  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.
