Separation Cutoffs for Random Walk on Irreducible Representations

Abstract

Random walk on the irreducible representations of the symmetric and general linear groups is studied. A separation distance cutoff is proved and the exact separation distance asymptotics are determined. A key tool is a method for writing the multiplicities in the Kronecker tensor powers of a fixed representation as a sum of non-negative terms. Connections are made with the Lagrange-Sylvester interpolation approach to Markov chains.

Mixing Time for a Random Walk on Rooted Trees

“…Note that it takes n − 2 iterations of the Markov chain K to move from t to t ′ . By Proposition 4.1, K has n − 1 distinct eigenvalues (one more than the Markov chain distance between t and t ′ ), so it follows from Proposition 5.1 of [16] that…”

Tensor quasi-random groups

“…In Section 4 we specialize our results to the tensor product Markov chains studied in [Ful08,Ful04,Ful10,BDLT19]. The result is that the tensor quasi-randomness of a group G characterizes whether certain tensor product chains mix in constant time.…”

Tensor Powers of the Defining Representation of $$S_n$$ S n

“…Corollary 2.2 is very similar to Proposition 2 of [8], but our result is cleaner, as it does not involve associated Stirling numbers of the second kind. For another approach to the decomposition of tensor powers of ̺, see [6].…”