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Let $\mathfrak{g}$ be an Lie algebra of type G2. Are there some combinatorial ways to describe a basis of a $\mathfrak{g}$-module? For classical types, there is a method used tableaux. Thank you very much.

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Do you have hypotheses on your field, e.g. algebraically closed, characteristic 0? –  M Turgeon Oct 18 '11 at 21:01
    
@Turgeon, thank you. The field is $\mathbb{C}$. –  Jianrong Li Oct 19 '11 at 0:33
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3 Answers

up vote 8 down vote accepted

This paper constructs the representations of $G_2$ explicitly in terms of the Schur functor construction and multilinear operations: http://www.ams.org/journals/proc/1999-127-03/S0002-9939-99-04583-9/home.html

It doesn't give you a preferred basis in terms of tableaux though. For that, you might want to look at the book "Standard monomial theory" by Lakshmibai and Raghavan. But this is much less explicit. This paper of Littelmann does in much greater generality: http://www.ams.org/jams/1998-11-03/S0894-0347-98-00268-9/S0894-0347-98-00268-9.pdf

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@Steven, thank you very much. –  Jianrong Li Oct 19 '11 at 15:03
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Assuming you are working in the classical setting over a field like $\mathbb{C}$, I'm unaware of any elementary way to identify a natural basis in each irreducible finite dimensional module. But using creative indirect methods via quantum groups, Lusztig was able to define such canonical bases for all Lie types (and Kashiwara arrived at related methods independently). While the actual construction of such bases in individual cases like $G_2$ is not easy to implement, it is in principle "combinatorial" and is "canonical" in a strong sense. Lusztig's 1990 paper is freely available online here. See in particular 0.6 in the introduction.

P.S. As Steven points out, Littelmann's work (in many papers worth consulting) provides a new combinatorial framework for many of the standard problems in characteristic 0 about representations of semisimple Lie algebras; much of this generalizes to symmetrizable Kac-Moody algebras as well. In particular, his path method provides an algorithmic way to work out bases and characters, though the best results are obtained in type A. Other powerful methods are those of Lusztig (canonical basis) and Kashiwara (crystal basis), which are closely related to each other. All of this work probably has more theoretical than practical interest. Whatever approach is used, even in type $G_2$ the dimensions of irreducible representations (computed readily from Weyl's formula) grow so rapidly that it's questionable how far one can go in practice toward finding explicit bases of weight spaces or gaining extra insight from them.

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@Jim, thank you very much. –  Jianrong Li Oct 19 '11 at 0:37
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Another direction, that works (only?) for groups of rank at most 2, is Kuperberg's "spider" model: http://arxiv.org/abs/q-alg/9712003 . The web diagrams he constructs generalize tableaux, in the sense that the $A_1$ and $A_2$ versions are in bijection with tableaux (at least of rectangular shape). This is not quite the same as constructing representations via Schur functors, but it has a combinatorial appeal.

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I have to say that seeing your paper on $G_2$-Schubert loci led me to learn about the paper I mention in my answer, and in retrospect, seemed to be an important discovery in my grad school development. So thanks! –  Steven Sam Oct 21 '11 at 12:59
    
@Steven, thanks, glad to know it was useful! –  Dave Anderson Oct 21 '11 at 19:29
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