Symmetries of plane partitions

AbstractWe introduce a new symmetry operation, called complementation, on plane partitions whose three-dimensional diagram is contained in a given box. This operation was suggested by work of Mills, Robbins, and Rumsey. There then arise a total of ten inequivalent problems concerned with the enumeration of plane partitions with a given symmetry. Four of these ten problems had been previously considered. We survey what is known about the ten problems and give a solution to one of them. The proof is based on the theory of Schur functions, in particular the Littlewood-Richardson rule. Of the ten problems, seven are now solved while the remaining three have conjectured simple solutions