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Distributive Lattices Defined for Representations of Rank Two Semisimple Lie Algebras

2009
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SIAM Journal on Discrete Mathematics
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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).

doi:10.1137/070689887
fatcat:zafhcn3aprhyvchns2of7whrmu