Solitary bases for representations of semisimple Lie algebras

In this talk we will describe a program for studying weight bases of representations of semisimple Lie algebras from a combinatorial point of view. A "supporting graph" is an edge-colored directed graph which encodes information about the actions of Lie algebra generators on a given weight basis for a representation. For irreducible representations, the supporting graph for any weight basis is the Hasse diagram for a rank symmetric, rank unimodal, and strongly Sperner poset. We seek weight bases whose supporting graphs are "efficient" in a certain sense and have a "nice" combinatorial structure. We take particular interest in those weight bases which are "uniquely" identified by their supporting graphs; we call these solitary bases. At most a finite number of weight bases for a given representation are solitary. The Gelfand-Tsetlin bases, some of the bases recently obtained by Molev, and many of the bases we have constructed enjoy this rare property.

We will present recent results for several families of representations. We have used supporting graphs to construct all "efficient" bases for the adjoint representations, to classify all representations which have only one weight basis up to scalar changes, and to construct certain odd orthogonal representations (joint work with Scott Lewis and Robert Pervine at Murray State University). These latter constructions are used to resolve a conjecture concerning a certain family of distributive lattices found by Reiner and Stanton.