t ) s (3) inside(t )

60 The 3-Schur expansion of H spin H 2211 = t 4 s (3) 33 + t3 s (3) t2 s (3) t s(3) t s(3) s(3) 2211.

61 The 3-Schur expansion of H spin 4 inside = 321 spin = H 2211 = t 4 s (3) 33 + t3 s (3) t2 s (3) t s(3) t s(3) s(3) 2211.

62 k-littlewood Richardson coefficients Def. The k-littlewood Richardson coefficients c λ µν are defined by s (k) µ (x; 1)s (k) ν (x; 1) = λ cµν λ s (k) λ (x; 1). Theorem (Lam) The k-schur functions at t = 1 represent Schubert classes in the homology of the affine Grassmannian Gr SLk+1. Hence the structure constants for H (Gr SLk+1 ) in the Schubert basis are the k-littlewood Richardson coefficients.

63 Quantum equals Affine Theorem (Peterson) There is a ring isomorphism between a localization of H (Gr SLk+1 ) and a localization of the quantum cohomology ring QH (Fl k+1 ), which matches the Schubert bases. The 3-point Gromov-Witten invariants of genus 0 are the structure constants for QH (Fl k+1 ) in the Schubert basis. They contain the Schubert structure constants as a special case. Corollary The 3-point Gromov-Witten invariants of genus 0 agree with the k-littlewood Richardson coefficients, and these nonnegative integers. Open Problem. Find a positive combinatorial formula for the k-littlewood Richardson coefficients and Gromov-Witten invariants.

64 Quantum equals Affine Theorem (Peterson) There is a ring isomorphism between a localization of H (Gr SLk+1 ) and a localization of the quantum cohomology ring QH (Fl k+1 ), which matches the Schubert bases. The 3-point Gromov-Witten invariants of genus 0 are the structure constants for QH (Fl k+1 ) in the Schubert basis. They contain the Schubert structure constants as a special case. Corollary The 3-point Gromov-Witten invariants of genus 0 agree with the k-littlewood Richardson coefficients, and these nonnegative integers. Open Problem. Find a positive combinatorial formula for the k-littlewood Richardson coefficients and Gromov-Witten invariants.

65 Quantum equals Affine Theorem (Peterson) There is a ring isomorphism between a localization of H (Gr SLk+1 ) and a localization of the quantum cohomology ring QH (Fl k+1 ), which matches the Schubert bases. The 3-point Gromov-Witten invariants of genus 0 are the structure constants for QH (Fl k+1 ) in the Schubert basis. They contain the Schubert structure constants as a special case. Corollary The 3-point Gromov-Witten invariants of genus 0 agree with the k-littlewood Richardson coefficients, and these nonnegative integers. Open Problem. Find a positive combinatorial formula for the k-littlewood Richardson coefficients and Gromov-Witten invariants.

66 Schur times k-schur into k-schur SSYT θ (r) = semistandard Young tableaux of shape θ with entries from {1,..., r}. B µ = Shimozono-Zabrocki generalized Hall-Littlewood vertex operator, which is multiplication by s µ at t = 1. Theorem (B.-Morse-Pun-Summers) Let µ be a partition of length r with µ 1 k r + 1, and ν a partition such that µν is a partition. Set R = (k r + 1) r. Then B µ s (k) ν = s (k) Rν u colword(t ). T SSYT R/µ (r) Example. Let k = 6, r = 3, µ = 432, ν = 22. Then R = 444. { SSYT R/µ (r) = B µ s (k) ν = s (k) Rν (u u u u u u u u 332 ) }.

67 Schur times k-schur into k-schur Example. 6-Schur expansion of a t-analog of s 432 s 22. B 432 s (6) 22 = s(6) (u u u u u u u u 332 ) B 432 s (6) 22 = t3 s (6) t2 s (6) t2 s (6) t1 s (6) t1 s (6) s(6)

68 Schur times k-schur into k-schur Example. 6-Schur expansion of a t-analog of s 432 s 22. B 432 s (6) 22 = s(6) (u u u u u u u u 332 ) inside = spin = = 2 B 432 s (6) 22 = t3 s (6) t2 s (6) t2 s (6) t1 s (6) t1 s (6) s(6)

69 Gromov-Witten invariants A word is cyclically increasing if some rotation of it is increasing. Inv i (w) = {j > i : w i > w j }. θ : S k+1 k-bounded partitions Corollary Let u, v, w S k+1 and d Z k 0. Suppose u has only one descent at position j and v m+1 v k+1 is cyclically increasing, where m is the maximum index such that Inv 1 (u) = = Inv m (u). Then the Gromov-Witten invariant is given by u, v, w w d = 1, T SSYT R/θ(u) (r) S SMT k (colword(t ) ;Rθ(v)) inside(s)=λ where r = k + 1 j Inv 1 (u), R = (k r + 1) r, and λ is determined from θ(w), d, and the descent sets of v, w.

70 Gromov-Witten invariants Let u, v, w S k+1 and d Z k 0. Suppose u has only one descent at position j and v m+1 v k+1 is cyclically increasing, where m is the maximum index such that Inv 1 (u) = = Inv m (u). Example. k = 6, u = , v = The only descent of u is at position j = 4, Inv 1 (u) = Inv 2 (u) = 0, and v 3 v 7 = is cyclically increasing. θ(u) = 432 and θ(v) = B 432 s (6) 21 5 = s (6) t 2 s (6) t 2 s (6) t s (6) t s (6) t 3 s (6) ( ) σ u σ v =σ σ σ q 2 q 3 q 4 q 5 q 6 σ σ σ where σ u σ v = u, v, w w d σ w d N k w S k+1

71 Gromov-Witten invariants Let u, v, w S k+1 and d Z k 0. Suppose u has only one descent at position j and v m+1 v k+1 is cyclically increasing, where m is the maximum index such that Inv 1 (u) = = Inv m (u). Example. k = 6, u = , v = The only descent of u is at position j = 4, Inv 1 (u) = Inv 2 (u) = 0, and v 3 v 7 = is cyclically increasing. θ(u) = 432 and θ(v) = B 432 s (6) 21 5 = s (6) t 2 s (6) t 2 s (6) t s (6) t s (6) t 3 s (6) ( ) σ u σ v =σ σ σ q 2 q 3 q 4 q 5 q 6 σ σ σ where σ u σ v = u, v, w w d σ w d N k w S k+1

72 Thank you! Happy birthday Sergey!