Positive $m$-divisible non-crossing partitions and their cyclic sieving

Christian Krattenthaler (U of Vienna)

Feb 17. 2025, 16:00 — 16:45

Buan, Reiten and Thomas (implicitly) defined positive $m$-divisible non-crossing partitions, by setting up a bijection between the facets of the generalized cluster complex of Fomin and Reading and m-divisible non-crossing partitions, positive clusters corresponding to positive $m$-divisible non-crossing partitions. We embark on a finer enumerative study of these combinatorial objects associated with finite reflection groups. In particular, we define a cyclic action on them, which - together with the "obvious" q-analogue of positive Fu{\ss}--Catalan numbers - satisfies the cyclic sieving phenomenon of Reiner, Stanton and White. Our proof is a - lengthy - case-by-case verification. Crucial in the proof in the classical types are a combinatorial realisation of the positive m-divisible non-crossing as certain classical m-divisible non-crossing partitions on the one hand and the combinatorial description of the action as some kind of pseudo-rotation on the other hand.

Further Information
Venue:
ESI Boltzmann Lecture Hall
Recordings:
Recording
Files:
Slides
Associated Event:
Recent Perspectives on Non-crossing Partitions through Algebra, Combinatorics, and Probability (Workshop)
Organizer(s):
Adrian Celestino Rodriguez (TU Graz)
Kurusch Ebrahimi-Fard (NTNU, Trondheim)
James Mingo (Queen's U, Kingston)
Martin Rubey (TU Vienna)
Eleni Tzanaki (U of Crete)
Yannic Vargas (CUNEF U, Madrid)