# On the combinatorics of $G_2$

## ABSTRACT

Abstract: To what extent can the representation theory of a simple Lie algebra $\mathcal{L}$ be recovered from straighforward (say, combinatorial) manipulations of information from the associated root system? Among other things, we might seek out a procedure for obtaining "labels" which can be used to generate characters for the irreducible representations of $\mathcal{L}$. For another, we might try to describe actions of Lie algebra elements on these labels.

Lately, Littelmann (following work of Lakshmibai, Seshadri, and others) has prescribed a method for generating labels by manipulating paths in the weight lattice for $\mathcal{L}$. We will present another way to obtain labels for the rank two simple Lie algebras $A_2$, $B_2$, and $G_2$ (joint work with Norman Wildberger). This method realizes labels as order ideals in certain partially ordered sets. In many cases, the distributive lattices obtained from these ideals can be used to realize actions of Lie algebra generators (joint work with Scott J.\ Lewis and Robert Pervine). In particular, we obtain new realizations of an infinte family of irreducible representations of $G_2$.

Speaker's Contact Info: donnelly(at-sign)math.mursuky.edu