Crystals and Their Generalizations

General information

This 5 day workshop will focus on Kashiwara's theory of crystals and its generalizations to other areas such as K-theory, geometric crystals, and Lie superalgebras. There will be a limited number of talks to encourage discussions among the participants.


March 25–29, 2019


The workshop will take place at OCAMI at Osaka City University. Talks will take place at the Faculty of Science (see 12 in the campus map) in the following rooms:

Sugimoto campus map and Access map


If you would like to register for this free workshop, please contact Travis Scrimshaw (tcscrims {at}


If you have any questions or issues fining accommodations, please contact Travis Scrimshaw (tcscrims {at}



Time Speaker Title
Monday 1:00 PM Takeshi Ikeda Set-valued decomposition tableaux
The set of decomposition tableaux (DT) introduced by Serrano is the image of a canonical section of Haiman–Serrano’s mixed insertion algorithm which is a map from the set of words to the set of marked shifted tableaux (MST) of Sagan and Worley. The decomposition tableaux was used by Cho who gave a combinatorial description of the multiplicative structure constants for Schur P-functions. The aim of talk is to explain an adaptation of this result to K-theory setting. In fact, we can introduce K-theory analogue of P-functions called GP-functions, which may be called stable Grothendieck polynomials in type B. Our main conjecture is the existence of a bijection between the set of set-valued marked shifted tableaux (SVMST) and the set of set-valued decomposition tableaux (SVDT). Finally we formulate a rule for the multiplicative structure constants for GP-functions in terms of SVDT. This talk is based on a joint work with Soojin Cho and Maki Nakasuji.
Monday 3:00 PM Daisuke Sagaki Chevalley type and Monk type formulas for level-zero Demazure modules
I'd like to explain some Chevalley type and Monk type formulas for the graded characters of Demazure modules in the level-zero extremal weight modules over quantum affine algebras. Namely,
  1. Let \(\mu \in P\) be an integral weight, and \(\lambda \in P_+\) a dominant integral weight (such that “\(\lambda \gg \mu\)”). Let \(w\) be an element of the finite Weyl group \(W_{fin}\). Denote by \(V_w^-(\lambda+\mu)\) the (opposite) Demazure module of lowest weight \(w(\lambda+\mu)\) in the level-zero extremal weight module \(V(\lambda+\mu)\). I'd like to explain a Chavalley type formula for the expansion of the graded character of \(V_w^-(\lambda+\mu)\) as a \(\mathbb{Z}[P][[q^{-1}]]\)-linear combination of the graded characters of \(V_x^-(\lambda), x \in W_{af}\), where \(W_{af}\) is the affine Weyl group.
  2. I'd like to explain a Monk type formula for the expansion of the graded character of \(V_w^-(\lambda)\) multiplied by \(e^{\mu}\) as a \(\mathbb{Z}[q]\)-linear combination of the graded characters of \(V_x^-(\nu)\), \(x \in W_{af}\) and \(\nu \in P\).
This talk is based on joint works (in progress) with Lenart, Naito, and Orr.
Tuesday 10:00 AM Jae-Hoon Kwon RSK correspondence of type D and affine crystals slides
In this talk, we consider the crystal of quantum nilpotent subalgebra of \(U_q(D_n)\) associated to a maximal Levi subalgebra of type \(A_{n-1}\). We show that it has an affine crystal structure of type \(D_n^{(1)}\) isomorphic to a limit of perfect Kirillov–Reshetikhin crystal \(B^{n,s}\) for \(s \geq 1\), and give a new polytope realization of \(B^{n,s}\). Moreover, we show that an analogue of RSK correspondence for type \(D\) due to Burge is an isomorphism of affine crystals and present a generalization of Greene's formula to type \(D\). This is a joint work with I.-S. Jang.
Tuesday 2:00 PM Yuki Kanakubo Positivity condition of Polyhedral realizations of crystal bases slides

Nakashima and Zelevinsky invented ‘polyhedral realization’, which is a kind of description of crystal bases as lattice points in some polyhedral convex cone.

To construct the polyhedral realization, we need an infinite sequence \(\iota\) of indices. In the case \(\iota\) satisfies a ‘positivity condition’, Nakashima and Zelevinsky also found a method to obtain an explicit form of the polyhedral realization associated with \(\iota\). However, it seems to be difficult to check whether \(\iota\) satisfies the positivity condition or not.

In this talk, I will give a sufficient condition of \(\iota\) for the positivity condition and an explicit form of the polyhedral realization in the case \(\iota\) satisfies the sufficient condition. This is a joint work with Toshiki Nakashima.

Wednesday 10:00 AM Toshiki Nakashima Half Potential on Geometric Crystals and the Crystal \(B(\infty)\)
First we review the full potential(=Berenstein–Kazhdan decoration) on some geometric crystals and the relation to the Kashiwara's crystal \(B(\lambda)\) applying tropicalization to this potential in the semi-simple setting. Next, we introduce the half potential on the geometric crystals and describe the crystal \(B(\infty)\) applying tropicalizaion to this half potential. We also present the relation to the polyhedral realizations. Finally, as an application, we shall show the connectedness of the cell crystal \(B_{i_1i_2\cdots i_k}\) associated with the reduced word \(i_1i_2\cdots i_k\). This is a joint work with Y.Kanakubo.
Wednesday 2:00 PM Dinakar Muthiah Constructions of affine Mirković–Vilonen polytopes

The theory of Mirković–Vilonen (MV) polytopes provides a beautiful combinatorial model of crystals that connects various disparate constructions in representation theory. By work of many people, we have explicit isomorphisms between the crystal structures on MV polytopes and on (1) Mirković–Vilonen cycles, (2) PBW bases in quantum groups, (3) preprojective varieties, and (4) KLR modules. In the past ten years, a flurry of activity has birthed a theory of affine MV polytopes. In particular, the connections between affine MV polytopes and the affine versions of (2), (3), and (4) have been established.

I will review the current state of affairs, and I will present my work with Peter Tingley that establishes connections (2) and (3) in the affine case. However, connection (1) between affine MV polytopes and affine MV cycles remains frustratingly open. I will discuss my current work that establishes the connection between affine MV cycles and “undecorated” affine MV polytopes in type A. The argument builds on my earlier work that established an isomorphism between type A affine MV cycles and the Naito–Sagaki–Saito crystal.

Thursday 10:00 AM Hiroyuki Yamane Weyl groupoids and generalized quantum groups with Kharchenko PBW theorem

Coxeter introduced Coxeter groups in 1934, and he classified the finite Coxeter groups in 1935. Those are classified into \({\mathrm{A}}_n\), \({\mathrm{B}}_n = {\mathrm{C}}_n\), \({\mathrm{D}}_n\), \({\mathrm{F}}_4\), \({\mathrm{E}}_6\), \({\mathrm{E}}_7\), \({\mathrm{E}}_8\), \({\mathrm{G}}_2\), \({\mathrm{I}}_n\), \({\mathrm{H}}_3\) and \({\mathrm{H}}_4\). Coxeter groups appear in many areas of Algebra and Geometry. One of the areas is the representation theory of Lie algebras. Perhaps from 1970's, many researchers have considered that they needed ‘Coxeter groupoids’ being applied for study of the representation theory of Lie superalgebras. In 2000's, ‘Coxeter groupoids’ also became necessary for study of Hopf algebras called ‘Generalized quantum groups (GQGs)’. A GQG can be a quantum group, a multi-parameter quantum group, a quantum group at a root of unity, a quantum superalgebra, or the quantum double of a Nichols algebra of diagonal-type.

In this talk, we mainly explain Bruhat order of Coxeter groupoids, Kharchenko PBW theorem and some important equations in the positive part of the \(A^{(1)}_1\) type affine quantum group.

Thursday 2:00 PM Satoshi Naito A description of the \(\mathbb{Z}[P]\)-module structure of the K-theory of the finite-dimensional flag manifold in terms of a generalization of LS paths
In this talk, we give a description of the canonical \(\mathbb{Z}[P]\)-module structure of the torus-equivariant K-theory \(K_{H}(G/B)\) of the (ordinary) finite-dimensional flag manifold \(G/B\), where \(P\) denotes the integral weight lattice of a simply-connected simple algebraic group \(G\) over \(\mathbb{C}\) with Borel subgroup \(B\) and maximal torus \(H\). We note that the description of the canonical \(\mathbb{Z}[P]\)-module structure of \(K_{H}(G/B)\) is equivalent to the Chevalley formula for products in \(K_{H}(G/B)\) of line bundles (associated to arbitrary integral weights) over \(G/B\) and structure sheaves of Schubert varieties; also, the description, in terms of the alcove path model, of the Chevalley formula in \(K_{H}(G/B)\) for line bundles associated to arbitrary integral weights is obtained by Lenart–Postnikov. Therefore, our result can be thought of as an interpretation of their result in terms of a generalization of (ordinary) Lakshmibai–Seshadri paths. This talk is based on a joint work (in progress) with T. Kouno and D. Orr.
Friday 10:00 AM Hiroshi Naruse Hook formula and equivariant \(K\)-theory
From the point of view of Schubert calculus, the hook formula for the number of standard tabtleaux and its generalizations are explained using localizations of equivariant cohomology or \(K\)-theory classes. In this talk I will explain the case of the generating function of reverse plane partitions on d-complete poset, which is a joint work with S. Okada. We used equivariant \(K\)-theory Chevalley rule, obtained by Lenart–Shimozono, which is very much related to LS paths and crystals. I also plan to mention some related recent progress on a (cojectured) generalization of Nakada's colored hook formula using Motivic Chern classes. This part is a joint work (in progress) with L. C. Mihalcea and C. Su.