The Annals of Probability

Spectra of random linear combinations of matrices defined via representations and Coxeter generators of the symmetric group

Steven N. Evans

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We consider the asymptotic behavior as n→∞ of the spectra of random matrices of the form

\[\frac{1}{\sqrt{n-1}}\sum_{k=1}^{n-1}Z_{nk}\rho_{n}\bigl ((k,k+1)\bigr),\]

where for each n the random variables Znk are i.i.d. standard Gaussian and the matrices ρn((k, k+1)) are obtained by applying an irreducible unitary representation ρn of the symmetric group on {1, 2, …, n} to the transposition (k, k+1) that interchanges k and k+1 [thus, ρn((k, k+1)) is both unitary and self-adjoint, with all eigenvalues either +1 or −1]. Irreducible representations of the symmetric group on {1, 2, …, n} are indexed by partitions λn of n. A consequence of the results we establish is that if λn,1λn,2≥⋯≥0 is the partition of n corresponding to ρn, μn,1μn,2≥⋯≥0 is the corresponding conjugate partition of n (i.e., the Young diagram of μn is the transpose of the Young diagram of λn), limn→∞λn,i/n=pi for each i≥1, and limn→∞μn,j/n=qj for each j≥1, then the spectral measure of the resulting random matrix converges in distribution to a random probability measure that is Gaussian with random mean θZ and variance 1−θ2, where θ is the constant ∑ipi2−∑jqj2 and Z is a standard Gaussian random variable.

Article information

Ann. Probab., Volume 37, Number 2 (2009), 726-741.

First available in Project Euclid: 30 April 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 15A52 60F99: None of the above, but in this section
Secondary: 20C30: Representations of finite symmetric groups

Random matrix eigenvalue irreducible representation transposition Coxeter generator Hermite polynomial Wiener integral character ratio Young’s orthogonal representation domino tiling


Evans, Steven N. Spectra of random linear combinations of matrices defined via representations and Coxeter generators of the symmetric group. Ann. Probab. 37 (2009), no. 2, 726--741. doi:10.1214/08-AOP418.

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