# A relationship between Gelfand-Tsetlin bases and Chari-Loktev bases for irreducible finite dimensional representations of special linear Lie algebras

## TL;DR

This paper explores the relationship between Gelfand-Tsetlin and Chari-Loktev bases in finite-dimensional irreducible representations of special linear Lie algebras, revealing a triangular transition matrix structure.

## Contribution

It establishes a partial order on basis parametrizations and explicitly describes the transition matrix between the two bases.

## Key findings

- Transition matrix is triangular with respect to the partial order.
- Explicit formulas for the diagonal elements of the transition matrix.
- Introduction of the 'row-wise dominance' partial order.

## Abstract

We consider two bases for an arbitrary finite dimensional irreducible representation of a complex special linear Lie algebra: the classical Gelfand-Tsetlin basis and the relatively new Chari-Loktev basis. Both are parametrized by the set of (integral Gelfand-Tsetlin) patterns with a fixed bounding sequence determined by the highest weight of the representation. We define the "row-wise dominance" partial order on this set of patterns, and prove that the transition matrix between the two bases is triangular with respect to this partial order. We write down explicit expressions for the diagonal elements of the transition matrix.