Modular Representation Theory of Symmetric Groups

TL;DR

This paper reviews recent developments in the modular representation theory of symmetric groups, highlighting connections with Khovanov-Lauda-Rouquier algebras, gradings, categorification, and dualities, advancing understanding of algebraic structures and their representations.

Contribution

It synthesizes recent advances in modular representation theory, emphasizing new connections with categorification, quantum groups, and dualities that deepen the theoretical framework.

Findings

Connections between symmetric groups and KLR algebras clarified

Gradings and cellular structures on group algebra blocks analyzed

New dualities and categorification approaches established

Abstract

We review some recent advances in modular representation theory of symmetric groups and related Hecke algebras. We discuss connections with Khovanov-Lauda-Rouquier algebras and gradings on the blocks of the group algebras $F\Sigma_n$, which these connections reveal; graded categorification and connections with quantum groups and crystal bases; modular branching rules and the Mullineaux map; graded cellular structure and graded Specht modules; cuspidal systems for affine KLR algebras and imaginary Schur-Weyl duality, which connects representation theory of these algebras to the usual Schur algebras of smaller rank.