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 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.
