Representation theory of projective Clifford groups via isocategoricality

Abstract

The representation theory of the projective Clifford group $C(A)$, attached to a finite abelian group $A$, is closely related to the symplectic action on $V_A=A\oplus\widehat A$. We make this relation precise by constructing an explicit tensor isomorphism between the representation category of $C(A)$ and the representation category of the affine symplectic group $\operatorname{ASp}(A)=\operatorname{Sp}(V_A)\ltimes\widehat{V_A}$. Thus $C(A)$ and $\operatorname{ASp}(A)$ are isocategorical, although they need not be isomorphic. The isomorphism transfers the little-group method from $\operatorname{ASp}(A)$ to $C(A)$, giving a uniform description of the irreducible representations of $C(A)$. The same approach gives conjugacy-class parameters, class-size formulas, and character formulas. Thus the character theory of $C(A)$ is reduced to ordinary character tables of stabilizers, affine centralizer orbits, and the scalar factors appearing in the Clifford action. In particular, $C(A)$ and $\operatorname{ASp}(A)$ have identical ordinary character tables, up to relabeling. Finally, the tensor isomorphism identifies the twisted group algebra determined by the Weyl commutation relations with the tensor transport of the ordinary group algebra $\mathbb C[V_A]$. It also transports the Clifford adjoint-action commutants to affine symplectic orbit algebras, where they admit an orbit basis with orbit-intersection structure constants.