Seminar on Discrete Mathematics: A proof of Kawanaka's Macdonald polynomial conjecture 

Speaker：Michael Schlosser (University of Vienna)

Time：2025-11-10 10:00

Location：Conference Room C503 at Administration Building at Haiyun Campus

Abstract:

In the 1980s, Macdonald introduced a very important family of multivariate orthogonal polynomials, the nowadays called "Macdonald polynomials". These are symmetric functions in a set of variables X with coefficients that are rational functions in two parameters q and t. They reduce in special cases to important families of symmetric functions including the Schur functions. The latter play a fundamental role in the representation theory of the symmetric group and possess very nice combinatorial properties. Many of these nice properties carry over to the Macdonald polynomials. When Macdonald introduced his family of polynomials, he proved some fundamental properties for them but also proposed some conjectures, known as the famous Macdonald conjectures. These were eventually proved (for all root systems) by Cherednik in 1995. In 1999, Kawanaka conjectured a formal identity involving Macdonald polynomials that can be regarded as a product formula for a quadratic type of generating function for the Macdonald polynomials. We establish a quite general multivariate theta function identity and utilize a special case of it to prove Kawanaka's Macdonald polynomial conjecture. This talk is on joint work with Robin Langer and Ole Warnaar. Full details are given in https://doi.org/10.3842/SIGMA.2009.055