The hyperoctahedral group, symmetric group representations and the moments of the real Wishart distribution (Q1776115)

Let \(r\) be a positive integer, \(\Omega\) be the cone of the real symmetric positive definite \(r\times r\) matrices, \(\bar\Omega\) be the cone of the nonnegative matrices, \(e\) be the unit matrix. If \[ p\in\Lambda=\left\{{1\over 2}, {2\over 2}, \dots,{{r-1}\over 2}\right\}\cup\left({{r-1}\over 2}, \infty\right) \] and if \(\sigma\in\Omega\), then the Wishart distribution \(\gamma_{p, \sigma}\) on \(\bar\Omega\) is defined by its Laplace transform as follows: for \(-\theta\in\Omega\), the Laplace transform of \(\gamma_{p, \sigma}\) is \( \int_{\bar\Omega}\exp \text{tr}(\theta s)\gamma_{p, \sigma}(ds)=(\text{det}(e-\theta_\sigma))^{-p}. \) The authors compute all the moments of the real Wishart distribution. To do so, they use the Gelfand pair \((S_{2k}, H)\), where \(H\) is the hyperoctahedral group, \(H\)-spherical representations of the symmetric group \(S_{2k}\) and some techniques based on graphs.