2
$\begingroup$

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?

$\endgroup$
  • 1
    $\begingroup$ 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
  • 2
    $\begingroup$ @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
5
$\begingroup$

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".

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.