Equivariant Combinatorics

Information

This Summer School and Workshop is part of the 2017 Thematic Session on Algebra and Words in Combinatorics organized by the CRM and LaCIM (UQAM). (Note that this is the organizer-maintained website for this event; there is also an official website hosted by the CRM.)

Update: financial support. A limited amount of funding will be available for students and junior researchers. All requests for financial support must be made through the official conference website hosted by the CRM. The deadline for financial support requests is April 30, 2017.

Dates:

  • School: June 12 - 16, 2017
  • Workshop: June 19 - 23, 2017

Organization Committee:

Contact: equivariant-combinatorics@lacim.ca

Overview

Recent years have seen a remarkable expansion of profound interactions between Algebraic Combinatorics, Algebraic Geometry, and Algebraic Topology, exhibiting sometimes surprising ties to Theoretical Physics. These interactions involve objects such as: Operator Algebras on Symmetric Functions; Macdonald Symmetric Functions; Graded Representations of the Symmetric and General Linear Groups; Cohomology Rings of Grassmann and Flag Manifolds; Rational Cherednik Algebras; Representation Stability; etc. Even more recently, Rectangular Catalan Combinatorics has been developed in relation with several subjects covering a wide range of areas of mathematics, including: Representation Theory of the Sn-modules of Diagonal Harmonic Polynomials or Diagonal Coinvariant Spaces; Flag Bundles over the Hilbert Scheme of Points in the Plane (or higher dimension spaces); Affine Springer Fibres; Coloured Khovanov-Rozansky Homology of (m, n)-Torus Knots; etc. Because of the intimate ties that all of these subjects share with algebraic combinatorics, and to underline that they typically involve actions of (perhaps deformed) groups, we have coined the term Equivariant Combinatorics to speak of them.

Among the questions that we plan to explore in this workshop are several refined extensions of the "Shuffle conjecture", which link explicit combinatorial formulas coming from rectangular Catalan combinatorics to realizations of the Elliptic Hall Algebra in terms of creation operators. On one side, the various shuffle conjectures involve sums of combinatorial data tied to "parking functions" associated to given families of lattice paths (which depend on the version considered); while on the other side, the effect of specific creation operators lead to explicit formulas for this enumeration. We have just started to understand which operators relate to specific choices of families of paths; and we are coming to understand how this relates to representation theory, in terms of Sn-modules of multivariate polynomials. Among the recent exciting developments, techniques inspired by Knot Theory seem to help settle some of the most advanced conjectures in the field.

We plan to explore all these avenues, and to expand our understanding of the various links to other areas where the interplay between algebraic combinatorics and actions of groups and algebras play a central role.

Program

Minicourses. There will be a one-week preparatory school that will consist of mini-courses on the following selected topics from equivariant combinatorics.

  • Combinatorics of Schubert Calculus

    Maria Gillespie (University of California, Davis)

    Abstract. Given four fixed lines in three dimensions, how many lines pass through all four of them? This mini-course will serve as a broad introduction to the theory of Schubert calculus, a collection of combinatorial and algebraic techniques for solving linear intersection problems such as these. We will begin with the basics of the partition theory and symmetric function theory that describes Schubert calculus in the type A Grassmannian and complete flag variety. Time permitting, we will discuss the combinatorics of Schubert calculus in other Lie types, in equivariant or K-theoretic Schubert calculus, and in other variants.

  • Algebraic combinatorics and representations of Cherednik algebras

    Stephen Griffeth (Universidad de Talca)

    Abstract. This short course will provide an introduction to some of the interaction between Cherednik algebras and algebraic combinatorics. A number of rings of interest in algebraic combinatorics and mathematical physics arise as irreducible representations of Cherednik algebras. The most significant examples are the diagonal coinvariant ring (and suitable versions of it for other complex reflection groups), the Garsia-Haiman modules arising in the study of Macdonald polynomials, and the Verlinde algebras.

  • The Combinatorics of Symmetric Functions

    Jeff Remmel (University of California, San Diego)

    Abstract. We will give a series of talks that develops the combinatorics of symmetric functions, quasisymmetric functions, and plethsym that is needed to understand the combinatorics of Macdonald polynomials and some of the combinatorics behind the various generalization of the shuffle conjecture.

Workshop. The workshop will consist of a program of approximately twenty talks, with approximately four one-hour lectures per day. There will be free time for discussions between conference attendees, and rooms will be available to facilitate such discussions.

List of confirmed participants as of 2017-04-24

Logistics

For all logistical information (venue, accommodations, travel directions, etc.), please see the official conference website hosted by the CRM

Funding

A limited amount of funding will be available for students and junior researchers. All requests for financial support must be made through the official conference website hosted by the CRM. The deadline for financial support requests is April 30, 2017.