Distributive Lattices Defined for Representations of Rank Two Semisimple Lie Algebras

L. Wyatt Alverson, Robert G. Donnelly, Scott J. Lewis, Marti McClard, Robert Pervine, Robert A. Proctor, N. J. Wildberger
2009 SIAM Journal on Discrete Mathematics  
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).
more » ... , 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.
doi:10.1137/070689887 fatcat:zafhcn3aprhyvchns2of7whrmu