On the structures of hive algebras and tensor product algebras for general linear groups of low rank

The tensor product algebra TA(n) for the complex general linear group GL(n), introduced by Howe et al., describes the decomposition of tensor products of 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.