The University of Michigan Combinatorics Seminar


Abstract 

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 edgecolored 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 GelfandTsetlin bases, some of the
bases recently obtained by Molev, and many of the bases we have constructed
enjoy this rare property. 