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

http://dx.doi.org/10.4153/CMB-2009-002-1
Canad. Math. Bull. 52(2009), 9-17

  Published:2009-03-01
 Printed: Mar 2009

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 $n\rightarrow \infty$.