Specialized Macdonald polynomials, quantum K-theory, and Kirillov-Reshetikhin crystals

Transcription

1 Specialized Macdonald polynomials, quantum K-theory, and Kirillov-Reshetikhin crystals Cristian Lenart Max-Planck-Institut für Mathematik, Bonn State University of New York at Albany Geometry Seminar, EPF Lausanne March 2014

2 Macdonald polynomials P ( x;q, 0) λ Braverman Finkelberg 12 Ion 03 q Whittaker functions LNSSS Braverman Finkelberg 11 (graded characters of) Kirillov Reshetikhin modules same combinatorial model L. Postnikov 07 quantum K theory of flag varieties

3 Complex simple and affine Lie algebras g complex simple Lie algebra (type A n 1 G 2 ).

4 Complex simple and affine Lie algebras g complex simple Lie algebra (type A n 1 G 2 ). Main example. Type A n 1 (n 1): sl n = { n by n matrices of trace 0 }.

5 Complex simple and affine Lie algebras g complex simple Lie algebra (type A n 1 G 2 ). Main example. Type A n 1 (n 1): sl n = { n by n matrices of trace 0 }. The (untwisted) affine algebras ĝ, of type A (1) n 1,..., G (1) 2 :

6 Complex simple and affine Lie algebras g complex simple Lie algebra (type A n 1 G 2 ). Main example. Type A n 1 (n 1): sl n = { n by n matrices of trace 0 }. The (untwisted) affine algebras ĝ, of type A (1) n 1,..., G (1) 2 : ĝ = C[t, t 1 ] g Cc Cd.

7 Complex simple and affine Lie algebras g complex simple Lie algebra (type A n 1 G 2 ). Main example. Type A n 1 (n 1): sl n = { n by n matrices of trace 0 }. The (untwisted) affine algebras ĝ, of type A (1) n 1,..., G (1) 2 : ĝ = C[t, t 1 ] g Cc Cd. Chevalley generators: e i, f i (i = 1,..., r for g of type X r, and i = 0,..., r for ĝ of type X (1) r ).

8 Complex simple and affine Lie algebras g complex simple Lie algebra (type A n 1 G 2 ). Main example. Type A n 1 (n 1): sl n = { n by n matrices of trace 0 }. The (untwisted) affine algebras ĝ, of type A (1) n 1,..., G (1) 2 : ĝ = C[t, t 1 ] g Cc Cd. Chevalley generators: e i, f i (i = 1,..., r for g of type X r, and i = 0,..., r for ĝ of type X (1) r ). Structure encoded by the corresponding root system Φ.

9 Root systems Φ V = R r ; for root α Φ, reflection s α.

10 Root systems Φ V = R r ; for root α Φ, reflection s α. Coroot: α := 2α/ α, α.

11 Root systems Φ V = R r ; for root α Φ, reflection s α. Coroot: α := 2α/ α, α. Simple roots: α 1,..., α r Φ.

12 Root systems Φ V = R r ; for root α Φ, reflection s α. Coroot: α := 2α/ α, α. Simple roots: α 1,..., α r Φ. Simple reflections: s i := s αi.

13 Root systems Φ V = R r ; for root α Φ, reflection s α. Coroot: α := 2α/ α, α. Simple roots: α 1,..., α r Φ. Simple reflections: s i := s αi. Fundamental weights: ω i, where ω i, αj = δ ij.

14 Root systems Φ V = R r ; for root α Φ, reflection s α. Coroot: α := 2α/ α, α. Simple roots: α 1,..., α r Φ. Simple reflections: s i := s αi. Fundamental weights: ω i, where ω i, αj = δ ij. Weights: i c iω i, c i Z; dominant if c i 0.

15 Root systems Φ V = R r ; for root α Φ, reflection s α. Coroot: α := 2α/ α, α. Simple roots: α 1,..., α r Φ. Simple reflections: s i := s αi. Fundamental weights: ω i, where ω i, αj = δ ij. Weights: i c iω i, c i Z; dominant if c i 0. Example. Type A n 1. V = (ε ε n ) in R n = ε 1,..., ε n (r = n 1). Φ = {α ij = ε i ε j = (i, j) : 1 i j n}.

16 Root systems Φ V = R r ; for root α Φ, reflection s α. Coroot: α := 2α/ α, α. Simple roots: α 1,..., α r Φ. Simple reflections: s i := s αi. Fundamental weights: ω i, where ω i, α j = δ ij. Weights: i c iω i, c i Z; dominant if c i 0. Example. Type A n 1. V = (ε ε n ) in R n = ε 1,..., ε n (r = n 1). Φ = {α ij = ε i ε j = (i, j) : 1 i j n}. Weights / dominant weights / fundamental weights: compositions (λ 1, λ 2,...) / partitions λ 1 λ 2... / columns ω k = (1 k ) = ε ε k.

17 The Weyl group W = s α : α Φ = s i : i = 1,..., r.

18 The Weyl group W = s α : α Φ = s i : i = 1,..., r. Length: l(w) = min {k : w = s i1... s ik }.

19 The Weyl group W = s α : α Φ = s i : i = 1,..., r. Length: l(w) = min {k : w = s i1... s ik }. Example. Type A n 1. W = S n, s εi ε j is the transposition t ij, s i = t i,i+1.

20 Representations Representation of g (or ĝ) on V : compatibly with the Lie bracket. X g, ĝ act as X : V V

21 Representations Representation of g (or ĝ) on V : X g, ĝ act as X : V V compatibly with the Lie bracket. Weight space decomposition (by restriction to the Cartan subalgebra h): V = V µ. µ weight

22 Representations Representation of g (or ĝ) on V : compatibly with the Lie bracket. X g, ĝ act as X : V V Weight space decomposition (by restriction to the Cartan subalgebra h): V = V µ. µ weight The highest weight representations of g (and ĝ) are indexed by dominant weights λ; denoted V (λ).

23 Representations Representation of g (or ĝ) on V : compatibly with the Lie bracket. X g, ĝ act as X : V V Weight space decomposition (by restriction to the Cartan subalgebra h): V = V µ. µ weight The highest weight representations of g (and ĝ) are indexed by dominant weights λ; denoted V (λ). The corresponding (irreducible) characters are invariant under the action of the Weyl group W.

24 Macdonald polynomials λ: dominant weight for a finite root system.

25 Macdonald polynomials λ: dominant weight for a finite root system. P λ (x; q, t): Weyl group invariant polynomials, orthogonal, generalizing the corresponding irreducible characters = P λ (x; 0, 0).

26 Macdonald polynomials λ: dominant weight for a finite root system. P λ (x; q, t): Weyl group invariant polynomials, orthogonal, generalizing the corresponding irreducible characters = P λ (x; 0, 0). Hecke algebra: q-deformation of the group algebra of the Weyl group (w T w, for w W ).

27 Macdonald polynomials λ: dominant weight for a finite root system. P λ (x; q, t): Weyl group invariant polynomials, orthogonal, generalizing the corresponding irreducible characters = P λ (x; 0, 0). Hecke algebra: q-deformation of the group algebra of the Weyl group (w T w, for w W ). Double affine Hecke algebra (DAHA): double affinization of the Hecke algebra (X λ, Y µ, for weights λ, µ).

28 Macdonald polynomials λ: dominant weight for a finite root system. P λ (x; q, t): Weyl group invariant polynomials, orthogonal, generalizing the corresponding irreducible characters = P λ (x; 0, 0). Hecke algebra: q-deformation of the group algebra of the Weyl group (w T w, for w W ). Double affine Hecke algebra (DAHA): double affinization of the Hecke algebra (X λ, Y µ, for weights λ, µ). P λ (x; q, t) defined in the DAHA setup, as common eigenfunctions of the Cherednik operators Y µ.

29 Macdonald polynomials λ: dominant weight for a finite root system. P λ (x; q, t): Weyl group invariant polynomials, orthogonal, generalizing the corresponding irreducible characters = P λ (x; 0, 0). Hecke algebra: q-deformation of the group algebra of the Weyl group (w T w, for w W ). Double affine Hecke algebra (DAHA): double affinization of the Hecke algebra (X λ, Y µ, for weights λ, µ). P λ (x; q, t) defined in the DAHA setup, as common eigenfunctions of the Cherednik operators Y µ. Recursive construction procedure (for the non-symmetric Macdonald polynomials E µ (x; q, t)), based on Cherednik s intertwiners I i.

30 Braverman-Finkelberg q-whittaker functions Ψ λ (x; q): eigenfunctions of the quantum difference Toda integrable system (Etingof, Sevostyanov).

31 Braverman-Finkelberg q-whittaker functions Ψ λ (x; q): eigenfunctions of the quantum difference Toda integrable system (Etingof, Sevostyanov). Let Ψ λ (x; q) := Ψ λ (x; q) i I λ,α i (1 q r ). r=1

32 Braverman-Finkelberg q-whittaker functions Ψ λ (x; q): eigenfunctions of the quantum difference Toda integrable system (Etingof, Sevostyanov). Let Ψ λ (x; q) := Ψ λ (x; q) i I λ,α i r=1 (1 q r ). Theorem (Braverman-Finkelberg, Ion) We have P λ (x; q, t = 0) = Ψ λ (x; q).

33 Schubert calculus Flag variety G/B, Schubert variety X w = B wb/b, for w W.

34 Schubert calculus Flag variety G/B, Schubert variety X w = B wb/b, for w W. H (G/B) and K(G/B) have bases of Schubert classes;

35 Schubert calculus Flag variety G/B, Schubert variety X w = B wb/b, for w W. H (G/B) and K(G/B) have bases of Schubert classes; for K-theory, they are the classes [O w ] = [O Xw ] of structure sheaves of X w.

36 Schubert calculus Flag variety G/B, Schubert variety X w = B wb/b, for w W. H (G/B) and K(G/B) have bases of Schubert classes; for K-theory, they are the classes [O w ] = [O Xw ] of structure sheaves of X w. The quantum cohomology algebra QH (G/B) still has the Schubert basis, but over C[q 1,..., q r ].

37 Schubert calculus Flag variety G/B, Schubert variety X w = B wb/b, for w W. H (G/B) and K(G/B) have bases of Schubert classes; for K-theory, they are the classes [O w ] = [O Xw ] of structure sheaves of X w. The quantum cohomology algebra QH (G/B) still has the Schubert basis, but over C[q 1,..., q r ]. The structure constants (for multiplying Schubert classes) are the 3-point Gromov-Witten (GW) invariants.

38 Schubert calculus Flag variety G/B, Schubert variety X w = B wb/b, for w W. H (G/B) and K(G/B) have bases of Schubert classes; for K-theory, they are the classes [O w ] = [O Xw ] of structure sheaves of X w. The quantum cohomology algebra QH (G/B) still has the Schubert basis, but over C[q 1,..., q r ]. The structure constants (for multiplying Schubert classes) are the 3-point Gromov-Witten (GW) invariants. A k-point GW invariant (of degree d) counts curves of degree d passing through k given Schubert varieties.

39 Quantum K-theory Givental and Lee defined K-theoretic GW invariants by applying the K-theory Euler characteristic when the space of curves (through given Schubert varieties) is infinite.

40 Quantum K-theory Givental and Lee defined K-theoretic GW invariants by applying the K-theory Euler characteristic when the space of curves (through given Schubert varieties) is infinite. The structure constants for the quantum K-theory QK(G/B) are defined based on the 2- and 3-point invariants (complex formula).

41 Quantum K-theory Givental and Lee defined K-theoretic GW invariants by applying the K-theory Euler characteristic when the space of curves (through given Schubert varieties) is infinite. The structure constants for the quantum K-theory QK(G/B) are defined based on the 2- and 3-point invariants (complex formula). The K-theoretic J-function is the generating function of 1-point K-theoretic GW invariants.

42 Quantum K-theory Givental and Lee defined K-theoretic GW invariants by applying the K-theory Euler characteristic when the space of curves (through given Schubert varieties) is infinite. The structure constants for the quantum K-theory QK(G/B) are defined based on the 2- and 3-point invariants (complex formula). The K-theoretic J-function is the generating function of 1-point K-theoretic GW invariants. Theorem (Braverman-Finkelberg) In simply-laced types, the q-whittaker function Ψ λ (x; q) (viewed as a function of λ) coincides with the K-theoretic J-function.

43 Kirillov-Reshetikhin (KR) modules W r,s : finite dimensional modules for ĝ (r I, s 1).

44 Kirillov-Reshetikhin (KR) modules W r,s : finite dimensional modules for ĝ (r I, s 1). Let p = (p 1,..., p k ) be a composition, and W p = W p1,1... W pk,1, λ = ω p ω pk.

45 Kirillov-Reshetikhin (KR) modules W r,s : finite dimensional modules for ĝ (r I, s 1). Let p = (p 1,..., p k ) be a composition, and W p = W p1,1... W pk,1, λ = ω p ω pk. X λ (x; q): the (graded) character of W p.

46 Kirillov-Reshetikhin (KR) modules W r,s : finite dimensional modules for ĝ (r I, s 1). Let p = (p 1,..., p k ) be a composition, and W p = W p1,1... W pk,1, λ = ω p ω pk. X λ (x; q): the (graded) character of W p. Main Theorem (L.-Naito-Sagaki-Schilling-Shimozono) For all untwisted affine root systems A (1) n 1 G (1) 2, we have P λ (x; q, 0) = X λ (x; q).

47 Kirillov-Reshetikhin (KR) modules W r,s : finite dimensional modules for ĝ (r I, s 1). Let p = (p 1,..., p k ) be a composition, and W p = W p1,1... W pk,1, λ = ω p ω pk. X λ (x; q): the (graded) character of W p. Main Theorem (L.-Naito-Sagaki-Schilling-Shimozono) For all untwisted affine root systems A (1) n 1 G (1) 2, we have P λ (x; q, 0) = X λ (x; q). Remarks. (1) The proof is based on two combinatorial models for W p, derived based on earlier work of Naito-Sagaki.

48 Kirillov-Reshetikhin (KR) modules W r,s : finite dimensional modules for ĝ (r I, s 1). Let p = (p 1,..., p k ) be a composition, and W p = W p1,1... W pk,1, λ = ω p ω pk. X λ (x; q): the (graded) character of W p. Main Theorem (L.-Naito-Sagaki-Schilling-Shimozono) For all untwisted affine root systems A (1) n 1 G (1) 2, we have P λ (x; q, 0) = X λ (x; q). Remarks. (1) The proof is based on two combinatorial models for W p, derived based on earlier work of Naito-Sagaki. (2) The result is believed to extend to the twisted types.

49 History Ion 03 Macdonald polynomials P ( x;q, 0) λ affine Demazure characters ={types A D E} Fourier Littelmann 06 Fourier Schilling Shimozono 07 LNSSS (graded characters of) Kirillov Reshetikhin modules

50 History Ion 03 Macdonald polynomials P ( x;q, 0) λ affine Demazure characters ={types A D E} Fourier Littelmann 06 Fourier Schilling Shimozono 07 LNSSS (graded characters of) Kirillov Reshetikhin modules Applications of LNSSS. [Chari, Ion], [Chari, Schneider, Shereen, Wand]

51 The underlying combinatorics All the mentioned structures are described in terms of the quantum alcove model:

52 The underlying combinatorics All the mentioned structures are described in terms of the quantum alcove model: the specialized Macdonald polynomials P λ (x; q, 0) and the q-whittaker functions (Ram-Yip, L.);

53 The underlying combinatorics All the mentioned structures are described in terms of the quantum alcove model: the specialized Macdonald polynomials P λ (x; q, 0) and the q-whittaker functions (Ram-Yip, L.); the quantum K-theory of G/B (conjecture by L.-Postnikov; evidence by L.-Maeno);

54 The underlying combinatorics All the mentioned structures are described in terms of the quantum alcove model: the specialized Macdonald polynomials P λ (x; q, 0) and the q-whittaker functions (Ram-Yip, L.); the quantum K-theory of G/B (conjecture by L.-Postnikov; evidence by L.-Maeno); the tensor products of one-column KR modules [LNSSS].

55 The underlying combinatorics All the mentioned structures are described in terms of the quantum alcove model: the specialized Macdonald polynomials P λ (x; q, 0) and the q-whittaker functions (Ram-Yip, L.); the quantum K-theory of G/B (conjecture by L.-Postnikov; evidence by L.-Maeno); the tensor products of one-column KR modules [LNSSS]. The model is uniform for all Lie types A n 1 G 2.

56 The quantum Bruhat graph The combinatorics is based on the quantum Bruhat graph on W, denoted QBG(W ).

57 The quantum Bruhat graph The combinatorics is based on the quantum Bruhat graph on W, denoted QBG(W ). This is the directed graph with labeled edges w α ws α, where l(ws α ) = l(w) + 1 (covers of Bruhat order), or l(ws α ) = l(w) 2ht(α ) + 1. (If α = i c iα i, then ht(α ) := i c i.)

58 The quantum Bruhat graph The combinatorics is based on the quantum Bruhat graph on W, denoted QBG(W ). This is the directed graph with labeled edges w α ws α, where l(ws α ) = l(w) + 1 (covers of Bruhat order), or l(ws α ) = l(w) 2ht(α ) + 1. (If α = i c iα i, then ht(α ) := i c i.) It originates in the quantum cohomology of flag varieties [Fulton and Woodward].

59 Hasse diagram of the Bruhat order for S 3 : α α α α α 23 α α 12 α

60 Quantum Bruhat graph for S 3 : α α α α α 23 α α 12 α 13 α

61 The quantum alcove model Given a dominant weight λ, we associate with it a sequence of roots, called a λ-chain: Γ = (β 1,..., β m ).

62 Example. Type A 2, λ = (3, 1, 0) = 3ε 1 + ε 2, Γ = ( (1, 2), (1, 3), (2, 3), (1, 3), (1, 2), (1, 3) ). α 13 α 23 ε 1 α 12 ε 2 ε 3 r 6 =s α13, 2 r =s 3 λ α23, 0

63 The quantum alcove model (cont.) Given Γ = (β 1,..., β m ), let r i := s βi.

64 The quantum alcove model (cont.) Given Γ = (β 1,..., β m ), let r i := s βi. The objects of the model: subsets of positions in Γ J = (j 1 <... < j s ) {1,..., m}.

65 The quantum alcove model (cont.) Given Γ = (β 1,..., β m ), let r i := s βi. The objects of the model: subsets of positions in Γ J = (j 1 <... < j s ) {1,..., m}. For w W and J, construct the chain π(w, J) of elements in W : w 0 = w,..., w i := wr j1... r ji,..., w s = end(w, J).

66 The quantum alcove model (cont.) Given Γ = (β 1,..., β m ), let r i := s βi. The objects of the model: subsets of positions in Γ J = (j 1 <... < j s ) {1,..., m}. For w W and J, construct the chain π(w, J) of elements in W : w 0 = w,..., w i := wr j1... r ji,..., w s = end(w, J). Important structures: A q (Γ, w) := {J : π(w, J) path in QBG(W )}, A (Γ, w) := {J : π(w, J) saturated chain in (W, <)}.

67 The quantum alcove model (cont.) Given Γ = (β 1,..., β m ), let r i := s βi. The objects of the model: subsets of positions in Γ J = (j 1 <... < j s ) {1,..., m}. For w W and J, construct the chain π(w, J) of elements in W : w 0 = w,..., w i := wr j1... r ji,..., w s = end(w, J). Important structures: A q (Γ, w) := {J : π(w, J) path in QBG(W )}, A (Γ, w) := {J : π(w, J) saturated chain in (W, <)}. Let A q (Γ) := A q (Γ, 1 W ) and A (Γ) := A (Γ, 1 W ).

68 Macdonald polynomials: the Ram-Yip formula Given a dominant weight λ, consider a λ-chain Γ := (β 1,..., β m ).

69 Macdonald polynomials: the Ram-Yip formula Given a dominant weight λ, consider a λ-chain Γ := (β 1,..., β m ). Given J A q (Γ), we associate with it a weight weight(j),

70 Macdonald polynomials: the Ram-Yip formula Given a dominant weight λ, consider a λ-chain Γ := (β 1,..., β m ). Given J A q (Γ), we associate with it a weight weight(j), a statistic height(j), which measures the down steps w i 1 > w i in the path π(1 W, J) in QBG(W ).

71 Macdonald polynomials: the Ram-Yip formula Given a dominant weight λ, consider a λ-chain Γ := (β 1,..., β m ). Given J A q (Γ), we associate with it a weight weight(j), a statistic height(j), which measures the down steps w i 1 > w i in the path π(1 W, J) in QBG(W ). Theorem (Ram-Yip, L.) P λ (X ; q, 0) = J A q(γ) q height(j) x weight(j).

72 Macdonald polynomials: the Ram-Yip formula Given a dominant weight λ, consider a λ-chain Γ := (β 1,..., β m ). Given J A q (Γ), we associate with it a weight weight(j), a statistic height(j), which measures the down steps w i 1 > w i in the path π(1 W, J) in QBG(W ). Theorem (Ram-Yip, L.) P λ (X ; q, 0) = J A q(γ) q height(j) x weight(j). Remark. For q = 0, we recover the LS-gallery/alcove models (Gaussent-Littelmann, L.-Postnikov), which are discrete versions of the Littelmann path model: P λ (X ; 0, 0) = ch(v (λ)) = x weight(j). J A (Γ)

73 K(G/B) and QK(G/B): Chevalley formulas Recall: K(G/B) and QK(G/B) have bases of Schubert classes [O Xw ] = [O w ], w W.

74 K(G/B) and QK(G/B): Chevalley formulas Recall: K(G/B) and QK(G/B) have bases of Schubert classes [O Xw ] = [O w ], w W. Let Γ rev = reverse of an ω k -chain (ω k a fundamental weight).

75 K(G/B) and QK(G/B): Chevalley formulas Recall: K(G/B) and QK(G/B) have bases of Schubert classes [O Xw ] = [O w ], w W. Let Γ rev = reverse of an ω k -chain (ω k a fundamental weight). Theorem (L.-Postnikov, L.-Shimozono) In K(G/B) (finite-type or Kac-Moody), we have [O w ] [O sk ]= J A (Γ rev,w)\{ } ( 1) J 1 [O end(w,j) ].

76 K(G/B) and QK(G/B): Chevalley formulas Recall: K(G/B) and QK(G/B) have bases of Schubert classes [O Xw ] = [O w ], w W. Let Γ rev = reverse of an ω k -chain (ω k a fundamental weight). Theorem (L.-Postnikov, L.-Shimozono) In K(G/B) (finite-type or Kac-Moody), we have [O w ] [O sk ]= Conjecture (L.-Postnikov) J A (Γ rev,w)\{ } In QK(G/B) (finite-type), we have: [O w ] [O sk ]? = J A q(γ rev,w)\{ } ( 1) J 1 [O end(w,j) ]. ( 1) J 1 q 1... q r [O end(w,j) ].

77 K(G/B) and QK(G/B): Chevalley formulas Recall: K(G/B) and QK(G/B) have bases of Schubert classes [O Xw ] = [O w ], w W. Let Γ rev = reverse of an ω k -chain (ω k a fundamental weight). Theorem (L.-Postnikov, L.-Shimozono) In K(G/B) (finite-type or Kac-Moody), we have [O w ] [O sk ]= Conjecture (L.-Postnikov) J A (Γ rev,w)\{ } In QK(G/B) (finite-type), we have: [O w ] [O sk ]? = J A q(γ rev,w)\{ } ( 1) J 1 [O end(w,j) ]. ( 1) J 1 q 1... q r [O end(w,j) ]. Remark. Restricting the RHS, we retrieve the Chevalley formula in QH (G/B) [Fulton-Woodward].

78 Evidence for the conjectured formula in QK(G/B) Computer experiments (A. Buch).

79 Evidence for the conjectured formula in QK(G/B) Computer experiments (A. Buch). The work of Braverman-Finkelberg connecting QK(G/B) to specialized Macdonald polynomials.

80 Evidence for the conjectured formula in QK(G/B) Computer experiments (A. Buch). The work of Braverman-Finkelberg connecting QK(G/B) to specialized Macdonald polynomials. [L.-Maeno] Based on some relations in QK(SL n /B) discovered by Kirillov-Maeno, we constructed polynomials G w (x; q), called quantum Grothendieck polynomials.

81 Evidence for the conjectured formula in QK(G/B) Computer experiments (A. Buch). The work of Braverman-Finkelberg connecting QK(G/B) to specialized Macdonald polynomials. [L.-Maeno] Based on some relations in QK(SL n /B) discovered by Kirillov-Maeno, we constructed polynomials G w (x; q), called quantum Grothendieck polynomials. - They specialize to the usual polynomial representatives in K(SL n /B) and QH (SL n /B).

82 Evidence for the conjectured formula in QK(G/B) Computer experiments (A. Buch). The work of Braverman-Finkelberg connecting QK(G/B) to specialized Macdonald polynomials. [L.-Maeno] Based on some relations in QK(SL n /B) discovered by Kirillov-Maeno, we constructed polynomials G w (x; q), called quantum Grothendieck polynomials. - They specialize to the usual polynomial representatives in K(SL n /B) and QH (SL n /B). - They multiply as in the conjectured Chevalley formula.

83 Evidence for the conjectured formula in QK(G/B) Computer experiments (A. Buch). The work of Braverman-Finkelberg connecting QK(G/B) to specialized Macdonald polynomials. [L.-Maeno] Based on some relations in QK(SL n /B) discovered by Kirillov-Maeno, we constructed polynomials G w (x; q), called quantum Grothendieck polynomials. - They specialize to the usual polynomial representatives in K(SL n /B) and QH (SL n /B). - They multiply as in the conjectured Chevalley formula. - They are conjectured to represent Schubert classes [O w ] in QK(SL n /B).

84 Kirillov-Reshetikhin (KR) modules/crystals Recall the KR modules for ĝ: W r,s and W p = W p 1,1... W p k,1.

85 Kirillov-Reshetikhin (KR) modules/crystals Recall the KR modules for ĝ: W r,s and W p = W p 1,1... W p k,1. The corresponding KR crystals partially encode the structure of these representations.

86 Kashiwara s crystals Encode representations V of the corresponding quantum group U q (g) (or U q (ĝ)) as q 0.

87 Kashiwara s crystals Encode representations V of the corresponding quantum group U q (g) (or U q (ĝ)) as q 0. Kashiwara (crystal) operators: ẽ i, f i.

88 Kashiwara s crystals Encode representations V of the corresponding quantum group U q (g) (or U q (ĝ)) as q 0. Kashiwara (crystal) operators: ẽ i, f i. Fact. V has a crystal basis B: in the limit q 0 we have fi, ẽ i : B B {0}, fi (b) = b ẽ i (b ) = b.

89 Kashiwara s crystals Encode representations V of the corresponding quantum group U q (g) (or U q (ĝ)) as q 0. Kashiwara (crystal) operators: ẽ i, f i. Fact. V has a crystal basis B: in the limit q 0 we have fi, ẽ i : B B {0}, fi (b) = b ẽ i (b ) = b. Encode as colored directed graph: fi (b) = b b i b.

90 Kashiwara s crystals Encode representations V of the corresponding quantum group U q (g) (or U q (ĝ)) as q 0. Kashiwara (crystal) operators: ẽ i, f i. Fact. V has a crystal basis B: in the limit q 0 we have fi, ẽ i : B B {0}, fi (b) = b ẽ i (b ) = b. Encode as colored directed graph: fi (b) = b b i b. Remark. Crystals encode enough information for: inductions and restrictions, tensor product decompositions etc.

91 Example. Crystal for irreducible representation of sl 4 of highest weight λ = (3, 3, 1) denoted B(λ).

92 Example. Crystal for irreducible representation of sl 4 of highest weight λ = (3, 3, 1) denoted B(λ). blue: f 1, green: f 2, red: f 3.

93 Kirillov Reshetikhin (KR) crystals W r,s B r,s, W p = W p 1,1... W p k,1 B p = B p 1,1... B p k,1.

94 Kirillov Reshetikhin (KR) crystals W r,s B r,s, W p = W p 1,1... W p k,1 B p = B p 1,1... B p k,1. The corresponding crystals have arrows f 0, f 1,..., but there is no highest weight vertex.

95 Kirillov Reshetikhin (KR) crystals W r,s B r,s, W p = W p 1,1... W p k,1 B p = B p 1,1... B p k,1. The corresponding crystals have arrows f 0, f 1,..., but there is no highest weight vertex. Remarks. (1) The tensor product of crystals is constructed via a specific rule (tensor product rule).

96 Kirillov Reshetikhin (KR) crystals W r,s B r,s, W p = W p 1,1... W p k,1 B p = B p 1,1... B p k,1. The corresponding crystals have arrows f 0, f 1,..., but there is no highest weight vertex. Remarks. (1) The tensor product of crystals is constructed via a specific rule (tensor product rule). (2) B p is connected with the 0-arrows, but disconnected as a classical crystal (remove 0-arrows).

97 Kirillov Reshetikhin (KR) crystals W r,s B r,s, W p = W p 1,1... W p k,1 B p = B p 1,1... B p k,1. The corresponding crystals have arrows f 0, f 1,..., but there is no highest weight vertex. Remarks. (1) The tensor product of crystals is constructed via a specific rule (tensor product rule). (2) B p is connected with the 0-arrows, but disconnected as a classical crystal (remove 0-arrows). (3) A KR crystal B r,1 of classical type A D is realized in terms of fillings of columns of height r with integers.

98 Kirillov Reshetikhin (KR) crystals W r,s B r,s, W p = W p 1,1... W p k,1 B p = B p 1,1... B p k,1. The corresponding crystals have arrows f 0, f 1,..., but there is no highest weight vertex. Remarks. (1) The tensor product of crystals is constructed via a specific rule (tensor product rule). (2) B p is connected with the 0-arrows, but disconnected as a classical crystal (remove 0-arrows). (3) A KR crystal B r,1 of classical type A D is realized in terms of fillings of columns of height r with integers. (4) The highest weight crystals for ĝ can be constructed in terms of infinite tensor products with p 1 = p 2 =... (Kyoto path model).

99 The quantum alcove model for KR crystals Given p = (p 1,..., p k ) and an arbitrary Lie type, let λ = ω p ω pk.

100 The quantum alcove model for KR crystals Given p = (p 1,..., p k ) and an arbitrary Lie type, let λ = ω p ω pk. Let Γ be an arbitrary λ-chain, and consider A q (Γ).

101 The quantum alcove model for KR crystals Given p = (p 1,..., p k ) and an arbitrary Lie type, let λ = ω p ω pk. Let Γ be an arbitrary λ-chain, and consider A q (Γ). Main Theorem (L.-Naito-Sagaki-Schilling-Shimozono) The tensor product of KR crystals B p is realized as a directed graph with vertices A q (Γ).

102 KR crystals in classical types Fact. We have as classical crystals (remove 0-arrows): B p,1 B(ω p ) in types A (1) (1) n 1 and C n ;

103 KR crystals in classical types Fact. We have as classical crystals (remove 0-arrows): B p,1 B(ω p ) in types A (1) (1) n 1 and C n ; but B p,1 B(ω p ) B(ω p 2 )... in types B (1) n and D (1) n.

104 KR crystals in classical types Fact. We have as classical crystals (remove 0-arrows): B p,1 B(ω p ) in types A (1) (1) n 1 and C n ; but B p,1 B(ω p ) B(ω p 2 )... in types B (1) n and D (1) n. Remark. Combinatorial models for B(ω l ) in terms of increasing fillings of a column of height l (Kashiwara-Nakashima columns);

105 KR crystals in classical types Fact. We have as classical crystals (remove 0-arrows): B p,1 B(ω p ) in types A (1) (1) n 1 and C n ; but B p,1 B(ω p ) B(ω p 2 )... in types B (1) n and D (1) n. Remark. Combinatorial models for B(ω l ) in terms of increasing fillings of a column of height l (Kashiwara-Nakashima columns); in type A, increasing fillings with 1,..., n, but more involved in types B, C, D.

106 Type A (1) n 1 Consider B p := B p 1,1... B pm,1 where

107 Type A (1) n 1 Consider B p := B p 1,1... B pm,1 where B p,1 B(ω p ), and ω p = (1,..., 1, 0,..., 0) = (1 p ). }{{}}{{} p n p

108 Type A (1) n 1 Consider B p := B p 1,1... B pm,1 where B p,1 B(ω p ), and ω p = (1,..., 1, 0,..., 0) = (1 p ). }{{}}{{} p n p Vertices of B p represented as column-strict fillings with integers 1,..., n of a diagram p with columns of heights p 1,..., p m.

109 Type A (1) n 1 Consider B p := B p 1,1... B pm,1 where B p,1 B(ω p ), and ω p = (1,..., 1, 0,..., 0) = (1 p ). }{{}}{{} p n p Vertices of B p represented as column-strict fillings with integers 1,..., n of a diagram p with columns of heights p 1,..., p m. Example. n = 5 b = B 3,1 B 1,1 B 3,1 B 3,2.

110 Type A (1) n 1 Consider B p := B p 1,1... B pm,1 where B p,1 B(ω p ), and ω p = (1,..., 1, 0,..., 0) = (1 p ). }{{}}{{} p n p Vertices of B p represented as column-strict fillings with integers 1,..., n of a diagram p with columns of heights p 1,..., p m. Example. n = 5 b = The action of the crystal operators: B 3,1 B 1,1 B 3,1 B 3,2. 1 f 1 2 f n 1 fn 1 n f 0 1.

111 Another example Type A (1) 2, B (1,1,1) = B(1) B(1) B(1).

112 Another example Type A (1) 2, B (1,1,1) = B(1) B(1) B(1). Fact. We have, as a classical crystal: B (1,1,1) B(3) B(2, 1) B(2, 1) B(1 3 ).

113

114

115 The energy function It originates in the theory of exactly solvable lattice models.

116 The energy function It originates in the theory of exactly solvable lattice models. Let B := B p = B p 1,1... B pm,1.

117 The energy function It originates in the theory of exactly solvable lattice models. Let B := B p = B p 1,1... B pm,1. The energy function defines a grading on the classical components of B (Schilling and Tingley).

118 The energy function It originates in the theory of exactly solvable lattice models. Let B := B p = B p 1,1... B pm,1. The energy function defines a grading on the classical components of B (Schilling and Tingley). More precisely, D B : B Z 0 satisfies the following conditions: it is constant on classical components;

119 The energy function It originates in the theory of exactly solvable lattice models. Let B := B p = B p 1,1... B pm,1. The energy function defines a grading on the classical components of B (Schilling and Tingley). More precisely, D B : B Z 0 satisfies the following conditions: it is constant on classical components; it decreases by 1 along the f 0 arrows which are not at the end of a 0-string.

120

121 The quantum alcove model for KR crystals In terms of the quantum alcove model, we can do the following, uniformly for all affine types A (1) n 1 G (1) 2 :

122 The quantum alcove model for KR crystals In terms of the quantum alcove model, we can do the following, uniformly for all affine types A (1) n 1 G (1) 2 : (1) give a combinatorial construction for tensor products of (column shape) KR crystals;

123 The quantum alcove model for KR crystals In terms of the quantum alcove model, we can do the following, uniformly for all affine types A (1) n 1 G (1) 2 : (1) give a combinatorial construction for tensor products of (column shape) KR crystals; (2) define a statistic which computes the energy function based only on the combinatorial data associated with a crystal vertex;

124 The quantum alcove model for KR crystals In terms of the quantum alcove model, we can do the following, uniformly for all affine types A (1) n 1 G (1) 2 : (1) give a combinatorial construction for tensor products of (column shape) KR crystals; (2) define a statistic which computes the energy function based only on the combinatorial data associated with a crystal vertex; (3) give an explicit construction of the combinatorial R-matrix, i.e., the (unique) affine crystal isomorphism between X Y and Y X ;

125 The quantum alcove model for KR crystals In terms of the quantum alcove model, we can do the following, uniformly for all affine types A (1) n 1 G (1) 2 : (1) give a combinatorial construction for tensor products of (column shape) KR crystals; (2) define a statistic which computes the energy function based only on the combinatorial data associated with a crystal vertex; (3) give an explicit construction of the combinatorial R-matrix, i.e., the (unique) affine crystal isomorphism between X Y and Y X ; (4) verify the [HKOTY] conjecture regarding the perfectness of B r,1 and its graded classical decomposition in exceptional types.

126 The quantum alcove model for KR crystals In terms of the quantum alcove model, we can do the following, uniformly for all affine types A (1) n 1 G (1) 2 : (1) give a combinatorial construction for tensor products of (column shape) KR crystals; (2) define a statistic which computes the energy function based only on the combinatorial data associated with a crystal vertex; (3) give an explicit construction of the combinatorial R-matrix, i.e., the (unique) affine crystal isomorphism between X Y and Y X ; (4) verify the [HKOTY] conjecture regarding the perfectness of B r,1 and its graded classical decomposition in exceptional types. Remark. (2) and (3) were expressed in type A in terms of fillings, but little was known beyond type A.

127 The quantum alcove model for KR crystals Given p = (p 1,..., p k ) and an arbitrary Lie type, let λ = ω p ω pk.

128 The quantum alcove model for KR crystals Given p = (p 1,..., p k ) and an arbitrary Lie type, let λ = ω p ω pk. Let Γ be an arbitrary λ-chain, and consider A q (Γ).

129 The quantum alcove model for KR crystals Given p = (p 1,..., p k ) and an arbitrary Lie type, let λ = ω p ω pk. Let Γ be an arbitrary λ-chain, and consider A q (Γ). Construction. [L. and Lubovsky, generalization of Gaussent-Littelmann, L.-Postnikov] We defined combinatorial crystal operators f 1,..., f r and f 0 on A q (Γ).

130 The quantum alcove model for KR crystals Given p = (p 1,..., p k ) and an arbitrary Lie type, let λ = ω p ω pk. Let Γ be an arbitrary λ-chain, and consider A q (Γ). Construction. [L. and Lubovsky, generalization of Gaussent-Littelmann, L.-Postnikov] We defined combinatorial crystal operators f 1,..., f r and f 0 on A q (Γ). Main Theorem (L.-Naito-Sagaki-Schilling-Shimozono) The (combinatorial) crystal A q (Γ) is isomorphic to the tensor product of KR crystals B p.

131 Ingredients in the proof Lifts of QB(W J ) to

132 Ingredients in the proof Lifts of QB(W J ) to Littelmann s poset of level 0 weights W af λ (describing level 0 LS paths)

133 Ingredients in the proof Lifts of QB(W J ) to Littelmann s poset of level 0 weights W af λ (describing level 0 LS paths) = description of covers;

134 Ingredients in the proof Lifts of QB(W J ) to Littelmann s poset of level 0 weights W af λ (describing level 0 LS paths) = description of covers; the Bruhat order on W af ( parabolic quantum to affine, cf. Peterson 97, Lam Shimozono 10).

135 Example in type A 2. p = (1, 2, 2, 1) = ; λ = ω 1 + ω 2 + ω 2 + ω 1 = (4, 2, 0).

136 Example in type A 2. p = (1, 2, 2, 1) = ; λ = ω 1 + ω 2 + ω 2 + ω 1 = (4, 2, 0). A λ-chain as a concatenation of ω 1 -, ω 2 -, ω 2 -, and ω 1 -chains: Γ = ( (1, 2), (1, 3) (2, 3), (1, 3) (2, 3), (1, 3) (1, 2), (1, 3) ).

137 Example. Let J = {1, 2, 3, 6, 7, 8}. ( (1, 2), (1, 3) (2, 3), (1, 3) (2, 3), (1, 3) (1, 2), (1, 3) ).

138 Example. Let J = {1, 2, 3, 6, 7, 8}. ( (1, 2), (1, 3) (2, 3), (1, 3) (2, 3), (1, 3) (1, 2), (1, 3) ). Claim: J is admissible (belongs to A q (Γ)). Indeed, the corresponding path in the quantum Bruhat graph is < < < > < <

139 Example. Let J = {1, 2, 3, 6, 7, 8}. ( (1, 2), (1, 3) (2, 3), (1, 3) (2, 3), (1, 3) (1, 2), (1, 3) ). Claim: J is admissible (belongs to A q (Γ)). Indeed, the corresponding path in the quantum Bruhat graph is < < < > < < The corresponding element in B p = B 1,1 B 2,1 B 2,1 B 1,1 represented via a column-strict filling:

140 The energy via the quantum alcove model Consider J = {j 1 < j 2 <... < j s } in A q (Γ) for Γ = (β 1,..., β m ), i.e., we have a path in the quantum Bruhat graph β j1 β j2 β js 1 W = w 0 w1... ws.

141 The energy via the quantum alcove model Consider J = {j 1 < j 2 <... < j s } in A q (Γ) for Γ = (β 1,..., β m ), i.e., we have a path in the quantum Bruhat graph β j1 β j2 β js 1 W = w 0 w1... ws. Let J := {j i : w i 1 > w i }.

142 The energy via the quantum alcove model Consider J = {j 1 < j 2 <... < j s } in A q (Γ) for Γ = (β 1,..., β m ), i.e., we have a path in the quantum Bruhat graph β j1 β j2 β js 1 W = w 0 w1... ws. Let J := {j i : w i 1 > w i }. The height statistic is defined by height(j) := j J h j.

143 The energy via the quantum alcove model Consider J = {j 1 < j 2 <... < j s } in A q (Γ) for Γ = (β 1,..., β m ), i.e., we have a path in the quantum Bruhat graph β j1 β j2 β js 1 W = w 0 w1... ws. Let J := {j i : w i 1 > w i }. The height statistic is defined by height(j) := h j. j J Theorem (L.-Naito-Sagaki-Schilling-Shimozono) Given J A q (Γ), which is identified with B p, we have D B (J) = height(j).

144 Example. Consider the running example: λ = ω 1 + ω 2 + ω 2 + ω 1 in type A 2.

145 Example. Consider the running example: λ = ω 1 + ω 2 + ω 2 + ω 1 in type A 2. We considered the λ-chain Γ and J = {1, 2, 3, 6, 7, 8} A(Γ): Γ = ( (1, 2), (1, 3) (2, 3), (1, 3) (2, 3), (1, 3) (1, 2), (1, 3) ), (h i ) = ( 2, 4 2, 3 1, 2 1, 1 ).

146 Example. Consider the running example: λ = ω 1 + ω 2 + ω 2 + ω 1 in type A 2. We considered the λ-chain Γ and J = {1, 2, 3, 6, 7, 8} A(Γ): Γ = ( (1, 2), (1, 3) (2, 3), (1, 3) (2, 3), (1, 3) (1, 2), (1, 3) ), (h i ) = ( 2, 4 2, 3 1, 2 1, 1 ). We have height(j) = 2.

147 Example. Consider the running example: λ = ω 1 + ω 2 + ω 2 + ω 1 in type A 2. We considered the λ-chain Γ and J = {1, 2, 3, 6, 7, 8} A(Γ): Γ = ( (1, 2), (1, 3) (2, 3), (1, 3) (2, 3), (1, 3) (1, 2), (1, 3) ), (h i ) = ( 2, 4 2, 3 1, 2 1, 1 ). We have height(j) = 2. Remarks. (1) In type A, the height statistic translates into the Lascoux Schützenberger charge statistic on Young tableaux (L.).

148 Example. Consider the running example: λ = ω 1 + ω 2 + ω 2 + ω 1 in type A 2. We considered the λ-chain Γ and J = {1, 2, 3, 6, 7, 8} A(Γ): Γ = ( (1, 2), (1, 3) (2, 3), (1, 3) (2, 3), (1, 3) (1, 2), (1, 3) ), (h i ) = ( 2, 4 2, 3 1, 2 1, 1 ). We have height(j) = 2. Remarks. (1) In type A, the height statistic translates into the Lascoux Schützenberger charge statistic on Young tableaux (L.). (2) A similar charge statistic was defined in type C (L. and Schilling), and one is being developed in type B (Briggs and L.).

149 The combinatorial R-matrix It is the (unique) affine crystal isomorphism which commutes factors in a tensor product of KR crystals (the swap a b b a is not a crystal isomorphism!).

150 The combinatorial R-matrix It is the (unique) affine crystal isomorphism which commutes factors in a tensor product of KR crystals (the swap a b b a is not a crystal isomorphism!). In type A, it is realized by Schützenberger s jeu de taquin (sliding algorithm) on two columns.

151 The combinatorial R-matrix It is the (unique) affine crystal isomorphism which commutes factors in a tensor product of KR crystals (the swap a b b a is not a crystal isomorphism!). In type A, it is realized by Schützenberger s jeu de taquin (sliding algorithm) on two columns. Example. Realize the isomorphism B 1,1 B 2,1 B 2,1 B 1,1 B 1,1 B 2,1 B 1,1 B 2,1.

152 The combinatorial R-matrix It is the (unique) affine crystal isomorphism which commutes factors in a tensor product of KR crystals (the swap a b b a is not a crystal isomorphism!). In type A, it is realized by Schützenberger s jeu de taquin (sliding algorithm) on two columns. Example. Realize the isomorphism B 1,1 B 2,1 B 2,1 B 1,1 B 1,1 B 2,1 B 1,1 B 2,1. Commute the last two factors as follows: = =

153 Goal We will give a uniform realization of the combinatorial R-matrix (in arbitrary type) by extending to the quantum alcove model the Yang Baxter moves in the alcove model (L.).

154 Goal We will give a uniform realization of the combinatorial R-matrix (in arbitrary type) by extending to the quantum alcove model the Yang Baxter moves in the alcove model (L.). We will call them quantum Yang Baxter moves.

155 Main ingredient for the quantum Yang Baxter moves Consider the quantum Bruhat graph on a dihedral Weyl group, that is, a Weyl group of type A 1 A 1, A 2, B 2, C 2, or G 2 (B 2 and C 2 are not isomorphic!).

156 Main ingredient for the quantum Yang Baxter moves Consider the quantum Bruhat graph on a dihedral Weyl group, that is, a Weyl group of type A 1 A 1, A 2, B 2, C 2, or G 2 (B 2 and C 2 are not isomorphic!). A reflection order on the positive roots (α := α 1, β := α 2 ): β 1 := α, β 2 := s α (β), β 3 := s α s β (α),..., β q := β.

157 Main ingredient for the quantum Yang Baxter moves Consider the quantum Bruhat graph on a dihedral Weyl group, that is, a Weyl group of type A 1 A 1, A 2, B 2, C 2, or G 2 (B 2 and C 2 are not isomorphic!). A reflection order on the positive roots (α := α 1, β := α 2 ): β 1 := α, β 2 := s α (β), β 3 := s α s β (α),..., β q := β. Label the edges of W by numbers 1,..., q.

158 Main ingredient for the quantum Yang Baxter moves Consider the quantum Bruhat graph on a dihedral Weyl group, that is, a Weyl group of type A 1 A 1, A 2, B 2, C 2, or G 2 (B 2 and C 2 are not isomorphic!). A reflection order on the positive roots (α := α 1, β := α 2 ): β 1 := α, β 2 := s α (β), β 3 := s α s β (α),..., β q := β. Label the edges of W by numbers 1,..., q. Theorem. (Brenti, Fomin, Postnikov shellability of the quantum Bruhat graph) Between any two elements in W, there is a unique increasing and a unique decreasing (directed) path, both of the same length.

159 Main ingredient for the quantum Yang Baxter moves Consider the quantum Bruhat graph on a dihedral Weyl group, that is, a Weyl group of type A 1 A 1, A 2, B 2, C 2, or G 2 (B 2 and C 2 are not isomorphic!). A reflection order on the positive roots (α := α 1, β := α 2 ): β 1 := α, β 2 := s α (β), β 3 := s α s β (α),..., β q := β. Label the edges of W by numbers 1,..., q. Theorem. (Brenti, Fomin, Postnikov shellability of the quantum Bruhat graph) Between any two elements in W, there is a unique increasing and a unique decreasing (directed) path, both of the same length. (In fact, this is true for any Weyl group W and any reflection ordering.)

160 Example. Type G 2. s 2 s 1 s 2 s 1 : 1, 2, 5, 6 ; 6, 3, 2, 1. s 2 s 1 s 2 s 1 s 2 s s 2 s 1 s 2 s 1 s 2 s 1 s 2 s 1 s 2 s s 2 s 1 s 2 s 1 s 1 s 2 s 1 s s 2 s 1 s 2 s 1 s 2 s s 2 s 1 s 1 s s 2 s

161 The quantum Yang Baxter moves via the running example. Realize the isomorphism B 1,1 B 2,1 B 2,1 B 1,1 B 1,1 B 2,1 B 1,1 B 2,1.

162 The quantum Yang Baxter moves via the running example. Realize the isomorphism B 1,1 B 2,1 B 2,1 B 1,1 B 1,1 B 2,1 B 1,1 B 2,1. ( (1, 2), (1, 3) (2, 3), (1, 3) (2, 3), (1, 3) (1, 2), (1, 3) ), ( (1, 2), (1, 3) (2, 3), (1, 3) (1, 2), (1, 3) (2, 3), (1, 3) ).

163 The quantum Yang Baxter moves via the running example. Realize the isomorphism B 1,1 B 2,1 B 2,1 B 1,1 B 1,1 B 2,1 B 1,1 B 2,1. ( (1, 2), (1, 3) (2, 3), (1, 3) (2, 3), (1, 3) (1, 2), (1, 3) ), ( (1, 2), (1, 3) (2, 3), (1, 3) (1, 2), (1, 3) (2, 3), (1, 3) ).

164 The quantum Yang Baxter moves via the running example. Realize the isomorphism B 1,1 B 2,1 B 2,1 B 1,1 B 1,1 B 2,1 B 1,1 B 2,1. ( (1, 2), (1, 3) (2, 3), (1, 3) (2, 3), (1, 3) (1, 2), (1, 3) ), ( (1, 2), (1, 3) (2, 3), (1, 3) (1, 2), (1, 3) (2, 3), (1, 3) ). 321 =(1,2)(1,3)(2,3) α α 13 α α α 23 α 12 α α α

165 The quantum Yang Baxter moves via the running example. Realize the isomorphism B 1,1 B 2,1 B 2,1 B 1,1 B 1,1 B 2,1 B 1,1 B 2,1. ( (1, 2), (1, 3) (2, 3), (1, 3) (2, 3), (1, 3) (1, 2), (1, 3) ), ( (1, 2), (1, 3) (2, 3), (1, 3) (1, 2), (1, 3) (2, 3), (1, 3) ). 321 =(1,2)(1,3)(2,3) α α 13 α α α 23 α 12 α α α

166 The quantum Yang Baxter moves via the running example. Realize the isomorphism B 1,1 B 2,1 B 2,1 B 1,1 B 1,1 B 2,1 B 1,1 B 2,1. ( (1, 2), (1, 3) (2, 3), (1, 3) (2, 3), (1, 3) (1, 2), (1, 3) ), ( (1, 2), (1, 3) (2, 3), (1, 3) (1, 2), (1, 3) (2, 3), (1, 3) ). 321 =(1,2)(1,3)(2,3) α α 13 α α α 23 α 12 α α α

167 The quantum Yang Baxter moves via the running example. Realize the isomorphism B 1,1 B 2,1 B 2,1 B 1,1 B 1,1 B 2,1 B 1,1 B 2,1. ( (1, 2), (1, 3) (2, 3), (1, 3) (2, 3), (1, 3) (1, 2), (1, 3) ), ( (1, 2), (1, 3) (2, 3), (1, 3) (1, 2), (1, 3) (2, 3), (1, 3) ). The Bruhat chain corresponding to the second case: < < < > > <

168 The quantum Yang Baxter moves via the running example. Realize the isomorphism B 1,1 B 2,1 B 2,1 B 1,1 B 1,1 B 2,1 B 1,1 B 2,1. ( (1, 2), (1, 3) (2, 3), (1, 3) (2, 3), (1, 3) (1, 2), (1, 3) ), ( (1, 2), (1, 3) (2, 3), (1, 3) (1, 2), (1, 3) (2, 3), (1, 3) ). The Bruhat chain corresponding to the second case: < < < > > < So we have

169 The general case Let p = (p 1, p 2,...), and p = (p 1, p 2,...) a permutation of p.

170 The general case Let p = (p 1, p 2,...), and p = (p 1, p 2,...) a permutation of p. Let λ := ω p1 + ω p = ω p 1 + ω p , B := B p 1,1 B p 2,1..., B := B p 1,1 B p 2,1....

171 The general case Let p = (p 1, p 2,...), and p = (p 1, p 2,...) a permutation of p. Let λ := ω p1 + ω p = ω p 1 + ω p , B := B p 1,1 B p 2,1..., B := B p 1,1 B p 2,1.... Fix an ω i -chain Γ i, for each i.

172 The general case Let p = (p 1, p 2,...), and p = (p 1, p 2,...) a permutation of p. Let λ := ω p1 + ω p = ω p 1 + ω p , B := B p 1,1 B p 2,1..., B := B p 1,1 B p 2,1.... Fix an ω i -chain Γ i, for each i. Construct λ-chains Γ, Γ : Γ := Γ p 1 Γ p 2..., Γ := Γ p 1 Γ p 2....

173 The general case Let p = (p 1, p 2,...), and p = (p 1, p 2,...) a permutation of p. Let λ := ω p1 + ω p = ω p 1 + ω p , B := B p 1,1 B p 2,1..., B := B p 1,1 B p 2,1.... Fix an ω i -chain Γ i, for each i. Construct λ-chains Γ, Γ : Γ := Γ p 1 Γ p 2..., Γ := Γ p 1 Γ p Recall that A(Γ) and A(Γ ) are models for B and B.

174 The general case Let p = (p 1, p 2,...), and p = (p 1, p 2,...) a permutation of p. Let λ := ω p1 + ω p = ω p 1 + ω p , B := B p 1,1 B p 2,1..., B := B p 1,1 B p 2,1.... Fix an ω i -chain Γ i, for each i. Construct λ-chains Γ, Γ : Γ := Γ p 1 Γ p 2..., Γ := Γ p 1 Γ p Recall that A(Γ) and A(Γ ) are models for B and B. Theorem. (L. and Lubovsky) There is a sequence of moves Γ = Γ 0 Γ 1 Γ 2... Γ t = Γ, where each move consists of reversing a segment of roots which is a reflection order for some rank 2 root subsystem.

175 The general case Let p = (p 1, p 2,...), and p = (p 1, p 2,...) a permutation of p. Let λ := ω p1 + ω p = ω p 1 + ω p , B := B p 1,1 B p 2,1..., B := B p 1,1 B p 2,1.... Fix an ω i -chain Γ i, for each i. Construct λ-chains Γ, Γ : Γ := Γ p 1 Γ p 2..., Γ := Γ p 1 Γ p Recall that A(Γ) and A(Γ ) are models for B and B. Theorem. (L. and Lubovsky) There is a sequence of moves Γ = Γ 0 Γ 1 Γ 2... Γ t = Γ, where each move consists of reversing a segment of roots which is a reflection order for some rank 2 root subsystem. The (unique) affine crystal isomorphism between A(Γ i ) and A(Γ i+1 ) is realized by a quantum Yang Baxter move (replace the label increasing chain in QBG(X 2 ) with the decreasing chain).

176 Proof. Based on showing the commutation between the quantum Yang Baxter moves and the crystal operators, including f 0.

177 Future work Extend Ion s result, identifying P λ (x; q, 0) with an affine Demazure character, to the types C n (1), F (1) 4, G (1) 2

178 Future work Extend Ion s result, identifying P λ (x; q, 0) with an affine Demazure character, to the types C n (1), F (1) 4, G (1) 2 (P λ (x; q, 0) will be a sum of affine Demazure characters; type has been solved in [Chari et al.]). B (1) n

179 Future work Extend Ion s result, identifying P λ (x; q, 0) with an affine Demazure character, to the types C n (1), F (1) 4, G (1) 2 (P λ (x; q, 0) will be a sum of affine Demazure characters; type has been solved in [Chari et al.]). B (1) n Extend the realization of tensor products of B p,1 to B p,q (new methods and models needed).

180 Future work Extend Ion s result, identifying P λ (x; q, 0) with an affine Demazure character, to the types C n (1), F (1) 4, G (1) 2 (P λ (x; q, 0) will be a sum of affine Demazure characters; type has been solved in [Chari et al.]). B (1) n Extend the realization of tensor products of B p,1 to B p,q (new methods and models needed). Prove the conjectured multiplication formula (Chevalley) in the quantum K-theory of flag varieties, in terms of the quantum alcove model (with A. Buch and L. Mihalcea).