# Monomial bases and branching rules

## TL;DR

This paper develops a method for constructing monomial bases for representations of reductive Lie algebras, connecting these bases to classical bases like Gelfand-Tsetlin and Littelmann, and applying them to specific Lie types.

## Contribution

It introduces a new approach to construct monomial bases for multiplicity spaces in Lie algebra representations, linking them to existing bases and extending their properties.

## Key findings

- Constructed monomial bases for representations of general linear and symplectic Lie algebras.
- Established triangular transition matrices connecting new bases with Gelfand-Tsetlin and Littelmann bases.
- Demonstrated the basis properties extend to classical types A and C.

## Abstract

Following a question of Vinberg, a general method to construct monomial bases for finite-dimensional irreducible representations of a reductive Lie algebra was developed in a series of papers by Feigin, Fourier, and Littelmann. Relying on this method, we construct monomial bases of multiplicity spaces associated with the restriction of the representation to a reductive subalgebra. As an application, we produce monomial bases for representations of the general linear and symplectic Lie algebras associated with natural chains of subalgebras. We also show that our basis in type A is related to both the Gelfand-Tsetlin basis and the Littelmann basis via triangular transition matrices which implies that the triangularity property extends to the matrix connecting the Gelfand-Tsetlin and canonical bases. A similar relationship holds between our basis in type C and a suitably modified version of the basis constructed earlier by the first author.