Discrete series of fusion algebras

Abstract

We show that the left regular representation of a countably infinite (discrete) group admits no finite-dimensional invariant subspaces. We also discuss a consequence of this fact, and the reason for our interest in this statement. We then formally state, as a 'conjecture', a possible generalisation of the above statement to the context of fusion algebras. We prove the validity of this conjecture in the case of the fusion algebra arising from the dual of a compact Lie group. We finally show, by example, that our conjecture is false as stated, and raise the question of whether there is a 'good' class of fusion algebras, which contains (a) the two 'good classes' discussed above, namely, discrete groups and compact group duals, and (b) only contains fusion algebras for which the conjecture is valid.