# About the saturation of the Littlewood Richardson coefficients for rational representations

The Saturation Theorem as proved by A. Knutson and T. Tao states that for $N \in \mathbb{Z}_{> 0}$ and polynomial representations of $GL(n, \mathbb{C})$: $V_\alpha, V_\beta$ and $V_\gamma$ we have that $V_\gamma$ is isomorphic to a submodule of $V_\alpha \otimes V_\beta$ if and only if $V_{N\gamma}$ is isomorphic to a submodule of $V_{N\alpha} \otimes V_{N\beta}$. I was wondering why is the case that this only holds for polynomial representations and not rational ones? Can't one just multiply for rational representations with a tensor power of the determinant representations apply the result for polynomial representations and then multiply back with the appropriate tensor power of the dual of the determinant representation?

There are several combinatorial models one can use to compute LR coefficients. Knutson and Tau use the honeycomb model. What would be the obstruction in this model to count LR coefficients for rational representations?

• There is no obstruction; as you say, by tensoring with a large multiple of the determinant representation the questions of saturation for polynomial representations and saturation for rational representations are easily seen to be equivalent; also, honeycombs are capable of counting both. Which formulation one prefers is largely a matter of personal taste. Jun 18 '17 at 2:49