## Category Archives: Exercise

### Three little things I learnt recently

In no particular order, and with no relevance whatsoever to the beginning of the year, here are three mathematical facts I learnt in recent months which might belong to the “I should have known this” category: (1) Which finite fields have the property that there is a “square root” homomorphism i.e., a group homomorphism such [...]

### Another exercise with characters

While thinking about something else, I noticed recently the following result, which is certainly not new: Let be a compact topological group [ADDITIONAL ASSUMPTION pointed out by Y. Choi: connected, Lie group], and let be a finite-dimensional irreducible unitary continuous representation of on a vector space . Then the natural representation of on decomposes as [...]

### Zeros of Hermite polynomials

In my paper with É. Fouvry and Ph. Michel where we find upper bounds for the number of certain sheaves on the affine line over a finite field with bounded ramification, the combinatorial part of the argument involves spherical codes and the method of Kabatjanski and Levenshtein, and turns out to depend on the rather [...]

### Orthogonality of columns of integral unitary operators: a challenge

Given a unitary matrix of finite size, it is a tautology that the column vectors of are orthonormal, and in particular that for any \$j\not=k\$. This has an immediate analogue for a unitary operator , if is a separable Hilbert space: given any orthonormal basis of , we can define the “matrix” representing by and [...]

### On Weyl groups and gaussians

Am I the last person to notice that for , the even moment of a standard gaussian random variable (with expectation zero and variance one) is the same as the index of the Weyl group of inside the Weyl group of (in other words, the index of the groups of permutations of elements commuting with [...]