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Towards plethystic formulas for wreath Macdonald polynomials

Date 2024-09-25 Speaker Joshua Wen Dept. Universität Wien Room 27-325 Time 14:30-16:00

Defined by Haiman, the wreath Macdonald polynomials are a generalization of the modified Macdonald polynomials where the symmetric groups are replaced by their wreath products with a fixed cyclic group. Through a wreath analogue of the Frobenius characteristic, they correspond to partially-symmetric functions. Their existence and Schur positivity were proved by Bezrukavnikov--Finkelberg using geometric methods, and many of their combinatorial aspects remain unexplored. In this talk, I will introduce them from a symmetric-function-theoretic perspective, building up to a conjectural wreath generalization of the plethystic formula of Garsia--Haiman--Tesler (GHT). If time permits, I will discuss the interplay with quantum toroidal algebras, both in the original GHT formula and in the wreath setting.