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# Ordering the Representations of $S_n$ Using the Interchange Process

Published:2011-07-15
Printed: Mar 2013
• Gil Alon,
Division of Mathematics, The Open University of Israel, Raanana 43107, Israel
Department of Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel
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## Abstract

Inspired by Aldous' conjecture for the spectral gap of the interchange process and its recent resolution by Caputo, Liggett, and Richthammer, we define an associated order $\prec$ on the irreducible representations of $S_n$. Aldous' conjecture is equivalent to certain representations being comparable in this order, and hence determining the Aldous order'' completely is a generalized question. We show a few additional entries for this order.
 Keywords: Aldous' conjecture, interchange process, symmetric group, representations
 MSC Classifications: 82C22 - Interacting particle systems [See also 60K35] 60B15 - Probability measures on groups or semigroups, Fourier transforms, factorization 43A65 - Representations of groups, semigroups, etc. [See also 22A10, 22A20, 22Dxx, 22E45] 20B30 - Symmetric groups 60J27 - Continuous-time Markov processes on discrete state spaces 60K35 - Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]