Math Table: Partitions, Posets, and Lie algebras

Finite Young’s lattices L(m,n) (partitions in an m x n box) have striking properties such as rank-symmetry, rank-unimodality, and the strong Sperner property. Stanley conjectured that these admit a symmetric chain order. After a general discussion of posets and their properties of interest, I present a geometric model that identifies L(m,n) with a natural ordering on the integer points of a dilated simplex that arises as a weight diagram of type A_n in representation theory. With this, we also exhibit a visual reconstruction of Lindström’s symmetric chain decomposition for L(3,n). This work is joint with Robert W. Donley, Ammara Gondal, and Terrance Coggins.

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