## The complex Wishart distribution and the symmetric group.(English)Zbl 1019.62047

Summary: Let $$V$$ be the space of $$(r,r)$$ Hermitian matrices and let $$\Omega$$ be the cone of the positive definite ones. We say that the random variable $$S$$, taking its values in $$\overline\Omega$$, has the complex Wishart distribution $$\gamma_{p,\sigma}$$ if $$\mathbb{E}(\text{exp trace}(\theta S))=(\det(I_r -\sigma \theta))^{-p}$$, where $$\sigma$$ and $$\sigma^{-1} -\theta$$ are in $$\Omega$$, and where $$p=1,2,\dots, r-1$$ or $$p>r-1$$. We compute all moments of $$S$$ and $$S^{-1}$$. The techniques involve in particular the use of the irreducible characters of the symmetric group.

### MSC:

 62H05 Characterization and structure theory for multivariate probability distributions; copulas 20B30 Symmetric groups 20C30 Representations of finite symmetric groups 60E05 Probability distributions: general theory

GAP
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### References:

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