Colloquium: Tuesday Jan. 14, 2025. Speaker: Ron Adin (Bar Ilan). Title: “Cyclic descents”.
Speaker: Ron Adin (Bar Ilan)
Place: Room 614, Education and Sciences Building, University of Haifa
Date and Time: January 14, 2025, 14:00-15:00
Title: Cyclic descents
Abstract: Descents of permutations, studied since Euler, have also been defined for other combinatorial objects. They have many applications in combinatorics and representation theory. Cyclic descents of permutations were introduced by Klyachko and Cellini, but extensions to other objects are apparently very limited.
We use an axiomatic approach to define a cyclic extension of the descent set. This opens the question of the existence of such an extension. We resolve the existence question in two important cases: the set of all standard Young tableaux of a given shape, and an arbitrary conjugacy class of permutations. In both cases, an extension exists for all but a small and simply defined class of exceptions.
Tools involve nonnegativity properties of toric Schur polynomials, a combinatorial interpretation of certain Gromov-Witten invariants, and a new approach to a classical open problem of Thrall.
No preliminary knowledge of these topics is assumed.
The talk is based on joint works with Sergi Elizalde, Pál Hegedűs, Vic Reiner and Yuval Roichman.
