## On Weyl groups and gaussians

Am I the last person to notice that for $k\geq 0$, the even moment
$m_{2k}=\frac{(2k)!}{2^kk!}$
of a standard gaussian random variable (with expectation zero and variance one) is the same as the index of the Weyl group of $\mathrm{Sp}_{2k}$ inside the Weyl group of $\mathrm{GL}_{2k}$ (in other words, the index of the groups of permutations of $2k$ elements commuting with a fixed-point free involution among all permutations)?

If “Yes”, what else have I been missing in the same spirit?

### 3 Responses to “On Weyl groups and gaussians”

1. Qiaochu Yuan wrote:

I am having a hard time understanding what you mean by “in the same spirit.”

Thursday, November 8, 2012 at 1:02 | Permalink
2. Kowalski wrote:

Well, it could be about other probability distributions whose moments (or some of them) have algebraic interpretations, or cases where this interpretation gives easy proofs of convergence to the normal law….