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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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감사합니다.
이승빈 올림.
