Abstract:
We shall discuss the problem of constructing “canonical” weight bases in finite-dimensional representations of simple complex Lie algebras. E. B. Vinberg proposed a method for constructing such bases by applying the lowering operators (in a fixed order) to the highest weight vector.
Such bases are defined by a semigroup $\Sigma$, called the semigroup of essential signatures. This semigroup is determined by a numbering of positive roots and a monomial order on $\mathbb{Z}^N$ where $N$ is the number of positive roots. A question of interest is when $\Sigma$ possesses “good” properties: finite generation, saturatedness, etc.
We plan to discuss known results and show how one can construct a “good” basis for representations of Lie algebras of types $\mathsf D_n$ and $\mathsf B_n$.
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