Characters of odd degree of symmetric groups

TL;DR

This paper establishes a natural correspondence between odd-degree irreducible characters of symmetric groups and linear characters of their Sylow 2-subgroups, revealing structural insights especially when n is a power of 2.

Contribution

It introduces a bijection linking odd-degree characters of S_n to linear characters of P_n and characterizes when restrictions have a unique linear constituent.

Findings

Bijection between odd-degree irreducible characters and linear characters of P_n

Special case when n is a power of 2 with induced bijection

Characterization of irreducible characters with unique linear restriction

Abstract

We construct a natural bijection between odd-degree irreducible characters of S_n and linear characters of its Sylow 2-subgroup P_n. When n is a power of 2, we show that such a bijection is nicely induced by the restriction functor. We conclude with a characterization of the irreducible characters \chi of S_n such that the restriction of \chi to P_n has a unique linear constituent.