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# On the Spectrum of an $n!\times n!$ Matrix Originating from Statistical Mechanics

Published:2009-03-01
Printed: Mar 2009
• Dominique Chassé
• Yvan Saint-Aubin
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## Abstract

Let $R_n(\alpha)$ be the $n!\times n!$ matrix whose matrix elements $[R_n(\alpha)]_{\sigma\rho}$, with $\sigma$ and $\rho$ in the symmetric group $\sn$, are $\alpha^{\ell(\sigma\rho^{-1})}$ with $0<\alpha<1$, where $\ell(\pi)$ denotes the number of cycles in $\pi\in \sn$. We give the spectrum of $R_n$ and show that the ratio of the largest eigenvalue $\lambda_0$ to the second largest one (in absolute value) increases as a positive power of $n$ as $n\rightarrow \infty$.
 Keywords: symmetric group, representation theory, eigenvalue, statistical physics
 MSC Classifications: 20B30 - Symmetric groups 20C30 - Representations of finite symmetric groups 15A18 - Eigenvalues, singular values, and eigenvectors 82B20 - Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 82B28 - Renormalization group methods [See also 81T17]