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Let $G$ be a semisimple Lie algebra over $\mathbb{C}$. Let $V(\lambda)$ be the irreducible highest weight module for $G$ with highest weight $\lambda$. If $G$ is of type A, we can decompose $V(\lambda)\otimes V(\mu)$ using Littlewood-Richardson rule. In other types, if $\lambda$ and $\mu$ are given explicitly, we can use Weyl character formula to compute the decomposition. My question is can we have some rules similar as Littlewood-Richardson rule for other types (especially for type $G_2$)? Does Littelmann's path model work for this?

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4 Answers 4

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The search term you want to look for is "Klimyk's Formula." This formula boils down to the following:

Fix $G$ a compact complex semisimple Lie group. Suppose $V(\lambda)$ and $V(\mu)$ are irreducible representations with highest weights $\lambda$ and $\mu$ respectively. Let $W_\lambda = \{\lambda_1,\lambda_2,\ldots \lambda_d\}$ be the multiset of weights of $V(\lambda)$ with $d = dim(V(\lambda))$. Then the irreducible components of $V(\lambda)\otimes V(\mu)$ are given by $\{V(\mu+\lambda_i)\}_{i=1}^d$.

To apply this in practice, you need to be comfortable with the concept of defining $V(\lambda)$ when $\lambda$ is not a dominant weight (which sometimes causes modules to cancel when they appear with both positive and negative signs in the sum), but it applies to lots of groups (even beyond the scope of compact complex semisimple in some cases if im not mistaken), and Littlewood-Richardson is just the special case of this formula in type $A$.

An example for $G_2$ (since that is also my favorite compact semisimple Lie group) is to let $\lambda = [1,0]$ be the highest weight of the 7-dimensional representation and $\mu = [0,1]$ the highest weight of the 14-dimensional adjoint representation.

The seven weights of $V(\lambda)$ are $[1,0]$, $[-1,1]$, $[2,-1]$, $[0,0]$, $[-2,1]$, $[1,-1]$, and $[-1,0]$ so Klimyk tells us the 98-dimensional tensor product decomposes as:

$V([1,1]) \oplus V([-1,2]) \oplus V([2,0]) \oplus V([0,1]) \oplus V([-2,2]) \oplus V([1,0]) \oplus V([-1,1])$

This is where familiarity with interpreting modules with non-dominant highest weights comes in; $V[-1,2]$ and $V[-1,1]$ turn out to be 0-dimensional modules, while $V([-2,2]) \cong -V([0,1])$*. Thus the terms which do not disappear are $V([1,1])$ which is a 64-dimensional module, $V([2,0])$ which is a 27-dimensional module, and $V([1,0])$ which is the 7-dimensional defining representation, a total of 98 dimensions.

If you had instead chosen to switch $\lambda$ and $\mu$ and add the 14 weights of $V([0,1])$ to [1,0], you would have obtained 14 modules, but as before, some would have been zero and others would have cancelled in pairs ultimately leading to the same three modules as above being the only things left over. In my opinion, this reflexivity always holding is the coolest thing about Klimyk's formula.

One neat corollary to Klimyk's formula is that a tensor product of two irreducible modules cannot decompose into a sum of more than $d$ irreducibles where $d$ is the minimum of the dimensions of the two modules.

*EDIT: After posting, I decided to add a bit more about modules with non-dominant highest weight. Basically, the weights of a group $G$ are permuted via the Weyl group action on the weights. Weights are determined by integer $r$-tuples where $r$ is the rank of $G$; tuples containing a -1 lie in the walls of the Weyl chambers and so the modules with these highest weights end up being 0. There are a few other subspaces which also correspond to walls; weights $\mu$ not lying in the walls satisfy $w(\mu) = \lambda$ for some dominant weight $\lambda$ (all coordinates nonnegative) and a unique $w$ in the Weyl group (i.e. only one $w$ in the Weyl group will take $\mu$ to a dominant weight, so $\lambda$ is also uniquely determined). Then $V(\mu)$ is defined by the following relationship:

$V(\mu) = (-1)^w\cdot V(\lambda)$

Here $(-1)^w$ is the sign representation which appears with all Weyl groups; in the $A$-series whose Weyl groups are the $S_n$'s this is the ordinary sign representation.

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This answer is somewhat loose and, consequently, some statements are incorrect. Firstly, it is only true that the components of the tensor product $V(\lambda)\otimes V(\mu)$ are contained among $\{V(\lambda_i+\mu\}$ -- certainly the multiplicities can decrease and some components may not occur at all. (This is related to "cancellations" discussed further in the answer). (Continued) –  Victor Protsak Jul 16 '13 at 23:01
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Secondly, the action of the Weyl group relevant to the Klimyk's formula is the shifted action $$w\cdot \nu=w(\nu+\rho)-\rho,$$ where $\rho$ is the half-sum of positive roots. (This explains why certain weights are referred to as "singular" even though they have coefficient $-1$, and not $0$, when expanded in the basis of the fundamental weights; ordinarily, the walls of the Weyl chambers pass through $0.$) To avoid confusion, I suggest including a precise statement of Klimyk's formula from Humphreys' or Samelson's book. –  Victor Protsak Jul 16 '13 at 23:05

Since the decomposition of tensor products has a complicated history, it's worth adding some comments to ARupinski's answer.

1) Though Weyl's character formula is fundamental for finite dimensional representation theory, it doesn't lead immediately to a method for decomposing tensor products. However, Steinberg derived in 1961 such an elegant method (which however involves an impractical double summation over the Weyl group) here.

2) As early as 1937, Brauer wrote down the explicit recipe referred to here (and often in the applied literature directed toward physicists) as Klimyk's rule: this was a short article in C.R. Acad. Sci. Paris 204, more easily located in Brauer's collected papers. The selling point of this rule is that it requires only the knowledge of one highest weight along with the full character of the second module (given for example by Weyl or by Kostant's equivalent later method). These matters are discussed in Section 24 of my 1972 Springer graduate text, where a proof of Brauer's formula is formulated in the exercises.

3) Klimyk's early work appeared around 1967 in Russian, with an English translation soon after in an AMS volume:

MR0237712 (38 #5993) 22.60, Klimyk, A.U. Multiplicities of weights of representations and multiplicities of representations of semisimple Lie algebras. (Russian) Dokl. Akad. Nauk SSSR 177 (1967) 1001–1004.

4) It's important to realize that Klimyk's work has involved not just tensor products of finite dimensional representations, but also tensor products in which one factor is allowed to be infinite dimensional of various special types. This is part of a much broader program for Lie group representations involving Kostant and others.

5) From a purely computational point of view, getting explicit results for type $G_2$ is not at all easy because dimensions grow so fast. There used to be some explicit printed tables, which always stop when the going gets tough. The older methods of Brauer and Klimyk are anyway inherently inefficient, requiring a huge amount of cancellation as indicated by ARupinski. Combinatorists have found Littelmann's approach (generalizing Littlewood-Richardson for type A) more appealing, but I'm unaware of literature illustrating the method explicitly for type $G_2$.

6) It's also worth mentioning that interesting special features of tensor products have been studied using algebraic geometry by Kumar and others in response to the old "PRV Conjecture", but here the concern is about the occurrence of specific summands in a tensor product decomposition and not the entire picture. (To get some perspective on the range of "geometric" literature about tensor product decompositions, take a look at Kumar's 2010 ICM report. This is on his homepage but apparently not on arXiv: http://math.unc.edu/Faculty/kumar/papers/kumar60.pdf)

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Interesting history of the formulas Jim. In fact I think it almost certainly was in your book that I first came across the formula when I started needing to calculate decompositions for my dissertation work. –  ARupinski Jan 19 '12 at 23:43

The short answer is Yes.

Here are some of the details. You start with a crystal for the representation. You don't need to specify which model you are taking. When you take a tensor product a subset of the vertices of the crystal give highest weight vectors. The rule is a simple rule depending on the depth (or rise) of the vertex.

For the fundamental representation this is particularly straightforward.

If you want to get started with crystals, a good place to begin is:

MR1881971 (2002m:17012) Hong, Jin ; Kang, Seok-Jin . Introduction to quantum groups and crystal bases. Graduate Studies in Mathematics, 42. American Mathematical Society, Providence, RI, 2002. xviii+307 pp. ISBN: 0-8218-2874-6

This does not discuss the tensor product rule you asked for. The original reference for for the general tensor product rule is:

MR0225936 (37 #1526) Parthasarathy, K. R. ; Ranga Rao, R. ; Varadarajan, V. S. Representations of complex semi-simple Lie groups and Lie algebras. Ann. of Math. (2) 85 1967 383--429.

This tensor product rule requires further notation. The advantage is that multiplicities are calculated directly without any cancelation.

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@Bruce, thank you very much. I don't know much about crystals for a representation. Could you give some references about this? –  Jianrong Li Jan 13 '12 at 17:19

I think there is. Actually there is a version of Weyl's theory for $G_2$, i.e. all finite dimensional irreducible representations are given by Schur functors and hence are intimately related to representations of symmetric groups. See article by Jing-Song Huang for details.

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@r0b0t, it seems that this paper is only for the decomposition of $V(\omega_1)^{\otimes n}$ –  Jianrong Li Jan 13 '12 at 20:11
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My reasoning is that Littlewood-Richardson rule can be viewed as a statement about decompositions of Schur functions. These are related to representations of $\mathfrak{sl}$ via Schur-Weyl duality. Since the article by Huang proves this kind of duality for $G_2$ I expect that some kind of LW rule can be inferred from this. I suggest you look first at the case of $\mathfrak{so}$, where the contractions are surely more familiar operations. I think that quite a good reference is the book by Goodman and Wallach. –  Vít Tuček Jan 13 '12 at 22:45

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