The Representation Theory of 2-Sylow Subgroups of the Symmetric Group

TL;DR

This paper investigates the structure of Sylow 2-subgroups of symmetric groups using binary trees, revealing recursive and self-similar properties, and analyzing the multiplicities of irreducible characters.

Contribution

It introduces a recursive, self-similar model for the Bratteli diagram of Sylow 2-subgroups and computes character multiplicities for odd-dimensional symmetric group representations.

Findings

The Bratteli diagram is simple, recursive, and self-similar.

The subgraph of one-dimensional representations is contrasted with the Macdonald tree.

Explicit multiplicities of irreducible characters are derived for certain representations.

Abstract

We use binary trees to study the Bratteli diagram of Sylow 2-subgroups of symmetric groups. We show that it is simple, has a recursive structure, and self-similarities at all scales. We contrast its subgraph of one-dimensional representations with the Macdonald tree. We exploit the recursive structure to find the multiplicities of irreducible characters in the restriction to a Sylow 2-subgroup of odd-dimensional representations of the symmetric group $S_{2^k}$.