# Decomposing tensor products of modules for the orthogonal/symplectic groups in characteristic zero

I would like to know if there is a perfect analogue of the classical Littlewood-Richardson rule for decomposing tensor products of simple modules for the orthogonal/symplectic groups in characteristic zero, i.e. a "tableaux counting algorithm" which determines the solution.

I know that there are tableaux counting rules due to King and De Concini, there is the Littelmann path algorithm, and the theory of crystal bases and probably many more descriptions besides. There are also results due to Koike and others which express the answer in terms of a sum or products of Littlewood-Richardson coefficients (this isn't really what I'm looking for).

So my question is: is there some summary of how all these different methods coincide? Do any of these methods give "perfect solutions"? I know the Littelmann path approach gives a complete answer, but can it be expressed in terms of something like the classical tableaux counting procedure? Do the King/De Concini rules provide (positive) solutions in all cases? And is there some bible in which all these different approaches are compiled and compared?

Post Jim's comment:

The main question is: under what conditions (probably on $n$) do (any) of the tableaux-based approaches coincide with the Littelmann path approach to answer the problem?

And what is the best reference for these tableaux? I was told to look for

R. C. King: "Weight multiplicities for the classical groups"

but can't get a hold of it. Does anyone have a pdf? If not, does anyone have another good reference? There seems to be a LOT of literature to wade through

• I'm pretty sure the answer to your final question is no, though it would be helpful to connect all these parts of the literature. Aside from that it would be better to focus more on a single question, preferably highlighted using > at the start of a paragraph. Jul 8, 2013 at 20:27
• P.S. Concerning the article by R.C. King, it's unlikely that any legal PDF versions are floating around: Weight multiplicities for the classical groups. Group theoretical methods in physics (Fourth Internat. Colloq., Nijmegen, 1975), pp. 490–499. Lecture Notes in Phys., Vol. 50, Springer, Berlin, 1976. Many of his related papers are in J. Phys. A, maybe more accessible online via a library. Jul 9, 2013 at 16:00

## 1 Answer

I counted 8 questions asked in this post. I will answer one: Before Littelmann invented the LS-path mode, he worked out a generalized LR-rule using the language of generalized tableaux for most simple complex Lie algebras (icnluding all classical ones), see here. Since he was extending the work of Lakshmibai and Seshadri, I am pretty sure, the formalism Littelmann was using is equivalent to the LS-path model, just he did not have a geometric picture at the time.