The Standard Representation of the Symmetric Group $S_n$ over the Ring of Integers

TL;DR

This paper introduces a Casimir invariant for the symmetric group $S_n$ and provides explicit formulas for the standard representation matrices, enhancing the understanding of symmetric group representations.

Contribution

It presents the first explicit formulas for standard representation matrices of $S_n$ in terms of permutation matrices and introduces a new Casimir invariant.

Findings

Explicit formulas for standard representation matrices of $S_n$

Introduction of a Casimir invariant for $S_n$

Enhanced understanding of symmetric group representations

Abstract

In this paper we give a Casimir Invariant for the Symmetric group $S_n$. Furthermore we obtain and present, for the first time in the literature, explicit formulas for the matrices of the standard representation in terms of the matrices of the permutation representation.