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The moment matrix of a Sato-Tate group $G$ is the matrix of moments $\mathrm{E}[\chi_i\chi_j]$ with $\chi_i$ and $\chi_j$ ranging over the irreducible characters of the ambient group that contains $G$; The irreducible characters are ordered according to their highest weight, which can be written as a monomial in the elementary symmetric functions; these are ordered by (unweighted) degree and then by reverse lexicographic order of exponent vectors. The diagonal of the moment matrix is the vector $(\mathrm{E}[\chi_i^2])$.

In the symplectic case the irreducible characters of $\mathrm{USp}(2g)$ can be explicitly computed via the Brauer-Klimyk formula using the algorithm of Shieh [arXiv:1605.07743 , 10.1112/S1461157016000279 ]. Each character $\chi$ is then an integer polynomial in $a_1,\ldots a_g$ and the moments $\mathrm{E}[\chi_i\chi_j]$ can be computed as integer linear combinations of entries of the moment simplex corresponding to the terms of $\chi_i\chi_j$.

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• Last edited by Andrew Sutherland on 2022-01-21 13:56:33
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