  Let H be a subgroup of a finite group G. We use Markov chains to quantify how
large r should be so that the decomposition of the r tensor power of the
representation of G on cosets on H behaves (after renormalization) like the
regular representation of G. For the case where G is a symmetric group and H a
parabolic subgroup, we find that this question is precisely equivalent to the
question of how large r should be so that r iterations of a shuffling method
randomize the Robinson-Schensted-Knuth shape of a permutation. This equivalence
is rather remarkable, if only because the representation theory problem is
related to a reversible Markov chain on the set of representations of the
symmetric group, whereas the card shuffling problem is related to a
nonreversible Markov chain on the symmetric group. The equivalence is also
useful, and results on card shuffling can be applied to yield sharp results
about the decomposition of tensor powers.
