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PhD Exit Seminar: New Directions in Macdonald Theory
Algebra & Discrete MathematicsSpeaker: | Milo Bechtloff Weising, UC Davis |
Related Webpage: | https://sites.google.com/view/milobechtloffweising// |
Location: | 2112 MSB |
Start time: | Tue, Jun 4 2024, 11:00AM |
Macdonald polynomials have been central objects in algebraic combinatorics over the last few decades bridging together the fields of combinatorics, representation theory, and geometry in interesting and important ways. In this talk we will discuss some new directions in which to generalize Macdonald theory stemming from the proof of the Shuffle Theorem by Carlsson-Mellit and their introduction of the double Dyck path algebra. We will discuss how this algebra relates to Macdonald theory and to Cherednik's theory of double affine Hecke algebras (DAHA) using the work of Ion-Wu. Important to this picture is a generalization of the non-symmetric Macdonald polynomials for the space of almost symmetric functions which exhibit many exceptional properties. We will look at general machinery for studying representations of the double Dyck path algebra directly from the representation theory of DAHAs as well as a new family of explicit combinatorial representations generalizing the polynomial representation derived using the theory of vector-valued Macdonald polynomials of Dunkl-Luque. At the end of the talk we will discuss some areas for future research in Macdonald theory using this new machinery.