1. Symmetric groups and their young subgroups 2. Ordinary irreducible representations and characters of symmetric and alternating groups 3. Ordinary irreducible matrix representations of symmetric groups 4. Representations of wreath products 5. Applications to combinatories and representation theory 6. Modular representations 7. Representation theory of Sn over an arbitrary field 8. Representations of general linear groups Appendices Index.

The representation theory of the symmetric group

Introduction to lie algebras and representation theory

We construct a 2-category associated with a Kac-Moody algebra and we study its 2-representations. This generalizes earlier work with Chuang for type A. We relate categorifications relying on K_0 properties and 2-representations.

2-Kac-Moody algebras

/pdf/a-diagrammatic-approach-to-categorification-of-quantum-28fc3fedas.pdf

A diagrammatic approach to categorification of quantum groups II

We define and study sl2-categorifications on abelian categories. We show in particular that there is a self-derived (even homotopy) equivalence categorifying the adjoint action of the simple reflection. We construct categorifications for blocks of symmetric groups and deduce that two blocks are splendidly Rickard equivalent whenever they have isomorphic defect groups and we show

http://annals.math.princeton.edu/wp-content/uploads/annals-v167-n1-p05.pdf

Derived equivalences for symmetric groups and sl2-categorification

We construct an explicit isomorphism between blocks of cyclotomic Hecke algebras and (sign-modified) cyclotomic Khovanov-Lauda algebras in type A. These isomorphisms connect the categorification conjecture of Khovanov and Lauda to Ariki’s categorification theorem. The Khovanov-Lauda algebras are naturally graded, which allows us to exhibit a non-trivial ℤ-grading on blocks of cyclotomic Hecke algebras, including symmetric groups in positive characteristic.

/pdf/blocks-of-cyclotomic-hecke-algebras-and-khovanov-lauda-13rby7hieq.pdf

Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras

/pdf/kazhdan-lusztig-polynomials-and-character-formulae-for-the-5c0sk5y5jg.pdf

Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra (|)

We construct an explicit isomorphism between blocks of cyclotomic Hecke algebras and (sign-modified) Khovanov-Lauda algebras in type A. These isomorphisms connect the categorification conjecture of Khovanov and Lauda to Ariki's categorification theorem. The Khovanov-Lauda algebras are naturally graded, which allows us to exhibit a non-trivial Z-grading on blocks of cyclotomic Hecke algebras, including symmetric groups in positive characteristic.

We prove that integral blocks of parabolic category O associ- ated to the subalgebra glm(C) � gln(C) of glm+n(C) are Morita equivalent to quasi-hereditary covers of generalised Khovanov algebras. Although this result is in principle known, the existing proof is quite indirect, going via perverse sheaves on Grassmannians. Our new approach is completely alge- braic, exploiting Schur-Weyl duality for higher levels. As a by-product we get a concrete combinatorial construction of 2-Kac-Moody representations in the sense of Rouquier corresponding to level two weights in finite type A.

/pdf/highest-weight-categories-arising-from-khovanov-s-diagram-32l629ekbp.pdf

Highest weight categories arising from khovanov's diagram algebra iii: category o

https://pages.uoregon.edu/brundan/papers/lltfinal.pdf

Jonathan Brundan

Papers

Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras

Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra (|)

Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras

Highest weight categories arising from khovanov's diagram algebra iii: category o

Graded decomposition numbers for cyclotomic Hecke algebras