The group of symplectic matrices is explicitly parameterized and this description is applied to solve two types of problems. First, we describe several sets of structured symplectic matrices, i.e., sets of symplectic matrices that simultaneously have another structure. We consider unitary symplectic matrices, positive definite symplectic matrices, entrywise positive symplectic matrices, totally...

In this paper we construct a new variant of the Weil representation, associated with a symplectic vector space (V, ω) defined over a finite field of characteristic two. Our variant is a representation ρ : AMp (V ) → GL (H), where the group AMp (V ) is forth cover of a group ASp (V ), which is a nontrivial gluing of the symplectic group Sp (V ) and the dual group V ∗. In particular, the group AS...