Distributive Lattices Defined for Representations of Rank Two Semisimple Lie Algebras

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 A 1 ⊕ A 1 , A 2 , C 2 , and G 2 . 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.

Diamond-colored distributive lattices, move-minimizing games, and fundamental Weyl symmetric functions: The type $\mathsf{A}$ case

“…Most often, such a graph will either have its edges or its vertices colored by elements from some index set I (usually a set of positive integers). The conventions and notation we use concerning such structures largely borrow from [6], [19], [1], and [16]. The reader should note that in this paper, many proofs are omitted.…”

“…. , L p are all modular (respectively, distributive) lattices that are diamond-colored by a set I, with respective rank functions ρ (1) , ρ (2) , . .…”

“…This simple logic puzzle illustrates a couple of ideas that will be useful in what follows: (1) We will propose a game that uses domino tiles and pavings of certain shapes within a certain rectangular grid of squares, and (2) Although checkerboard-style coloring is not needed to state the possiblility/impossibility problems we address, it will be useful in formulating our solution.…”

“…However, at the outset of the paper in Section 2, we will concern ourselves with the technical and seemingly ancillary topic of diamond-colored modular and distributive lattices. These arise naturally in certain areas of algebraic combinatorics: as supporting graphs, which are combinatorial models for complex semi-simple Lie algebra representations (see for example [6]), and as splitting posets, which are poset models for so-called Weyl symmetric functions (see for example [1]).…”