On tensor product decomposition of \(\widehat{\mathfrak{sl}}(n)\) modules (Q2847701)

Let \(V(\Lambda_0)\) be the integrable highest weight module with highest weight \(\Lambda_0\) for the affine Lie algebra \(\widehat{\mathfrak{sl}}(n)\). The tensor product \(V(\Lambda_0)\otimes V(\Lambda_0)\) is completely reducible as a direct sum of modules \(V(\lambda)\), \(\lambda\in P_+\), with certain multiplicities. In this paper, the authors determine the highest weight of each irreducible summand and each corresponding multiplicity. For this, they use the theory of crystals. They use an explicit realization of the crystal \(\mathcal{B}(\Lambda_0)\) for \(V(\Lambda_0)\) in terms of colored Young diagrams. They find a one-to-one correspondence of the maximal elements of \(\mathcal{B}(\Lambda_0)\otimes \mathcal{B}(\Lambda_0)\) and integer partitions satisfying certain conditions, thus obtaining a value for the multiplicities. Moreover, they consider generating functions of these multiplicities and, using Jacobi triple product identity, obtain a number of partition identities, some of which appear to be new.