An introduction to the symmetric group algebra

TL;DR

This paper provides a comprehensive, elementary introduction to the symmetric group algebra, covering definitions, key elements, representations, and bases, suitable for graduate students with minimal prerequisites.

Contribution

It offers a detailed, accessible exposition of symmetric group algebra theory, including classical and less-known results, with a focus on elementary methods and computational approaches.

Findings

Characterization of irreducible representations in characteristic 0

Description of Specht modules and duals

Construction of Murphy cellular bases

Abstract

This is an introduction to the group algebras of the symmetric groups, written for a quarter-long graduate course. After recalling the definition of group algebras (and monoid algebras) in general, as well as basic properties of permutations, we introduce several families of elements in the symmetric group algebras $\mathbf{k}[S_n]$ such as the Young--Jucys--Murphy elements, the (sign-)integrals and the conjugacy class sums. Then comes a chapter on group actions and representations in general, followed by the core of this text: a study of the representations of symmetric groups (i.e., of left $\mathbf{k}[S_n]$-modules), including the classical theory of Young tableaux and Young symmetrizers. We prove in detail the main facts including the characterization of irreducible representations (in characteristic $0$), the Garnir relations, the standard basis theorem, the description of duals of Specht modules, and the hook length formula, as well as a number of less known results. Finally, we describe several bases of $\mathbf{k}[S_n]$ that arise from the study of Specht modules, including the Murphy cellular bases. The methods used are elementary and computational. We aim to assume as little as possible of the base ring $\mathbf{k}$, and to use as little as possible from representation theory (nothing more advanced than Maschke and Jordan--H\"older). Over 100 exercises (without solutions) are scattered through the text.