Rowmotion, Promotion and Homomesy via Toggleability Statistics

09.01.2024 15:15 - 16:45

Seline Unterwaditzer (University of Vienna)

Abstract: We are taking a closer look at the field of dynamical algebraic combinatorics which is the study of natural dynamical operators acting on objects familiar to algebraic combinatorics. A recurring theme in this field is homomesy, which occurs when a statistic on the set of objects has the same average along every orbit of the action. One of the most intensely studied operators is the rowmotion operator acting on order ideals of a (finite) poset. Recent research has focused on the impact of this rowmotion operator, aiming to identify homomesies. The paper by Colin Defant, Sam Hopkins, Svetlana Poznanovic and James Propp introduces a systematic technique, expressing relevant statistics as linear combinations of "toggleability statistics" plus a constant. This method recaptures most of the known homomesies and also extends to modified context. Expanding our exploration within dynamical algebraic combinatorics, we turn our attention to another prominent operator – the promotion operator. Specifically, our focus will be on examining the promotion action on Young Tableaux. The primary goal is to establish a correspondence between promotion and row-motion across various domains.

Organiser:
I. Fischer (U Wien), M. Schlosser (U Wien)
Location:

BZ 2, 2. OG., OMP 1