Distributive lattices defined for representations of rank two semisimple Lie algebras

TL;DR

This paper constructs uniform distributive lattices from elementary combinatorics for rank two semisimple Lie algebra representations, providing new insights into their structure and rank generating functions, answering longstanding questions.

Contribution

It introduces new combinatorial posets and lattices for rank two Lie algebra representations, offering explicit formulas and a new setting for Weyl characters, many of which are novel.

Findings

Provides quotient-of-products formulas for rank generating functions.

Establishes a new combinatorial framework for Weyl characters.

Most lattices introduced are new or related to known structures.

Abstract

For a rank two root system and a pair of nonnegative integers, using only elementary combinatorics we construct two posets. The constructions are uniform across the root systems A1+A1, A2, C2, and G2. Examples appear in Figures 3.2 and 3.3. We then form the distributive lattices of order ideals of these posets. Corollary 5.4 gives elegant quotient-of-products expressions for the rank generating functions of these lattices (thereby providing answers to a 1979 question of Stanley). Also, Theorem 5.3 describes how these lattices provide a new combinatorial setting for the Weyl characters of representations of rank two semisimple Lie algebras. Most of these lattices are new; the rest of them (or related structures) have arisen in work of Stanley, Kashiwara, Nakashima, Littelmann, and Molev. In a future paper, one author shows that the posets constructed here form a Dynkin diagram-indexed answer to a combinatorially posed classification question. In a companion paper, some of these lattices are used to explicitly construct some representations of rank two semisimple Lie algebras. This implies that these lattices are strongly Sperner.