Geodesic
Essential Signatures and Canonical Bases of Irreducible Representations of the Group $G_{2}$
Matematičeskie zametki, Tome 97 (2015) no. 1, pp. 35-47

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We consider representations of simple Lie algebras and the problem of constructing a “canonical” weight basis in an arbitrary irreducible finite-dimensional highest-weight module. Vinberg suggested a method for constructing such bases by applying the lowering operators corresponding to all negative roots to the highest-weight vector and put forward a number of conjectures on the parametrization and structure of such bases. It follows from papers by Feigin, Fourier, and Littelmann that these conjectures are true for the cases of $A_n$ and $C_{n}$. In the present paper, we prove these conjectures for the case of $G_2$ by using a different approach suggested by Vinberg.
Keywords: simple Lie algebra, irreducible representation, canonical base, essential signature, weight basis.
Mots-clés : group $G_{2}$ 
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     author = {A. A. Gornitskii},
     title = {Essential {Signatures} and {Canonical} {Bases} of {Irreducible} {Representations} of the {Group} $G_{2}$},
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A. A. Gornitskii. Essential Signatures and Canonical Bases of Irreducible Representations of the Group $G_{2}$. Matematičeskie zametki, Tome 97 (2015) no. 1, pp. 35-47. http://geodesic.mathdoc.fr/item/MZM_2015_97_1_a3/