Card shuffling and the decomposition of tensor products

• Autores: Jason Fulman
• Localización: Pacific journal of mathematics, ISSN 0030-8730, Vol. 217, Nº 2, 2004, págs. 247-262
• Idioma: inglés
• DOI: 10.2140/pjm.2004.217.247
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• Resumen

  • Let H be a subgroup of a .nite 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 .nd that this question is precisely equivalent to the question of how large r should be so that r iterations of a shu.ing method randomise 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 shu.ing problem is related to a nonreversible Markov chain on the symmetric group. The equivalence is also useful, and results on card shu.ing can be applied to yield sharp results about the decomposition of tensor powers.