# Littlewood Richardson rules for the orthogonal group SO(d)

I have a question related to tensor products of Young diagrams. More precisely, I know the Littlewood Richardson rules for the general linear group GL(d) and would like to know the restriction of these rules for the orthogonal group SO(d).

Explicitly, the Littlewood Richardson reads (as extracted from http://arxiv.org/abs/hep-th/0611263):

The decomposition of an outer product $\mu\cdot\nu$ of irreducible representations $\mu$ and $\nu$ of $\mathfrak{S}_{n_1}$ and $\mathfrak{S}_{n_2}$, respectively, into irreducible representations of $\mathfrak{S}_{n_1+n_2}$ can be determined by means of the following algorithm involving Young diagrams. The product is commutative, so it does not matter which factor is regarded as the right-hand'' one. [In practice, on should choose the simpler Young diagram for that role.]

(I) Label each box in the top row of the right-hand diagram, $\nu$, by $a$, each box in the second row by $b$, etc.

(II) Add the labeled boxes of $\nu$ to the left-hand diagram $\mu$, one at a time, first the $a$s, then the $b$s, ..., subject to these constraints:

(A) No two boxes in the same column are labeled with the same letter;

(B) At all stages the result is a legitimate Young diagram;

(C) At each stage, if the letters are read right-to-left along the rows, from top to bottom, one never encounters more $b$s than $a$s, more $c$s than $b$s, etc.

(III) Each of the distinct diagrams constructed in this way specifies an irreducible subrepresentation $\lambda$, appearing in the decomposition of the outer product. The same labeled Young diagram may arise in more than one way;the multiplicity of that representation must be counted accordingly.

Thanks already for any references or comments !

• Welcome to Math.SE! Since this is a pretty specialized question, it would help if you would include the LR rules. This makes sure that we are all talking about the same thing. Aug 25 '15 at 14:21
• @Hrodelbert: I just did! Thanks for your comment! Aug 25 '15 at 14:36
• Are you familiar with the work of Peter Littelmann? Berenstein and Zelevinsky? Aug 25 '15 at 14:55
• @studiosus I am not actually. Do you know of any specific work that might be relevant ? Aug 25 '15 at 15:10
• Aug 25 '15 at 16:59

## 2 Answers

Take a look for instance at papers by Peter Littelmann: here for a generalization of the LR rule in the case of representations of classical Lie algebras (including the orthogonal case, of course) and here for even more general combinatorial model covering all semisimple Lie algebras. He has more papers here.

A formula for the decomposition of a tensor product of finite dimensional irreducible representations of $$O(d)$$ into a direct sum of irreducibles, along with multiplicities, was first given by Littlewood in 1958:

Littlewood, D. E., Products and plethysms of characters with orthogonal, symplectic and symmetric groups, Can. J. Math. 10, 17-32 (1958). ZBL0079.03604.

The rule was not stated in terms of diagrams originally, but rather in terms of the Littlewood-Richardson coefficients, which themselves can be obtained diagrammatically, as the OP states in the question. The $$SO(d)$$ case is basically the same, up the the identification of some representations that are distinct under $$O(d)$$. In more modern language, the rule is summarized in this blog post along with a reference to the eventual diagrammatic treatment of the rule.