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Lecture Series on Representation Theory

Lecture 1: Geometric Crystals and Integral Polytopes Associated with Kirillov-Reshetikhin Modules

Speaker: Masato Okado (Osaka Metropolitan University)

Place and Dates: 129-406 (SNU) Talk 1: 09.15. (Tue) 10:00~11:30 Talk 2: 09.16. (Wed) 10:00~11:30 Talk 3: 09.17. (Thu) 10:00~11:30

Abstract: Among the finite-dimensional representations of quantum affine algebras, there exists a family of modules with desirable properties known as Kirillov-Reshetikhin (KR) modules. KR modules are parameterized by a vertex $r$ of the Dynkin diagram and a positive integer $l$, and they possess crystal structures in the sense of Kashiwara. Meanwhile, Berenstein and Kazhdan introduced the concept of "geometric crystals" as an algebraic-geometric analogue of crystals; it is conjectured that these exist in correspondence with sequences of KR modules where $r$ is fixed. Recently, we have obtained the "decoration functions" (in the sense of Berenstein and Kazhdan) for cases where the corresponding affine Lie algebra is of ADE type and $r$ corresponds to a minuscule weight representation. By applying ultradiscretization (tropicalization) to these geometric crystals with decoration functions, one obtains an integral convex polytope representation of the KR crystal—the crystal structure associated with the KR module—for any value of $l$.

In the talk, I will illustrate the construction procedures for crystals, geometric crystals, and integral convex polytopes using basic examples. For root systems of type A, these correspond respectively to (rectangular) Young tableaux, Grassmannians, and Gelfand-Tsetlin polytopes (corresponding to specific highest weight representations). This conjecture can be extended to all root systems, yet the area remains almost entirely unexplored for types other than type A.

Lecture 2: Representation theory of quantum symmetric pairs and related combinatorics

Speaker: Hideya Watanabe (Rikkyo University)

Place and Dates: 129-406 (SNU) Talk 1: 09.15. (Tue) 15:00~16:30 Talk 2: 09.16. (Wed) 15:00~16:30 Talk 3: 09.17. (Thu) 15:00~16:30

Abstract: Given a symmetric pair (g,k), a pair of complex reductive Lie algebras, one has its quantum deformation (U_q(g), U^i(k)), called a quantum symmetric pair (QSP for short). A QSP consists of a quantum group U_q(g) and its certain coideal subalgebra U^i(k), called an i-quantum group. Although the i-quantum group U^i(k) is a quantum deformation of the Lie algebra k, it is different from the quantum group U_q(k), in general. As a result, the representation theory of i-quantum groups has a different flavor than that of quantum groups. In these lectures, we focus on non-Levi branching rules, i.e., the k-module structure of various g-modules. Such problems cannot be solved by means of quantum groups because U_q(k) is not a subalgebra of U_q(g).

In the first part of the lectures, we review representation theory of quantum groups quickly. Then, we introduce QSPs and their integrable modules. We show that the integrable modules for a QSP leads us to a Peter-Weyl type decomposition of the dual of the i-quantum group.

In the remaining parts, we introduce some interesting type-dependent results. We focus on type AI, AII, BII, and DII; the corresponding symmetric pairs are (gl_n, so_n), (gl_2n, sp_2n), (so_2n+1, so_2n), and (so_2n, so_2n-1). For these pairs, we can understand the branching rules in terms of some combinatorial objects. Such combinatorial objects include semistandard Young tableaux, King's symplectic tableaux, Kashiwara-Nakashima's orthogonal tableaux, and Gelfand-Tsetlin patterns.

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