Self-complementary Plane Partitions by Proctor's Minuscule Method

Self-complementary Plane Partitions by Proctor's Minuscule Method

A method of Proctor [4] 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 parti...

(-1)-Enumeration of Self-Complementary Plane Partitions

Abstract. We prove a product formula for the remaining cases of the weighted enumeration of self–complementary plane partitions contained in a given box where adding one half of an orbit of cubes and removing the other half of the orbit changes the sign of the weight. We use nonintersecting lattice path families to express this enumeration as a Pfaffian which can be expressed in terms of the kn...

Totally Symmetric Self-Complementary Plane Partitions and Alternating Sign Matrices

We present multiresidue integral formulae for partial sums in the basis of link patterns of the polynomial solution to the level 1 Uq(ŝl2) quantum Knizhnik–Zamolodchikov equation at generic values of the quantum parameter q. These allow for rewriting and generalizing a recent conjecture [Di Francesco ’06] connecting the above to generating polynomials for weighted Totally Symmetric Self-Complem...

The Equivalence between Enumerating Cyclically Symmetric, Self-Complementary and Totally Symmetric, Self-Complementary Plane Partitions

We prove that the number of cyclically symmetric, self-complementary plane partitions contained in a cube of side 2n equals the square of the number of totally symmetric, self-complementary plane partitions contained in the same cube, without explicitly evaluating either of these numbers. This appears to be the first direct proof of this fact. The problem of finding such a proof was suggested b...

On Minuscule Representations, Plane Partitions and Involutions in Complex Lie Groups