Let $\frak{g}$ be a complex semi-simple Lie algebra, and $\lambda,\mu \in P^+$ two positive dominant weights with corresponding irreducible representations $V(\lambda)$ and $V(\mu)$. The tensor product $V(\lambda) \otimes V(\mu)$ has the well-known decomposition into irreducible components given by the Littlewood-Richardson rule.


My question is whether or not this decomposition also exists in the quantum case, in other words, for the quantized enveloping algebra $U\_q(\frak{g})$ with irreducible representations $V(\lambda)$ and $V(\mu)$, does $V(\lambda) \otimes V(\mu)$ have a "quantum Littlewood-Richardson" decomposition?



"Quantum Littlewood-Richardson" Rule?
Yes, there certainly is a quantum version of the Littlewood-Richardson decomposition (in the generic parameter case) for types $A,B,C,D$ (I don't know about the exceptional types). The generalized Littlewood-Richardson rule develops naturally out of a construction of crystal bases by semistandard tableaux (satisfying additional assumptions outside of type $A$). Taking tensor products then amounts to adding or subtracting boxes to your Young diagram, and as a result you get Littlewood-Richardson in type $A$ and similar decompositions in the other types.


This view was (I believe) first elaborated by Kashiwara and Nakashima in '94 in the paper "Crystal graphs for representations of the $q$-analogue of classical Lie algebras", but a nice exposition is given in Chapter 8 of Hong and Kang's textbook "Introduction to Quantum Groups and Crystal Bases".


