##### Department of Mathematics,

University of California San Diego

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### Math 269 - Combinatorics

## Rekha Biswal

#### Max Planck Institute

## Macdonald polynomials and level two Demazure modules for affine $\mathfrak{sl}_{n+1}$

##### Abstract:

An important result due to Sanderson and Ion says that characters of level one Demazure modules are specialized Macdonald polynomials. In this talk, I will introduce a new class of symmetric polynomials indexed by a pair of dominant weights in $\mathfrak{sl}_{n+1}$ which is expressed as a linear combination of specialized symmetric Macdonald polynomials with coefficients defined recursively. These polynomials arose in my own work while investigating the characters of higher level Demazure modules. Using representation theory, we will see that this new family of polynomials interpolates between characters of level one and level two Demazure modules for affine $\mathfrak{sl}_{n+1}$ and give rise to new results in the representation theory of current algebras as a corollary. This is based on joint work with Vyjayanthi Chari, Peri Shereen, and Jeffrey Wand.

Host: Brendon Rhoades

### February 4, 2020

### 1:00 PM

### AP&M 7321

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