Representation Theory Connections to (q,t)-Combinatorics (19w5131)


(York University)

(Université du Québec à Montréal)

(University of Pennsylvania)


The Banff International Research Station will host the "Representation Theory Connections to q, t-Combinatorics" workshop in Banff from January 20, 2019 to January 25, 2019.

It is well recognized that one of the most important topics in modern mathematics and mathematical physics is representation theory. Recent advances linking Macdonald polynomials (an object of central importance to algebraic combinatorics which depends on two parameters q,t) to geometry and integrable models of mathematical physics have resulted in a number of amazing new formulas potentially describing the refined decomposition into “irreducible” pieces of central representations for these two fields. However the formulas that we currently have are only expressed in terms of monomials, rather than "Schur functions”,which would be much preferable. Indeed, such expressions would explicitly describe the underlying structure of the representation, since irreducibles are directly encoded by Schur functions. Hence one would then be able to directly read off their decomposition from the corresponding expression in terms of Schur functions. From the physical perspective, this would make it possible to explicitly deduce properties of the models considered, directly from the formulas. Many of these formulas are also connected to exciting new combinatorial expressions tied to the study of knot invariants, with analogous encoding of properties in term of Schur functions. This workshop intends to bring together a diverse set of top researchers from algebraic combinatorics, knot theory, and algebra, with the aim of exploiting these recent advances in order to find the required explicit Schur expressions, and obtain new combinatorial expressions regarding the associated knot invariants.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT).