The Cyclic Sieving Phenomenon on the Alternating Sign Matrices

We first present a previously unpublished result of Stanton [11] that the group of order four generated by rotation by 90◦ acting on alternating sign matrices exhibits the CSP with the obvious q-analogue of |ASM(n)|. In [12], Wieland introduced a much larger cyclic group that acts on the set of alternating sign matrices. Unfortunately, it has a very complex orbit structure that does not exhibit the CSP with that polynomial and is not suggestive of CSP with any simple polynomial. However, we found smaller groups that do exhibit the phenomenon, and in the process discovered an extremely large group of maps on the alternating sign matrices. We finish by suggesting the existence of a group of order three on the ASMs that exhibits cyclic sieving and a class of subsets of the ASM(n) that exhibit cyclic sieving with gyration and a new set of polynomials.