E. Kowalski’s blog

Comments on mathematics, mostly.

On Weyl groups and gaussians

with 3 comments

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?

Written by Kowalski

November 7th, 2012 at 4:50 pm

Posted in Exercise,Mathematics

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3 Responses to 'On Weyl groups and gaussians'

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

    Qiaochu Yuan

    8 Nov 12 at 1:02

  2. 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….

    Kowalski

    8 Nov 12 at 11:03

  3. Are you looking for something like Theorem 1 in my paper “Averages over classical Lie groups, twisted by characters”
    http://user.math.uzh.ch/dehaye/papers/Averages_over_classical_Lie_groups,_twisted_by_characters.pdf
    ?
    The paper does the orthogonal case as well. (Most certainly there are other anterior references for what you asked, but this might be a starting point).

    Paul Dehaye

    8 Nov 12 at 23:14

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