Crystal structures on shifted tableaux

Transcription

1 Crystal structures on shifted tableaux Jake Levinson (UW) joint with Maria Gillespie (UC Davis), Kevin Purbhoo (Waterloo) Séminaire du LaCIM 5 Janvier 2018

2 Goal Crystal-like structure on (skew) semistandard shifted tableaux Crystal-like: Purely combinatorial Source from geometry, H pogpn, 2n ` 1qq, rather than representation theory Highest-weight elements are Type B, but crystal stucture resembles doubled Type A.

3 Notation and background in type A Skew shape: difference of two partition shapes λ{µ (below, λ p5, 3, 3q and µ p2, 1q) Semistandard Young tableau: Skew shape filled with positive integers, increasing down columns and weakly increasing across rows T P SSYT pλ{µq

4 Notation and background in type A Skew shape: difference of two partition shapes λ{µ (below, λ p5, 3, 3q and µ p2, 1q) Semistandard Young tableau: Skew shape filled with positive integers, increasing down columns and weakly increasing across rows T P SSYT pλ{µq Reading word: concatenate rows, bottom up ( )

5 Notation and background in type A Skew shape: difference of two partition shapes λ{µ (below, λ p5, 3, 3q and µ p2, 1q) Semistandard Young tableau: Skew shape filled with positive integers, increasing down columns and weakly increasing across rows T P SSYT pλ{µq Reading word: concatenate rows, bottom up ( ) Weight: The sequence pm 1, m 2,...q where m i is the number of i s in the tableau, p2, 2, 3, 1q above.

6 Jeu de taquin slides Inner slide:

7 Jeu de taquin slides Inner slide:

8 Jeu de taquin slides Inner slide:

9 Jeu de taquin slides Inner slide:

10 Jeu de taquin slides Outer slide:

11 Jeu de taquin slides Outer slide:

12 Jeu de taquin slides Outer slide:

13 Jeu de taquin slides Outer slide:

14 Jeu de taquin slides Outer slide: Rectification: Sequence of jeu de taquin slides leading to a straight shape tableau rectpt q.

15 Jeu de taquin slides Outer slide: Rectification: Sequence of jeu de taquin slides leading to a straight shape tableau rectpt q.

16 Jeu de taquin slides Outer slide: Rectification: Sequence of jeu de taquin slides leading to a straight shape tableau rectpt q.

17 Jeu de taquin slides Outer slide: Rectification: Sequence of jeu de taquin slides leading to a straight shape tableau rectpt q.

18 Jeu de taquin slides Outer slide: Rectification: Sequence of jeu de taquin slides leading to a straight shape tableau rectpt q.

19 Jeu de taquin slides Outer slide: Rectification: Sequence of jeu de taquin slides leading to a straight shape tableau rectpt q.

20 Jeu de taquin slides Outer slide: Rectification: Sequence of jeu de taquin slides leading to a straight shape tableau rectpt q.

21 Jeu de taquin slides Outer slide: Rectification: Sequence of jeu de taquin slides leading to a straight shape tableau rectpt q.

22 Jeu de taquin slides Outer slide: Rectification: Sequence of jeu de taquin slides leading to a straight shape tableau rectpt q.

23 Jeu de taquin slides Outer slide: Rectification: Sequence of jeu de taquin slides leading to a straight shape tableau rectpt q. T is Littlewood-Richardson or ballot if rectpt q is a highest weight tableau:

24 Combinatorial crystals: Type A Crystal operators E i, F i act on SSYT pλ{µq: F 1 ÝÑ, F 2 ÝÑ (F i : changes an i i ` 1) E i : partial inverse of F i Operators work on skew tableaux T and on (reading) words w. Highest-weight element: E i pt q for all i For straight shapes: connected, unique max: T

25 Universal properties (Type A) The operators E i, F i are uniquely determined by three properties: 1. Action of E i, F i is only on i, i ` 1 entries, and is the same as E 1, F 1 on 1, The action of E 1, F 1 on rectified tableaux is: F 1

26 Universal properties (Type A) The operators E i, F i are uniquely determined by three properties: 1. Action of E i, F i is only on i, i ` 1 entries, and is the same as E 1, F 1 on 1, The action of E 1, F 1 on rectified tableaux is: E F F F1 E E

27 Universal properties (Type A) The operators E i, F i are uniquely determined by three properties: 1. Action of E i, F i is only on i, i ` 1 entries, and is the same as E 1, F 1 on 1, The action of E 1, F 1 on rectified tableaux is: E F F F1 E E Coplacticity: E 1, F 1 commute with all jeu de taquin slides. This determines E 1, F 1 on general tableaux:

28 Universal properties (Type A) The operators E i, F i are uniquely determined by three properties: 1. Action of E i, F i is only on i, i ` 1 entries, and is the same as E 1, F 1 on 1, The action of E 1, F 1 on rectified tableaux is: E F F F1 E E Coplacticity: E 1, F 1 commute with all jeu de taquin slides. This determines E 1, F 1 on general tableaux: E F 1 F F1 E E Well-definedness + local description of E 1, F 1 are nontrivial!

29 Crystal operators E i, F i in general Local rule for skew shapes: In reading word, substitute 2 ( and 1 ), match parentheses p p q q q q p q p p q

30 Crystal operators E i, F i in general Local rule for skew shapes: In reading word, substitute 2 ( and 1 ), match parentheses p p q q q q p q p p q

31 Crystal operators E i, F i in general Local rule for skew shapes: In reading word, substitute 2 ( and 1 ), match parentheses p p q q q q p q p p q

32 Crystal operators E i, F i in general Local rule for skew shapes: In reading word, substitute 2 ( and 1 ), match parentheses p p q q q q p q p p q F 1 changes the rightmost unmatched 1 to 2 if possible.

33 Crystal operators E i, F i in general Local rule for skew shapes: In reading word, substitute 2 ( and 1 ), match parentheses p p q q q p p q p p q F 1 changes the rightmost unmatched 1 to 2 if possible. E 1 changes the leftmost unmatched 2 to 1 if possible.

34 Crystal operators E i, F i in general Local rule for skew shapes: In reading word, substitute 2 ( and 1 ), match parentheses p p q q q q p q p p q F 1 changes the rightmost unmatched 1 to 2 if possible. E 1 changes the leftmost unmatched 2 to 1 if possible. Consequence of crystal structure: LR rule for skew shapes, s λ{µ ÿ ν c λ ν,µs ν.

35 Alternately: Stembridge s axiomatic description View crystal as a Z n -weighted, edge-labeled (by 1,..., n 1) directed graph G. Basic structure:

36 Alternately: Stembridge s axiomatic description View crystal as a Z n -weighted, edge-labeled (by 1,..., n 1) directed graph G. Basic structure: Axiom. Following an edge w i ÝÑ x lowers the weight: wtpxq wtpwq α i p0..., 1, 1,... 0q.

37 Alternately: Stembridge s axiomatic description View crystal as a Z n -weighted, edge-labeled (by 1,..., n 1) directed graph G. Basic structure: Axiom. Following an edge w i ÝÑ x lowers the weight: wtpxq wtpwq α i p0..., 1, 1,... 0q. Axiom. For each i, the i-connected components are strings: ÝÑ i ÝÑ i i e i pwq ÝÑ w ÝÑ i i f i pwq ÝÑ Set ɛ i pwq #e i -steps available, ϕ i pwq #f i -steps. Above: ɛ i pwq 3 and ϕ i pwq 2.

38 Alternately: Stembridge s axiomatic description View crystal as a Z n -weighted, edge-labeled (by 1,..., n 1) directed graph G. Basic structure: Axiom. Following an edge w i ÝÑ x lowers the weight: wtpxq wtpwq α i p0..., 1, 1,... 0q. Axiom. For each i, the i-connected components are strings: ÝÑ i ÝÑ i i e i pwq ÝÑ w ÝÑ i i f i pwq ÝÑ Set ɛ i pwq #e i -steps available, ϕ i pwq #f i -steps. Above: ɛ i pwq 3 and ϕ i pwq 2. Axiom. For i j ą 1: edges commute:

39 Axioms relating arrows Axiom. Suppose w i 1 ÝÝÑx. Then i ÝÑ, ÝÝÑ i`1 pɛ i pwq ɛ i pxq, ϕ i pwq ϕ i pxqq p0, 1q or p1, 0q.

40 Axioms relating arrows Axiom. Suppose w i 1 ÝÝÑx. Then Two cases: i ÝÑ, ÝÝÑ i`1 pɛ i pwq ɛ i pxq, ϕ i pwq ϕ i pxqq p0, 1q or p1, 0q.

41 Axioms relating arrows Axiom. Suppose w i 1 ÝÝÑx. Then Two cases: i ÝÑ, ÝÝÑ i`1 pɛ i pwq ɛ i pxq, ϕ i pwq ϕ i pxqq p0, 1q or p1, 0q. Focus on change to ɛ (always 0 or 1).

42 Axioms relating arrows Axiom. Suppose w i ÝÑ, ÝÝÑ i`1 ÝÑ i x and wýýñy. i`1 Compare ɛ values: : pɛ i`1 pwq ɛ i`1 pxq, ɛ i pwq ɛ i pyqq p1, 1q, p1, 0q, p0, 1q or p0, 0q.

43 Axioms relating arrows Axiom. Suppose w i ÝÑ, ÝÝÑ i`1 ÝÑ i x and wýýñy. i`1 Compare ɛ values: : pɛ i`1 pwq ɛ i`1 pxq, ɛ i pwq ɛ i pyqq p1, 1q, p1, 0q, p0, 1q or p0, 0q. Then: p0, 0q p0, 0q Also: dual versions of same axioms (for going backwards).

44 Stembridge axioms Theorem (Stembridge 03) Let G be a finite connected graph satisfying the local axioms. Then G has a unique highest-weight element g, with wtpg q λ a partition, and canonically G SSYT pλ, nq.

45 Stembridge axioms Theorem (Stembridge 03) Let G be a finite connected graph satisfying the local axioms. Then G has a unique highest-weight element g, with wtpg q λ a partition, and canonically G SSYT pλ, nq. Corollary (without connected hypothesis): Weight generating function of G is Schur-positive! (each connected component Ø some SSYT pλ, nq).

46 Stembridge axioms Theorem (Stembridge 03) Let G be a finite connected graph satisfying the local axioms. Then G has a unique highest-weight element g, with wtpg q λ a partition, and canonically G SSYT pλ, nq. Corollary (without connected hypothesis): Weight generating function of G is Schur-positive! (each connected component Ø some SSYT pλ, nq). Morse-Schilling (2015): Crystal-theoretic proof of Schur positivity for Stanley symmetric functions

47 Type B: Shifted shapes and tableaux Strict partition: λ pλ 1 ą ą λ k q, displayed as a shifted Young diagram. Skew shape λ{µ: Difference of two shifted partitions. λ = (6, 4, 2, 1)

48 Type B: Shifted shapes and tableaux Strict partition: λ pλ 1 ą ą λ k q, displayed as a shifted Young diagram. Skew shape λ{µ: Difference of two shifted partitions. λ = (6, 4, 2, 1) µ = (3, 2)

49 Type B: Shifted shapes and tableaux Strict partition: λ pλ 1 ą ą λ k q, displayed as a shifted Young diagram. Skew shape λ{µ: Difference of two shifted partitions λ = (6, 4, 2, 1) µ = (3, 2) Shifted semistandard tableau: Filling by 1 1 ă 1 ă 2 1 ă 2 ă, allowing i 1 i 1 and i i. Southwesternmost i{i 1 must be unprimed. ShST pλ{µ, nq : tt of shape λ{µ, entries P t1 1, 1,..., n 1, nuu.

50 Type B: Shifted shapes and tableaux Strict partition: λ pλ 1 ą ą λ k q, displayed as a shifted Young diagram. Skew shape λ{µ: Difference of two shifted partitions λ = (6, 4, 2, 1) µ = (3, 2) Shifted semistandard tableau: Filling by 1 1 ă 1 ă 2 1 ă 2 ă, allowing i 1 i 1 and i i. Southwesternmost i{i 1 must be unprimed. ShST pλ{µ, nq : tt of shape λ{µ, entries P t1 1, 1,..., n 1, nuu. Reading word: along rows, from bottom: wordpt q Shifted jeu de taquin: slides are uniquely-determined to preserve semistandardness

51 Combinatorial crystal structure on (skew) shifted tableaux Shifted tableau crystals: two lowering operators F i, F 1 i for each i, acting only on the entries ti 1, i, i`1, i`1 1 u. Focus on i 1. For a straight shape λ with one or two rows, ShST pλ, 2q is simple:

52 Combinatorial crystal structure on (skew) shifted tableaux Shifted tableau crystals: two lowering operators F i, F 1 i for each i, acting only on the entries ti 1, i, i`1, i`1 1 u. Focus on i 1. For a straight shape λ with one or two rows, ShST pλ, 2q is simple:

53 Combinatorial crystal structure on (skew) shifted tableaux Shifted tableau crystals: two lowering operators F i, F 1 i for each i, acting only on the entries ti 1, i, i`1, i`1 1 u. Focus on i 1. For a straight shape λ with one or two rows, ShST pλ, 2q is simple: F 1 F 1 2 F 1 F F F 1 2 F

54 Combinatorial crystal structure on (skew) shifted tableaux Shifted tableau crystals: two lowering operators F i, F 1 i for each i, acting only on the entries ti 1, i, i`1, i`1 1 u. Focus on i 1. For a straight shape λ with one or two rows, ShST pλ, 2q is simple: F 1 F 1 2 F 1 F F 1 2 F 1 2 F F 1 F 1 F 1 F 1 F 1 F 1

55 Combinatorial crystal structure on (skew) shifted tableaux Shifted tableau crystals: two lowering operators F i, F 1 i for each i, acting only on the entries ti 1, i, i`1, i`1 1 u. Focus on i 1. For a straight shape λ with one or two rows, ShST pλ, 2q is simple: F 1 F 1 2 F 1 F F 1 2 F 1 2 F F 1 F 1 F F 1 F 1 F 1 F 1 By coplacticity: uniquely determines operators ÝÝÝÑ, i skew shifted tableaux. F i 1 on all

56 Example: the crystals Bp, 3q and Bp, 3q

57 Explicit description of F i, F 1 i Explicit descriptions of F i, F 1 i on B ShST pλ{µ, nq are known, but complicated: Theorem (Gillespie-L-Purbhoo 17) There are direct definitions of F i, Fi 1, depending only on w wordpt q, via first-quadrant lattice walks.

58 Explicit description of F i, F 1 i Explicit descriptions of F i, F 1 i on B ShST pλ{µ, nq are known, but complicated: Theorem (Gillespie-L-Purbhoo 17) There are direct definitions of F i, Fi 1, depending only on w wordpt q, via first-quadrant lattice walks. Type B enumerative structure: Each connected component of B has a unique maximum. The maxima are precisely the Littlewood-Richardson tableaux.

59 Explicit description of F i, F 1 i Explicit descriptions of F i, F 1 i on B ShST pλ{µ, nq are known, but complicated: Theorem (Gillespie-L-Purbhoo 17) There are direct definitions of F i, Fi 1, depending only on w wordpt q, via first-quadrant lattice walks. Type B enumerative structure: Each connected component of B has a unique maximum. The maxima are precisely the Littlewood-Richardson tableaux. Recovers the skew Littlewood-Richardson rule for Schur Q functions: Q λ{µ pxq ÿ fλµ ν Q ν. ν

60 Example: repeated F 1, followed by one F 1 1 : The full definition of F i on words is complicated. Goal: Describe Stembridge-like axioms for these crystals.

61 Example: repeated F 1, followed by one F 1 1 : The full definition of F i on words is complicated. Goal: Describe Stembridge-like axioms for these crystals.

62 Example: repeated F 1, followed by one F 1 1 : The full definition of F i on words is complicated. Goal: Describe Stembridge-like axioms for these crystals.

63 Example: repeated F 1, followed by one F 1 1 : The full definition of F i on words is complicated. Goal: Describe Stembridge-like axioms for these crystals.

64 Example: repeated F 1, followed by one F 1 1 : The full definition of F i on words is complicated. Goal: Describe Stembridge-like axioms for these crystals.

65 Example: repeated F 1, followed by one F 1 1 : The full definition of F i on words is complicated. Goal: Describe Stembridge-like axioms for these crystals.

66 Axioms for shifted tableau crystals G Z n -weighted digraph, edge labels t1 1, 1,..., n 1 1, n 1u. Basic structure:

67 Axioms for shifted tableau crystals G Z n -weighted digraph, edge labels t1 1, 1,..., n 1 1, n 1u. Basic structure: Axiom. Following an edge w i or i 1 ÝÝÝÑ x lowers the weight: wtpxq wtpwq α i p0..., 1, q. Axiom. For i j ą 1, all edges commute.

68 Axioms for shifted tableau crystals G Z n -weighted digraph, edge labels t1 1, 1,..., n 1 1, n 1u. Basic structure: Axiom. Following an edge w i or i 1 ÝÝÝÑ x lowers the weight: wtpxq wtpwq α i p0..., 1, q. Axiom. Axiom. For i j ą 1, all edges commute. The ti, i 1 u-connected components are doubled strings: or (iff wt i 0 at bottom)

69 Axioms for shifted tableau crystals G Z n -weighted digraph, edge labels t1 1, 1,..., n 1 1, n 1u. Basic structure: Axiom. Following an edge w i or i 1 ÝÝÝÑ x lowers the weight: wtpxq wtpwq α i p0..., 1, q. Axiom. Axiom. For i j ą 1, all edges commute. The ti, i 1 u-connected components are doubled strings: or (iff wt i 0 at bottom) Set ɛ i pwq, ϕ i pwq total # steps to top, bottom Also ɛ 1 i, ϕ1 i (#t u only) and pɛ i, pϕ i (#týñu only).

70 Axioms for shifted tableau crystals Remains to state axioms for i, i ` 1.

71 Axioms for shifted tableau crystals Remains to state axioms for i, i ` 1. Axiom. Suppose w i 1 or i 11 ÝÝÝÝÝÝÝÝÑ x. Then pɛ i pwq ɛ i pxq, ϕ i pwq ϕ i pxqq p0, 1q or p1, 0q. Same two possibilities as Type A:

72 Recall: Type A Axiom. Suppose w i 1 ÝÝÑ x. Then pɛ i pwq ɛ i pxq, ϕ i pwq ϕ i pxqq p0, 1q or p1, 0q. Two possibilities: Focus on change to ɛ (always 0 or 1).

73 1 or 11 2 or 21 Relations between ÝÝÝÑ, ÝÝÝÑ 1 or 11 2 or 21 Suppose we have two edges: wýýýýñx, wýýýýñy. Define: pɛ 2 pwq ɛ 2 pxq, ɛ 1 pwq ɛ 1 pyqq p1, 1q, p1, 0q, p0, 1q or p0, 0q.

74 1 or 11 2 or 21 Relations between ÝÝÝÑ, ÝÝÝÑ 1 or 11 2 or 21 Suppose we have two edges: wýýýýñx, wýýýýñy. Define: pɛ 2 pwq ɛ 2 pxq, ɛ 1 pwq ɛ 1 pyqq p1, 1q, p1, 0q, p0, 1q or p0, 0q. Axioms. For tf1 1, f 2 1u, tf 1, f2 1 u, tf 1 1, f 2u (assume f 2 f2 1): Conditions Axiom Conditions Axiom p0, 0q, ϕ 2 pwq 1, xϕ 2 pwq 0 pɛ 1 pwq ą 0

75 1 or 11 2 or 21 Relations between ÝÝÝÑ, ÝÝÝÑ 1 or 11 2 or 21 Suppose we have two edges: wýýýýñx, wýýýýñy. Define: pɛ 2 pwq ɛ 2 pxq, ɛ 1 pwq ɛ 1 pyqq p1, 1q, p1, 0q, p0, 1q or p0, 0q. Axioms. For tf1 1, f 2 1u, tf 1, f2 1 u, tf 1 1, f 2u (assume f 2 f2 1): Conditions Axiom Conditions Axiom p0, 0q, ϕ 2 pwq 1, xϕ 2 pwq 0 pɛ 1 pwq ą 0 Note: No axiom for tf 1, f 1 2 u when pɛ 1pwq 0!

76 Relations between 1 ÝÑ, 2 ÝÑ The interesting case: w 1 ÝÝÑx, w 2 ÝÝÑy.

77 Relations between 1 ÝÑ, 2 ÝÑ The interesting case: w 1 ÝÝÑx, w 2 ÝÝÑy. Observation. By applying previous axioms, reduce to case where f 1 1 ( ) not defined.

78 Relations between 1 ÝÑ, 2 ÝÑ The interesting case: w 1 ÝÝÑx, w 2 ÝÝÑy. Observation. By applying previous axioms, reduce to case where f 1 1 ( ) not defined. Then, the axioms for tf 1, f 2 u resemble Type A!

79 Relations between 1 ÝÑ, 2 ÝÑ Assume f1 1 is not defined at w, and f 2 f2 1 ( strict solid edge). Axioms for tf 1, f 2 u: Conditions Axiom Conditions Axiom p0, 1q, p1, 0q p1, 1q p0, 0q, pϕ 1 pf 2 pwqq ě 2 p0, 0q, pɛ 1 pwq pɛ 1 pyq

80 Uniqueness of shifted tableau crystals Theorem (Gillespie, ) Let G be a finite connected graph satisfying these local axioms. G has a unique maximal element g, and wtpg q σ is a strict partition. There is a canonical isomorphism ShSYT pσ, nq ÝÝÑ G. In particular, the generating function ÿ 2 lpwtpgqq x wtpgq is Schur Q positive. gpg

81 Coda: Geometry of (odd orthogonal) Grassmannians Fix V C 2n`1 a vector space, x, y a symmetric form on V. Isotropic subspaces W Ă V : xw, w 1 y 0 for all w, w 1 P W. (Odd) Orthogonal Grassmannian: OGpn, V q tw Ă V isotropicu Ă Grpn, V q.

82 Coda: Geometry of (odd orthogonal) Grassmannians Fix V C 2n`1 a vector space, x, y a symmetric form on V. Isotropic subspaces W Ă V : xw, w 1 y 0 for all w, w 1 P W. (Odd) Orthogonal Grassmannian: OGpn, V q tw Ă V isotropicu Ă Grpn, V q. For enumerative geometry, we studied certain real algebraic curves S Ă OGpn, V q (intersections of Schubert varieties): S " W Ă V : * W satisfies given incidence conditions w.r.t certain specified linear spaces.

83 Coda: Geometry of (odd orthogonal) Grassmannians Theorem (Gillespie,,Purbhoo) Combinatorial description of SpRq in terms of shifted jeu de taquin and the operators F i.

84 Thank you!