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arXiv · 0707.2421

Distributive lattices defined for representations of rank two semisimple Lie algebras

L. Wyatt Alverson II·Robert G. Donnelly·Scott J. Lewis·Marti McClard·Robert Pervine·Robert A. Proctor·N. J. Wildberger
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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.

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L. Wyatt Alverson II, Robert G. Donnelly, Scott J. Lewis, Marti McClard, Robert Pervine, Robert A. Proctor, N. J. Wildberger. 2007-07-17. Distributive lattices defined for representations of rank two semisimple Lie algebras. https://arxiv.org/abs/0707.2421

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