Cyclic descents, toric Schur functions and Gromov-Witten invariants

Descents of permutations have been studied for more than a century. This concept was vastly generalized, in particular to standard Young tableaux (SYT). More recently, cyclic descents of permutations were introduced by Cellini and further studied by Dilks, Petersen and Stembridge. Looking for a corresponding concept for SYT, Rhoades found a very elegant solution for rectangular shapes. In an attempt to extend the concept of cyclic descents, explicit combinatorial definitions for two-row and certain other shapes have been found, implying the Schur-positivity of various quasi-symmetric functions. In all cases, the cyclic descent set admits a cyclic group action and restricts to the usual descent set when the letter n is ignored. Consequently,

the existence of a cyclic descent set with these properties was conjectured for all shapes, even the skew ones. This talk will report on the surprising resolution of this conjecture: Cyclic descent sets exist for nearly all skew shapes, with an interesting small set of exceptions. The proof applies non-negativity properties of Postnikov's toric Schur polynomials and a new combinatorial interpretation of certain Gromov-Witten invariants. We shall also comment on issues of uniqueness. Based on joint works with Sergi Elizalde, Vic Reiner, Yuval Roichman.

Dr. Menachem Shlossberg invites you to a “haramat kosit” in celebration of obtaining a postdoctorate

at the University of Udine, Italy, at 1:30 PM next to the Colloquium Room

ד"ר מנחם שלוסברג מזמין אתכם להרמת כוסית לרגל קבלת משרת פוסט דוקטורט באונ' אודין באיטליה

 בשעה 13:30 בחדר ע"י חדר המחלקה