Rational Catalan combinatorics

December 17 to December 21, 2012

at the

American Institute of Mathematics, San Jose, California

organized by

Drew Armstrong, Stephen Griffeth, Victor Reiner, and Monica Vazirani

Original Announcement

This workshop will be devoted to understanding the interaction between new developments in algebra and combinatorics. In particular, it will focus on combinatorial objects counted by generalizations of Catalan numbers and their interaction with the representation theory of Cherednik algebras.

To each finite group generated by reflections (either real or pseudo-reflections) there is associated a family of generalizations of the classical Catalan numbers. These numbers have shown up in the study of: noncrossing partitions and braid groups; quotients of the root lattice and diagonal harmonics; Shi hyperplane arrangements and affine Weyl gruoups; and in cluster algebras and tropical geometry. The occurrence of the same numerology in different subjects suggests that there may be a deeper connection. There has been recent progress in connecting certain elements of this story, but a general encompassing framework is still missing.

Cherednik introduced the double affine Hecke algebra in the 1990s to solve Macdonald's constant term conjecture, and the more general "Cherednik algebras" have been quite influential. More recently, the generalized Catalan numbers mentioned above have appeared as Hilbert series for certain representations of "rational" Cherednik algebras, and their appearance in this context is uniform (i.e. it avoids Cartan-Killing type classification). Because the theory of Cherednik algebras is broad, powerful, and flexible, there is hope that it will provide a suitable unifying framework for Catalan combinatorics.

The cultures of Catalan combinatorics and Cherednik algebras are at present fairly disjoint. One of the purposes of this workshop is to bring the two cultures together with the hope that major breakthroughs will result.

The main goals of the workshop are:

Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

Papers arising from the workshop:

Affine permutations and rational slope parking functions
by  Eugene Gorsky, Mikhail Mazin, and Monica Vazirani,  Trans. Amer. Math. Soc. 368 (2016), no. 12, 8403-8445  MR3551576