  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.
