Representation theory of projective Clifford groups via isocategoricality

Abstract

We prove that the projective Clifford group $C(A)$ associated with a finite abelian group $A$ is isocategorical to the affine symplectic group $\mathrm{ASp}(A)=\mathrm{Sp}(V_A)\ltimes\widehat{V_A}$, where $V_A=A\oplus\widehat{A}$. This explicit isomorphism of tensor categories gives common parametrizations of their irreducible characters and conjugacy classes, under which their character tables agree entry by entry. It also gives an explicit basis for every tensor-power commutant of the projective Weil representation, indexed by symplectic orbits of tuples in $V_A$ whose entries sum to zero. For cyclic $A$, we classify these orbits by oriented two-generated submodules and obtain exact commutant dimensions in every tensor power.