Representations of symmetric groups with non-trivial determinant

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.

22 pages, 4 figures

References in corpus (1)

• Odd partitions in Young's lattice

Cited by in corpus (2)

• On the determinant of representations of generalized symmetric groups
• Macdonald trees and determinants of representations for finite Coxeter groups