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Search: MSC category 15A18 ( Eigenvalues, singular values, and eigenvectors )

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1. CMB 2009 (vol 52 pp. 9)

Chassé, Dominique; Saint-Aubin, Yvan
On the Spectrum of an $n!\times n!$ Matrix Originating from Statistical Mechanics
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
Categories:20B30, 20C30, 15A18, 82B20, 82B28

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