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?

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    $\begingroup$ The Littlewood-Richardson rule is only for type A. $\endgroup$ – Tobias Kildetoft Apr 25 '14 at 13:27
  • $\begingroup$ . . . a quantum version for $U_q(\frak{sl}_n)$ is fine for me. $\endgroup$ – Milan Bernolak Apr 25 '14 at 13:31
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    $\begingroup$ @Milan: The answer to your question (which only involves type $A$ directly, as Tobias points out) depends on what your parameter $q$ is. If it's an indeterminate, there is no difference between the decompositions in the classical and quantum cases, since the irreducibles are essentially the same. But if $q$ is a root of unity, you get into trickier issues including "fusion rules". I don't know any role there for Littlewood-Richardson as such. $\endgroup$ – Jim Humphreys Apr 25 '14 at 13:35
  • $\begingroup$ I am interested in $q$ a non-root of unity complex number. $\endgroup$ – Milan Bernolak Apr 25 '14 at 13:43
  • $\begingroup$ . . . so what is a good reference for this please? $\endgroup$ – Milan Bernolak Apr 25 '14 at 13:44

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".


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