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On the Spectrum of an n! × n! Matrix Originating from Statistical Mechanics
Published online by Cambridge University Press: 20 November 2018
Abstract
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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 
$n\,\to \,\infty$.
- Type
- Research Article
- Information
- Copyright
- Copyright © Canadian Mathematical Society 2009
References
[1] Langlands, R. P., The renormalization fixed point as a mathematical object, In: Twenty years of Bialowieza: Aspects of differential geometry methods in physics, World Scientific, Hacksensack, NJ, 2005, pp. 185–216. Downloadable from sunsite.ubc.ca/DigitalMathArchive/Langlands/intro.html.Google Scholar
[2] James, G. D.,The Representation Theory of the Symmetric Groups. Lecture Notes inMathematics, 682, Springer, Berlin, 1978.Google Scholar
[3] Sagan, B. E., The symmetric group. Representations, combinatorial algorithms, and symmetric functions.
2nd edition, Springer-Verlag, New York, 2001.Google Scholar
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