##### Department of Mathematics,

University of California San Diego

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### Math 269 -Combinatorics Seminar

## Tianyi Yu

#### UC San Diego

## Harmonic bases for generalized coinvariant algebras

##### Abstract:

S. Griffin introduced a quotient $R_{n,\lambda}$ of the polynomial ring $\mathbb{Q}$ with $\mathbb{Q}$. It simultaneously generalizes the Delta Conjecture coinvariant rings of Haglund-Rhoades-Shimozono and the cohomology rings of Springer fibers studied by Tanisaki and Garsia-Procesi. We describe the space $V_{n,\lambda}$ of harmonics attached to $R_{n,\lambda}$ and produce a harmonic basis of $R_{n,\lambda}$ indexed by certain ordered set partitions $\mathcal{OP}_{n,\lambda}$. Our description of $V_{n,\lambda}$ involves injective tableaux and Vandermonde determinants. Combinatorics of our harmonic basis is governed by a new extension of the Lemher code. \\ \\ This is a joint work with Brendon Rhoades and Zehong Zhao.

### June 3, 2021

### 3:00 PM

### https://ucsd.zoom.us/j/98781502962

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