  We describe bases for the morphism spaces of the Frobenius Heisenberg
categories associated to a symmetric graded Frobenius algebra, proving several
open conjectures. Our proof uses a categorical comultiplication and generalized
cyclotomic quotients of the category. We use our basis theorem to prove that
the Grothendieck ring of the Karoubi envelope of the Frobenius Heisenberg
category recovers the lattice Heisenberg algebra associated to the Frobenius
algebra.

Representation Theory;Category Theory