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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 ofirreducible polynomial representations of GL(n). Using the hive model for theLittlewood-Richardson coefficients, we provide a finite presentation of thealgebra TA(n) for n=2, 3, 4 in terms of generators and relations, therebygiving a description of highest weight vectors of irreducible representationsin the tensor products. We also compute the generating function of certain sumsof Littlewood-Richardson coefficients.

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