Two Propositions Involving the Standard Representation of $S_n$

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.