Representations of Symmetric Groups

A surprising theorem in the modular representation theory of symmetric groups uses induction and restriction functors to define an action of an affine Kac-Moody special linear algebra on the level of Grothendieck groups. This action identifies the direct sum of Grothendieck groups with an integrable highest weight module of the Kac-Moody algebra. The purpose of this write-up is to provide a gentle introduction to these theorems. This write-up, still a work in progress, is mostly expository with few proofs. No technical background is assumed — all unfamiliar terms in this paper are either explicitly defined or blackboxed. The aim of this write-up is to be accessible to the average graduate student with enough patience. The reader who is rusty or unfamiliar with the basics of rings, fields, and modules may need to review those concepts first. For the sake of presentation, a technical summary of the contents of this paper is included at the end instead of this introduction, in section 11. You may have heard that the irreducible representations of symmetric groups over complex numbers can be classified explicitly by Young Diagrams. Sections 1 through 3 begin by describing this classification via eigenspace decompositions of certain elements of the group algebra, along with relevant induction and restriction functors. We then give some examples in section 4 of how one can use this theory to compute irreducible representations. Section 5 describes some relations satisfied by the induction and restriction functors. We then elucidate in sections 6 through 8 how the same analysis generalizes to the characteristic p case. In this case, the theory allows us to compute a large class of representations, but it is unknown how to compute the irreducible representations. A later version of this paper will probably include some example computations in this case, similar to section 4. Sections 9 and 10 generalize the theory to related Cyclotomic Hecke algebras, which completes the picture of a major theorem which was conjectured by Lascoux, Leclerc, and Thibon and proved by Ariki in the 1990s. This paper is a work in progress. Sections 1 through 7 are mostly completed.