The cyclic sieving phenomenon

The cyclic sieving phenomenon, described by Reiner, Stanton and White in 2004, is a new way to discover and organise enumerative results on the orbit structure of cyclic group actions on finite sets. Instances of the cyclic sieving phenomenon were observed in diverse areas such as the combinatorics of Coxeter groups, cluster algebras and cluster categories but also for Schützenberger`s promotion on standard Young tableaux and its generalisation to crystal graphs. We currently know of two methods to establish these formulas. The first can be loosely described as a `guess- and-verify` procedure, the other transfers the problem into the realms of representation theory and boils down to finding an appropriate action on a vector space and evaluating the corresponding character. We plan to contribute in three ways: 1) by exploring the relations between invariant theory and combinatorics, in particular diagram algebras, Kuperberg`s web basis, the crystal basis and their connections to cluster algebras, 2) by developing a deductive process that is applicable in settings that resisted attempts of transfer to representation theory so far, and that avoids the `guess-and-verify` procedure, 3) and by devising computer algorithms to aid the guessing of formulas for cyclic sieving polynomials but also suitable representations.

The cyclic sieving phenomenon was first described in 2004 by Reiner, Stanton and White. The attribute "phenomenon" refers to the fact that, surprisingly often, and seemingly for no reason, one can obtain a formula for the lengths of the orbits of a "natural" cyclic group action by introducing a natural parameter into the counting formula for the family of sets under consideration. In this project, we studied the mechanisms underlying such "coincidences". A particularly interesting area in which this phenomenon occurs is invariant theory. In general, we study objects that are invariant under the action of a group, for example the group of rotations in space. More specifically, we studied objects on which a second, cyclic group action is defined. Our aim was to describe these objects as certain graphical diagrams, and the cyclic group action as rotation of these diagrams. One of the most important results of the project proves the existence of a natural set of diagrams in an important special case. This was surprising in two ways: Originally we assumed that we had to guess an exact definition of the diagrams involved, and we assumed that we would be able to do so. This was not the case. Moreover, an important part of the proof of existence was a numeric estimate for the number of certain other objects, which happens rather rarely in this branch of mathematics. Another result of the project are fundamental improvements to a publicly accessible database, https://findstat.org, for so-called statistics, or parameters, on combinatorial Objects such as permutations or finite graphs. Additional parameters often help to make mathematical problems more precise and therefore easier. In fact, with the help of this database, we were able to solve problems almost automatically.