Symmetry Classes of Alternating Sign Matrices

TL;DR

This paper explores symmetry classes of alternating sign matrices, identifying potential product formulas for their counts and revealing connections to cyclically symmetric plane partitions.

Contribution

It introduces eight symmetry classes of alternating sign matrices and provides evidence for simple enumeration formulas in six classes, highlighting unexpected links to plane partitions.

Findings

Six symmetry classes likely have simple product formulas for enumeration.

Connections between symmetry classes and cyclically symmetric plane partitions.

Evidence suggests new combinatorial identities involving these matrices.

Abstract

An alternating sign matrix is a square matrix satisfying (i) all entries are equal to 1, -1 or 0; (ii) every row and column has sum 1; (iii) in every row and column the non-zero entries alternate in sign. The 8-element group of symmetries of the square acts in an obvious way on square matrices. For any subgroup of the group of symmetries of the square we may consider the subset of matrices invariant under elements of this subgroup. There are 8 conjugacy classes of these subgroups giving rise to 8 symmetry classes of matrices. R. P. Stanley suggested the study of those alternating sign matrices in each of these symmetry classes. We have found evidence suggesting that for six of the symmetry classes there exist simple product formulas for the number of alternating sign matrices in the class. Moreover the factorizations of certain of their generating functions point to rather startling connections between several of the symmetry classes and cyclically symmetric plane partitions.