## Background

Irreducible finite dimensional representations of the group $GL_n$ are parameterized by the highest weights, that is by nonincreasing sequences of integers $$ \lambda_1 \ge \lambda_2 \ge \dots \ge \lambda_n. $$ Let us restrict to the case when $\lambda_n \ge 0$. Then one can encode a highest weight by a Young diagram with $\lambda_i$ boxes in the $i$-th row.

The Littlewood-Richardson rule describes the decomposition of a tensor product $V^\lambda \otimes V^\mu$ into a direct sum of irreducibles. It says that the multiplicity of $V^\nu$ in the tensor product $V^\lambda \otimes V^\mu$ is equal to the number of so-called Littlewood-Richardson tableux in the skew-diagram $\nu\setminus\lambda$ of weight $\mu$. See Littlewood-Richardson rule for precise definitions.

Note that the rule is not symmetric with respect to $\lambda$ and $\mu$!

On the other hand, the category of representations of $GL_n$ has a commutativity morphism: it is a bifunctorial isomorphism $$ c_{V,W}:V\otimes W \to W\otimes V, \qquad v\otimes w \mapsto w\otimes v. $$

## Question

Is there a possibility to make Littlewood-Richardson rule compatible with the commutativity morphism?

To be more precise, is there a way to associate with every Littlewood-Richardson tableau in a skew diagram $\nu\setminus\lambda$ of weight $\mu$ an embedding $V^\nu \to V^\lambda\otimes V^\mu$ such that the composition $V^\nu \to V^\lambda\otimes V^\mu \stackrel{c}\to V^\mu\otimes V^\lambda$ is the embedding associated to an appropriate Littlewood-Richardson tableau in a skew diagram $\nu\setminus\mu$ of weight $\lambda$?

Let me emphasize that I am asking about the $GL_n$ case, although this question has sense for any reductive group.