Workshop on Macdonald polynomials

About the workshop

The theory of symmetric polynomials is a classical centerpiece of mathematics. Its origins can be traced to the theory of equations and is at least three centuries old. This classical field received fresh impetus in the 1980s with Macdonald's introduction of a new family of symmetric functions P_λ(X; q, t) indexed by partitions λ and two parameters q, t. These included most previously known families of symmetric functions, such as Schur functions, Hall-Littlewood polynomials, Jack polynomials, and zonal spherical functions. This provided a framework that unified disparate results on special functions, orthogonal polynomials, constant term identities, etc.

More generally, Macdonald defined analogues of these polynomials for each root system. They form an orthogonal family of functions that is invariant under the corresponding Weyl group. He conjectured formulas for their norms, certain specializations, and symmetry properties. Many of these conjectures were established in special cases by numerous authors.

In the mid-1990s, Cherednik gave a unified proof of all these conjectures via his introduction of a new algebraic object - the "Double Affine Hecke Algebra" (DAHA), together with a canonical representation of the DAHA on a space of polynomials. Macdonald polynomials naturally became simultaneous eigenfunctions for a family of commuting elements of the DAHA acting on its polynomial representation. This point-of-view uncovered many more tools that have been used effectively over the last two decades to study Macdonald polynomials and their applications in other fields --representation theory, probablity theory etc.

This workshop aims to provide an account of the fundamentals of this theory. For a more detailed syllabus, please click here. This will be followed by a one-week conference focussing on the recent advances and applications (details can be found here).

Venue: ZOOM (online) Organizer: R. Venkatesh