Cyclic descent extensions on roots of unity in the symmetric group

The cyclic descent statistic that can be defined on certain permutation sets and other combinatorial objects, as a cyclic equivariant extension of the classical descent statistic. Cyclic descents were introduced by Klyachko and Cellini in the late 20th century, and further studied by many. An axiomatic approach was presented by Rhoades, further developed by Adin, Reiner and Roichman and others. 

In a recent work, Adin, Hegedus and Roichman characterize the conjugacy classes in the symmetric group, which carry a cyclic descent extension. A related permutation set is the set of roots of unity of fixed order in the symmetric group. In this talk we show that roots of unity of order $p^k$ in the symmetric group $S_n$ carry a cyclic descent extension if and only if $p$ and $n$ are not coprime. The proof involves a detailed study of unimodal permutations, their structure and enumeration.

Necessary background will be provided in the talk.