Sách Combinatorial Constructions of Weight Bases: The Gelfand-Tsetlin Basis

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 sln, the
celebrated Gelfand-Tsetlin basis is the only such basis known. Using the setup
of supporting graphs developed by Donnelly, we present a new interpretation and
a simple combinatorial proof of the Gelfand-Tsetlin formulas based on a rational
function identity (all the known proofs use more sophisticated algebraic tools). A
constructive approach to the Gelfand-Tsetlin formulas is then given, based on a
simple algorithm for solving certain equations on the lattice of semistandard Young
tableaux. This algorithm also implies certain extremal properties of the Gelfand-
Tsetlin basis.