Self-complementary plane partitions by Proctor's minuscule method

A method of Proctor [European J. Combin. 5 (1984), no. 4, 331-350] realizes the set of arbitrary plane partitions in a box and the set of symmetric plane partitions as bases of linear representations of Lie groups. We extend this method by realizing transposition and complementation of plane partitions as natural linear transformations of the representations, thereby enumerating symmetric plane partitions, self-complementary plane partitions, and transpose-complement plane partitions in a new way

Move-minimizing puzzles, diamond-colored modular and distributive lattices, and poset models for Weyl group symmetric functions

The move-minimizing puzzles presented here are certain types of one-player combinatorial games that are shown to have explicit solutions whenever they can be encoded in a certain way as diamond-colored modular and distributive lattices. Such lattices can also arise naturally as models for certain algebraic objects, namely Weyl group symmetric functions and their companion semisimple Lie algebra representations. The motivation for this paper is therefore both diversional and algebraic: To show how some recreational move-minimizing puzzles can be solved explicitly within an order-theoretic context and also to realize some such puzzles as combinatorial models for symmetric functions associated with certain fundamental representations of the symplectic and odd orthogonal Lie algebras

Minuscule posets from neighbourly graph sequences

AbstractWe construct minuscule posets, an interesting family of posets arising in Lie theory, algebraic geometry and combinatorics, from sequences of vertices of a graph with particular neighbourly properties