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Spectra of random linear combinations of matrices defined via representations and Coxeter generators of the symmetric group. (English) Zbl 1168.15017

Many works exist on various aspects of the distribution of the eigenvalues of large real symmetric or complex self-adjoint random matrices with simple stochastic structure. Recently, certain interest has been aroused in random matrices with more structure.
In this paper, the ensembles of highly structured random matrices defined in terms of representations of the symmetric group are investigated. The asymptotic behavior is analyzed as \(n \rightarrow \infty\) of the spectrum of the random matrices of the form
\[ \frac{1}{\sqrt{n-1}} \sum_{k=1}^{n-1}Z_{nk} \rho_n((k, k+1)), \]
where the random variables \(Z_{nk}\) are i.i.d. standard Gaussian and the matrices \(\rho_n((k,k+1))\) are obtained by applying an irreducible unitary representation \(\rho_n\) of the symmetric group of permutations on the set \(\{1,2, \dots, n \}\) to the transposition \((k,k+1)\) that interchanges \(k\) and \(k+1\) and satisfies the Coxeter relations. Irreducible representations of the symmetric group on \(\{1,2, \dots, n \}\) are indexed by a partition \(\lambda_n\) of \(n\).
The author establishes that if \(\lambda_{n,1}\geq \lambda_{n,2}\geq \dots \geq 0\) is a partition of \(n\) corresponding to \(\rho_n\), \(\mu_{n,1} \geq \mu_{n,2} \geq \dots \geq 0\) is the corresponding conjugate partition of \(n\), \(\lim_{n \rightarrow \infty} \frac{\lambda_{n,i}}{n}= \rho_i\) for each \(i \geq 1\), and \(\lim_{n \rightarrow \infty} \frac{\mu_{n,j}}{n}= q_j\) for each \(j \geq 1\), then the spectral measure of the resulting random matrix converges in distribution to a random probability measure that is Gaussian with random mean \(\theta Z\) and variance \(1-\theta^2\), where \(\theta\) is the constant \(\sum_i p_i^2 - \sum_j q_j^2\) and \(Z\) is a standard Gaussian random variable with mean \(0\) and variance \(\frac{1}{n-1}\).

MSC:

15B52 Random matrices (algebraic aspects)
20C30 Representations of finite symmetric groups
60F05 Central limit and other weak theorems
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