Representation Theory of Symmetric Groups and Related Hecke Algebras

TL;DR

This survey reviews recent advances in the representation theory of symmetric groups and related Hecke algebras, emphasizing connections with Lie theory, categorification, and algebraic structures like Khovanov-Lauda-Rouquier algebras.

Contribution

It compiles and explains recent developments in the field, highlighting the interplay between symmetric group representations and Lie-theoretic categorification methods.

Findings

Overview of branching rules and crystal graphs

Discussion of decomposition numbers and canonical bases

Connections with cyclotomic, affine Hecke algebras, and categorification

Abstract

This is an expository article. We survey some fundamental trends in representation theory of symmetric groups and related objects which became apparent in the last fifteen years. The emphasis is on connections with Lie theory via categorification. We present results on branching rules and crystal graphs, decomposition numbers and canonical bases, graded representation theory, connections with cyclotomic and affine Hecke algebras, Khovanov-Lauda-Rouquier algebras, category ${\mathcal O}$, $W$-algebras, ...