Two Propositions Involving the Standard Representation of $S_n$

TL;DR

This paper provides explicit formulas for tensor power decompositions of certain representations of the symmetric group and proves a fixed point averaging property for specific Markov chains on $S_n$, advancing understanding in algebraic combinatorics and Markov chain analysis.

Contribution

It introduces explicit formulas for tensor power decompositions of the defining and standard representations of $S_n$ and establishes a fixed point averaging property for class measure-based Markov chains.

Findings

Explicit formulas for tensor power decompositions of $S_n$ representations

Proof that certain Markov chains on $S_n$ always average exactly one fixed point

Abstract

We present here two standalone results from a forthcoming work on the analysis of Markov chains using the representation theory of $S_n$. First, we give explicit formulas for the decompositions of tensor powers of the defining and standard representations of $S_n$. Secondly, we prove that any Markov chain on $S_n$ starting with one fixed point and whose increment distributions are class measures will always average exactly one fixed point.