Solitary Bases for Irreducible Representations of Semisimple Lie Algebras∗

The main results of this paper were found while addressing the question: what do the “nice” bases for the irreducible representations of semisimple Lie algebras look like? Using the Gelfand-Zetlin bases for the irreducible representations of gl(n,C) as our model, we take a combinatorial approach to this question by associating a certain kind of directed graph to each weight basis for an irreducible representation of a semisimple Lie algebra. These directed graphs are of combinatorial interest in their own right, and we show that these are connected, rank symmetric, rank unimodal, and strongly Sperner posets. We will view these posets as discrete invariants on the set of all weight bases for a given representation. In this way we split the set of weight bases into smaller subsets. We will take particular interest in the smallest subsets, that is, those containing essentially only one weight basis (and its scalar multiples). We call such a basis a solitary basis. We show that the Gelfand-Zetlin bases are solitary, and then we describe solitary bases for the fundamental representations of sp(2n,C) and so(2n+ 1,C), and for the adjoint representations of the simple Lie algebras. These modest but somewhat striking preliminary results also suggest that there could be a deep relationship between solitary bases and the combinatorial structure and relative “efficiency” of their associated posets. One other compelling curiosity: we will see that the coefficients for the actions of the generators for the Lie algebras on these solitary bases are rational numbers.