# Manchester Algebra Seminar - Theo Douvropoulos

Dates: | 9 March 2023 |
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Times: | 13:00 - 14:00 |

What is it: | Seminar |

Organiser: | Department of Mathematics |

Who is it for: | University staff, Current University students |

The algebra seminar next week will be given by Theo Douvropoulos from the University of Massachusetts at Amherst. We draw attention to the fact that it will be on Thursday 9th March rather than Tuesday.

Date: Thursday 9th March Time: 13:00 Room: Frank Adams 1

Title: Decompositions of Parking Spaces, Reflection Laplacians, and Diagonal Coinvariants

Abstract: In the early 90's Haiman introduced the module of diagonal harmonics: certain co-invariants for the diagonal action of the symmetric group S_n on a set of 2n variables. Since then it has become a central object in algebraic combinatorics, representation theory, and algebraic geometry. After work of Gordon and Cherednik, similar objects have been defined for all finite Coxeter groups W and case-by-case techniques of the Coxeter Combinatorics community have produced combinatorial models for them, analogous --but weaker than-- the parking functions in the type A case). A long open problem has been to develop uniform, type-independent proofs and understanding for these combinatorial models.

I will discuss recent work, joint in part with Matthieu Josuat-Verges, that gives the first case-free proofs for a few of these questions. We prove a family of parabolic decompositions of the W-parking modules, and compare it to natural combinatorial recursions of the noncrossing partitions of W. Our main technical object is the spectral study of the W-Laplacian --an operator we introduced in previous work-- and its refinements. I will further present graded versions of these decompositions for the symmetric group S_n and state some open questions on other types.