Modular Representation Theory of Symmetric Groups

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.Comment: This is an expository paper for ICM proceeding

Representation theory and cohomology of Khovanov-Lauda-Rouquier algebras

This expository paper is based on the lectures given at the program `Modular Representation Theory of Finite and $p$-adic Groups' at the National University of Singapore. We are concerned with recent results on representation theory and cohomology of KLR algebras, with emphasis on standard module theory.Comment: arXiv admin note: text overlap with arXiv:1210.655