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 Ktheory, geometric crystals, and Lie superalgebras. There will be a limited number of talks to encourage discussions among the participants.
When:
March 25–29, 2019
Where:
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:
 March 2527: Bldg. F, Room F215 (2F)
 March 28,29: Bldg. E, Room E408 (4F)
Sugimoto campus map and Access map
Registration:
If you would like to register for this free workshop, please contact Travis Scrimshaw (tcscrims {at} gmail.com).
Accommodations:
If you have any questions or issues fining accommodations, please contact Travis Scrimshaw (tcscrims {at} gmail.com).
Speakers
 Takeshi Ikeda (Okayama University of Science)
 Yuki Kanakubo (Sophia University)
 JaeHoon Kwon (Seoul National University)
 Dinakar Muthiah (Kavli Institute for the Physics and Mathematics of the Universe)
 Satoshi Naito (Tokyo Institute of Technology)
 Toshiki Nakashima (Sophia University)
 Hiroshi Naruse (University of Yamanashi)
 Daisuke Sagaki (University of Tsukaba)
 Hiroyuki Yamane (University of Toyama)
Schedule
Time  Speaker  Title 

Monday 1:00 PM  Takeshi Ikeda  Setvalued 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 Pfunctions. The aim of talk is to explain an adaptation of this result to Ktheory setting. In fact, we can introduce Ktheory analogue of Pfunctions called GPfunctions, which may be called stable Grothendieck polynomials in type B. Our main conjecture is the existence of a bijection between the set of setvalued marked shifted tableaux (SVMST) and the set of setvalued decomposition tableaux (SVDT). Finally we formulate a rule for the multiplicative structure constants for GPfunctions 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 levelzero Demazure modules 
I'd like to explain some Chevalley type and Monk type formulas for the
graded characters of Demazure modules in the levelzero extremal
weight modules over quantum affine algebras. Namely,


Tuesday 10:00 AM  JaeHoon 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 \(A_{n1}\). 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 semisimple 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 multiparameter quantum group, a quantum group at a root of unity, a quantum superalgebra, or the quantum double of a Nichols algebra of diagonaltype. 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 Ktheory of the finitedimensional 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 torusequivariant Ktheory \(K_{H}(G/B)\) of the (ordinary) finitedimensional flag manifold \(G/B\), where \(P\) denotes the integral weight lattice of a simplyconnected 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 dcomplete 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. 
Organizers
 Travis Scrimshaw, The University of Queensland (tcscrims {at} gmail.com)
 Oliver Pechenik, University of Michigan (pechenik {at} umich.com)
 Masato Okado, Osaka City University (okado {at} sci.osakacu.ac.jp)