Four Symmetry Classes of Plane Partitions under One Roof

Abstract

In a previous paper, the author applied the permanent-determinant method of Kasteleyn and its non-bipartite generalization, the Hafnian–Pfaffian method, to obtain a determinant or a Pfaffian that enumerates each of the ten symmetry classes of plane partitions. After a cosmetic generalization of the Kasteleyn method, we identify the matrices in the four determinantal cases (plain plane partitions, cyclically symmetric plane partitions, transpose-complement plane partitions, and the intersection of the last two types) in the representation theory of sl(2, C). The result is a unified proof of the four enumerations.