We determine the necessary and sufficient combinatorial conditions for which the tensor product of two irreducible polynomial representations of $\GL(n,\mathbb{C})$ is isomorphic to another. As a consequence we discover families of Littlewood--Richardson coefficients that are non-zero, and a condition on Schur non-negativity.

Keywords:

polynomial representation, symmetric function, Littlewood--Richardson coefficient, Schur non-negative polynomial representation, symmetric function, Littlewood--Richardson coefficient, Schur non-negative

MSC Classifications:

05E05, 05E10, 20C30 show english descriptions Symmetric functions and generalizations
Combinatorial aspects of representation theory [See also 20C30]
Representations of finite symmetric groups 05E05 - Symmetric functions and generalizations
05E10 - Combinatorial aspects of representation theory [See also 20C30]
20C30 - Representations of finite symmetric groups

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