Essential Signatures and Canonical Bases of Irreducible Representations of the Group $G_{2}$

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.