##### Department of Mathematics,

University of California San Diego

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### Math 269 - Combinatorics Seminar

## Raymond Chou

#### UC Davis

## A descent basis for the Garsia-Procesi module

##### Abstract:

The Garsia-Procesi module $R_\lambda$ has a well known basis of Artin monomials indexed by λ-subYamanouchi words, which correspond to the inv-statistic of the Haglund-Haiman-Loehr combinatorial formula for the modified Macdonald polynomials $H_\lambda(X;q,t)$ at $t=0$. We introduce a new basis for $R_\lambda$ of Garsia-Stanton descent monomials, giving a major-index type formula of the modified Hall-Littlewood polynomial $H_\lambda(x;q,t)$, and discuss the subtle connection to $H_\lambda(x;q,t)$ at $q=0$ via Robinson-Schensted-Knuth insertion. Our formula was discovered while searching for a basis of the Garsia-Haiman module by extending a similar result of Carlsson and Oblomkov for the diagonal coinvariants $DH_n$. This is joint work with E. Carlsson.

Host: Brendon Rhoades

### November 7, 2022

### 4:00 PM

APM 5402

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