Representations of symmetric groups with non-trivial determinant

TL;DR

This paper provides formulas for counting partitions of n related to irreducible and permutation representations of symmetric groups with non-trivial determinants, including classifications of self-conjugate and hook partitions.

Contribution

It introduces a closed-form formula for counting partitions with non-trivial determinants of irreducible and permutation representations of symmetric groups, based on characterizing 2-core towers.

Findings

Formulas for the number of partitions with non-trivial determinants of irreducible representations.

Classification of such partitions into self-conjugate and hook types.

Characterization of 2-core towers for these partitions.

Abstract

We give a closed formula for the number of partitions $\lambda$ of $n$ such that the corresponding irreducible representation $V_\lambda$ of $S_n$ has non-trivial determinant. We determine how many of these partitions are self-conjugate and how many are hooks. This is achieved by characterizing the $2$-core towers of such partitions. We also obtain a formula for the number of partitions of $n$ such that the associated permutation representation of $S_n$ has non-trivial determinant.