Combinatorial constructions of weight bases: the Gelfand-Tsetlin basis
The electronic journal of combinatorics, Tome 17 (2010)
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 $\mathfrak{ 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 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.
@article{10_37236_305,
author = {Patricia Hersh and Cristian Lenart},
title = {Combinatorial constructions of weight bases: the {Gelfand-Tsetlin} basis},
journal = {The electronic journal of combinatorics},
year = {2010},
volume = {17},
doi = {10.37236/305},
zbl = {1228.17010},
url = {http://geodesic.mathdoc.fr/articles/10.37236/305/}
}
Patricia Hersh; Cristian Lenart. Combinatorial constructions of weight bases: the Gelfand-Tsetlin basis. The electronic journal of combinatorics, Tome 17 (2010). doi: 10.37236/305
Cité par Sources :