A. A. Gornitskii | Searcharxiv

arXiv2019

Essential Signatures and Monomial Bases for $B_n$ and $D_n$

In the representation theory of simple Lie algebras, we consider the problem of constructing a monomial basis in an arbitrary irreducible finite-dimensional highest weight module. We construct a PBW-type basis in every finite-dimensional representation of $B_n$ and $D_n$ and we describe the associated semigroup of essential signatures. These bases are parameterized by integer points in some polytopes. We give the inequalities defining these polytopes.

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arXiv2015

Essential Signatures and Canonical Bases for Irreducible Representations of $D_4$

In the representation theory of simple Lie algebras, we consider 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 positive roots to the highest weight vector. He proposed several conjectures on the parametrization and structure of such bases. It is already known that these conjectures are true for the cases $A_n$, $C_n$, $G_2$, $B_3$. In this paper, we prove these conjectures for the Lie algebra of type $D_4$.

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