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# On Tensor Products of Polynomial Representations

Published:2008-12-01
Printed: Dec 2008
• Kevin Purbhoo
• Stephanie van Willigenburg
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## Abstract

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
 MSC Classifications: 05E05 - Symmetric functions and generalizations 05E10 - Combinatorial aspects of representation theory [See also 20C30] 20C30 - Representations of finite symmetric groups