On the Spectrum of an n! × n! Matrix Originating from Statistical Mechanics

Chassé, Dominique ; Saint-Aubin, Yvan

Canadian mathematical bulletin, Tome 52 (2009) no. 1, pp. 9-17

Résumé

Let ${{R}_{n}}\left( \alpha\right)$ be the $n!\,\times \,n!$ matrix whose matrix elements ${{\left[ {{R}_{n}}\left( \alpha\right) \right]}_{\sigma p}}$ , with $\sigma$ and $p$ in the symmetric group ${{G}_{n}}$ , are ${{\alpha }^{\ell \left( \sigma {{p}^{-1}} \right)}}$ with $0\,<\,\alpha \,<\,1$ , where $\ell \left( \text{ }\!\!\pi\!\!\text{ } \right)$ denotes the number of cycles in $\text{ }\pi \text{ }\in {{G}_{n}}.$ We give the spectrum of ${{R}_{n}}$ and show that the ratio of the largest eigenvalue ${{\text{ }\!\!\lambda\!\!\text{ }}_{0}}$ to the second largest one (in absolute value) increases as a positive power of $n$ as $n\,\to \,\infty$ .

Détail

DOI : 10.4153/CMB-2009-002-1

Mots-clés : 20B30, 20C30, 15A18, 82B20, 82B28, symmetric group, representation theory, eigenvalue, statistical physics