Hive algebras and tensor product algebras for small GL(n)
For the complex general linear group GL(n), the GL(n) tensor product algebra TA(n) introduced by Howe et al. describes the decomposition of the tensor products of two irreducible polynomial representations of GL(n). Using the hive model for the Littlewood-Richardson coefficients, we provide a finite presentation of the algebra TA(n) for n=2, 3, 4 in terms of generators and relations, thereby giving a description of highest weight vectors of irreducible representations in the tensor products. We also compute the generating function of certain sums of Littlewood-Richardson coefficients.