Equivariant Combinatorics

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

Organizers : François Bergeron (Université du Québec à Montréal), Luc Lapointe (Universidad de Talca, Chile), Jennifer Morse (Drexel University, Philadelphia), Franco Saliola (Université du Québec à Montréal)

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.