Matrix generators for Weil representations

Abstract

Let $r$ be an odd prime and $\mathbb{F}$ a field containing a primitive $r$th root of unity. Then for all $\ell \ge 1$, there is a faithful representation $f:{\mathrm{Sp}}_{2\ell }(r)\to {\mathrm{GL}}_{r^{\ell }}(\mathbb{F})$ called the Weil representation. We provide explicit matrices generating ${\mathrm{Sp}}_{2\ell }(r)$ in ${\mathrm{GL}}_{r^{\ell }}(\mathbb{F})$, which we have implemented in Magma. We also describe such generators for the irreducible Weil representations of ${\mathrm{Sp}}_{2\ell }(r)$, which are of degree $(r^{\ell }\pm 1)/2$ and arise as irreducible constituents of the Weil representations.

11 pages