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


Published:20090301
Printed: Mar 2009
Dominique ChassÃ©
Yvan SaintAubin
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 $n\rightarrow \infty$.
MSC Classifications: 
20B30, 20C30, 15A18, 82B20, 82B28 show english descriptions
Symmetric groups Representations of finite symmetric groups Eigenvalues, singular values, and eigenvectors Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs Renormalization group methods [See also 81T17]
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]
