Young's Natural Representations of $\mathcal{S}_4$

TL;DR

This paper explicitly computes all irreducible representations of the symmetric group S4 using Young tableaux methods, providing detailed matrix representations for the group's elements.

Contribution

It offers a complete, explicit construction of S4's irreducible representations, enhancing understanding of symmetric group representations through concrete matrix forms.

Findings

All inequivalent irreducible representations of S4 are explicitly calculated.

Matrices for adjacent transpositions are specified in detail.

Methodology based on standard Young tableaux is demonstrated.

Abstract

We calculate all inequivalent irreducible representations of $\S_4$ by specifying the matrices for adjacent transpositions and indicating how to obtain general permutations in $\S_4$ from these transpositions. We employ standard Young tableaux methods as found in Sagan's \emph{The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions} (2001).