A unified approach to construction of Gelfand-Tsetlin-Zhelobenko base vectors for series $A$, $B$, $C$, $D$

TL;DR

This paper develops a unified method using Zhelobenko's approach to construct Gelfand-Tsetlin-Zhelobenko base vectors for series A, B, C, D Lie algebras, revealing relations between highest vector spaces across different series.

Contribution

It introduces a new explicit description of highest vector spaces and constructs bases that connect different algebra series, extending Gelfand-Tsetlin tableaux to a broader context.

Findings

Unified construction for all series A, B, C, D

Explicit description of highest vector spaces

Extension of Gelfand-Tsetlin tableaux

Abstract

Using the Zhelobenko's approach we investigate a branching of an irreducible representation of $g_n$ under the restriction of algebras $g_n\downarrow g_{n-1}$, where $g_n$ is a Lie algebra of type $B_n$, $C_n$, $D_n$ or a Lie algebra of type $A$, where in this case we put $g_{n}=\mathfrak{gl}_{n+1}$, $g_{n-1}=\mathfrak{gl}_{n-1}$. We give a new explicit description of the space of the $g_{n-1}$-highest vectors, then we construct a base in this space. The case $n=2$ is considered separately for different algebras, but a passage from $n=2$ to an arbitrary $n$ is the same for all series $A$, $B$, $C$, $D$. This new procedure has the following advantage: it establishes a relation between spaces of $g_{n-1}$-highest vectors for different series of algebras. This procedure describes an extension of Gelfand-Tsetlin tableaux to the left.