Matrix generators for Weil representations

TL;DR

This work delivers explicit matrix generators for the Weil representations of the finite symplectic groups $Sp_{2\ell}(r)$ over fields containing a primitive $r$th root of unity. By employing a tensor-constructed extraspecial $r$-group and a Jordan-type normalizer, the authors construct a subgroup $G$ that maps onto $Sp_{2\ell}(r)$ via the Weil representation and prove $G \cong Sp_{2\ell}(r)$ (with a special case treated separately when $(r,\ell)=(3,1)$). They also analyze the submodule structure to realize the irreducible Weil representations of degrees $(r^{\ell} \pm 1)/2$ and supply explicit generators for these modules, along with a Magma implementation. The results provide practical, computable tools for working with Weil representations and their submodules, with extensions to related groups and tensor constructions.

Abstract

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

Theorems & Definitions (26)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Remark 1.4
• Remark 1.5
• Remark 1.6
• Lemma 2.1
• proof
• Lemma 2.2
• proof