Combinatorial Constructions of Weight Bases: The Gelfand-Tsetlin Basis

TL;DR

This paper provides a combinatorial proof and new insights into the Gelfand-Tsetlin basis for irreducible representations of sl_n, enhancing understanding of weight bases with explicit Chevalley generator formulas.

Contribution

It introduces a simple combinatorial proof of Gelfand-Tsetlin formulas and explores properties of the basis using an algorithm on Young tableaux.

Findings

A combinatorial proof of Gelfand-Tsetlin formulas is established.

Properties of the Gelfand-Tsetlin basis are derived through an algorithm.

The approach simplifies understanding of weight bases for semisimple Lie algebra representations.

Abstract

This work is part of a project on weight bases for the irreducible representations of semisimple Lie algebras with respect to which the representation matrices of the Chevalley generators are given by explicit formulas. In the case of sl_n, the celebrated Gelfand-Tsetlin basis is the only such basis known. Using the setup of supporting graphs developed by Donnelly, we present a simple combinatorial proof of the Gelfand-Tsetlin formulas based on a rational function identity. Some properties of the Gelfand-Tsetlin basis are derived via an algorithm for solving certain equations on the lattice of semistandard Young tableaux.