## Annals of Probability

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

Steven N. Evans

#### Abstract

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

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

Dates
First available in Project Euclid: 30 April 2009

https://projecteuclid.org/euclid.aop/1241099927

Digital Object Identifier
doi:10.1214/08-AOP418

Mathematical Reviews number (MathSciNet)
MR2510022

Zentralblatt MATH identifier
1168.15017

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

#### Citation

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. https://projecteuclid.org/euclid.aop/1241099927