Crystals and KLR representations in type $A_1^{(1)}$

Abstract: Crystal bases are a powerful tool for studying representations of quantum groups, and the crystal $B(\infty)$ plays a central organizing role: it encodes the combinatorial skeleton of highest weight representations and arises naturally in the categorification program. I will discuss two important models for this crystal in the ‘easiest' affine type $A_1^{(1)}$: Kleshchev multipartitions, which describe branching rules for cyclotomic Hecke algebras, and affine MV polytopes, which encode PBW data for the affine $\mathfrak{sl}_2$ quantum group. I will sketch a combinatorial dictionary between these models. These two perspectives interact naturally in the realm of KLR algebras, where they govern different representation-theoretic regimes. Translating between regimes recovers some new results in the 2-modular representation theory of symmetric groups.

combinatoricsquantum algebrarings and algebrasrepresentation theory

Audience: researchers in the topic

OIST representation theory seminar

Series comments: Timings of this seminar may vary from week to week.