Littlewood-Richardson coefficients are both multiplicities of $GL_n$ tensor products, and of restrictions of $GL_{m+n}$ representations to $GL_m \times GL_n$. I want to turn this equality of numbers into an equality of maps of vector spaces, and generalize it to quantum groups. In more detail:

$\def\Part{\mathrm{Part}} \def\Hom{\mathrm{Hom}}$Let $\Part$ be the abelian category whose objects are collections of finite dimensional vectors spaces $(W_{\lambda})$, one for each partition $\lambda$, such that all but finitely many of the $W_{\lambda}$ are $0$. Here $$\Hom_{\Part}((W_{\lambda}), (X_{\lambda})) = \bigoplus_{\lambda} \Hom_{\mathrm{Vect}}(W_{\lambda}, X_{\lambda})$$ and composition is defined in the obvious manner. I'll write $[\lambda]$ for the element of $\Part$ which is $\mathbb{C}$ at $\lambda$ and $0$ everywhere else.

There are two potential tensor structures on $\Part$, which I will call $\otimes^T$ (for tensor) and $\otimes^R$ (for restriction).

Let $V_{\lambda}(n)$ denote the representation of $GL_n$ with highest weight $\lambda$. In $\otimes^T$, $$\left( [\lambda] \otimes^T [\mu] \right)_{\nu} = \Hom_{GL_n}(V_{\nu}(n), V_{\lambda}(n) \otimes V_{\mu}(n)) \ \mathrm{for} \ n\gg 0.$$ Part of the claim is that, for $n$ sufficiently large, the vector spaces which we say on the right hand side are isomorphic, and this isomorphism can be chosen to commute with the obvious commutator and associator. The definition of $\otimes^T$ for other objects of $\Part$ is forced by linearity.

Similarly, $$\left( [\lambda] \otimes^R [\mu] \right)_{\nu} = \Hom_{GL_m \times GL_n}(V_{\lambda}(m) \boxtimes V_{\mu}(n), V_{\nu}(m+n)) \ \mathrm{for}\ m,n \gg 0.$$

I believe it is not too bad to show that these two tensor structures are equivalent (including the commutator and associator) by identifying them both with the same thing in the representation theory of the symmetric group, through Schur-Weyl duality, although I haven't actually done this.

I think the same statement is true for $GL_n$ quantized at generic $q$. Am I right? Does someone know a reference for this?

Actually, what I really want is the corresponding statement in the category of crystals, so "at $q=0$". What this turns into concretely is that there is a bijection between

- $\lambda$-dominant tableaux of shape $\mu$ and content $\nu - \lambda$ and
- Yamnouchi tableaux of shape $\nu/\lambda$ and content $\mu$

which commutes with certain jdt moves. I need certain specific cases of this, and I have now checked most of the cases I need by hand -- but I'd much rather cite a general theorem about the category of crystals than write out a lot of tableaux bijections.

Thanks!