The Verlinde relations for the Grothendieck ring of the singlet algebra

The singlet vertex algebra $\mathcal{W}(2,2p-1)$ is considered together with its module category, which is expected to be a braided tensor category, or at least to contain a suitable braided tensor subcategory. Its Grothendieck ring is therefore available. Verlinde relations conjecture. The relations in Theorem also hold inside the Grothendieck ring. This would identify the previously computed character relations with fusion relations in the Grothendieck ring, despite the absence of a proof that the full module category is braided tensor.