The lowest discriminant ideal of central extensions of Abelian groups

Abstract: In a previous joint paper with Wu and Yakimov, we gave an explicit description of the lowest discriminant ideal in a Cayley-Hamilton Hopf algebra (H,C,tr) of degree d over an algebraically closed field k, char k $\notin[1, d]$ with basic identity fiber, i.e. all irreducible representations over the kernel of the counit of the central Hopf subalgebra C are one-dimensional. Using results developed in that paper, we compute relevant quantities associated with irreducible representations to explicitly describe the zero set of the lowest discriminant ideal in the group algebra of a central extension of the product of two arbitrary finitely generated Abelian groups by any finite Abelian group under some conditions. Over a fixed maximal ideal of C the representations are tensor products of representations each corresponding to a central extension of a subgroup isomorphic to the product of two cyclic groups of the same order. A description of the orbit of the identity, i.e. the kernel of the counit of C, under winding automorphisms is also given.