##### Department of Mathematics,

University of California San Diego

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### Combinatorics Seminar

## Cristian Lenart

#### Department of Mathematics - State University of New York, Albany

## Affine crystals, Macdonald polynomials, and combinatorics

##### Abstract:

Crystals are colored directed graphs encoding information about Lie algebra representations. Kirillov-Reshetikhin (KR) crystals correspond to certain finite-dimensional representations of affine Lie algebras. I will present a combinatorial model which realizes tensor products of (column shape) KR crystals uniformly across affine types. Some computational applications are discussed. A corollary states that the Macdonald polynomials (which generalize the irreducible characters of semisimple Lie algebras), upon a certain specialization, coincide with the graded characters of tensor products of KR modules. The talk is largely self-contained, and is based on a series of papers with A. Lubovsky, S. Naito, D. Sagaki, A. Schilling, and M. Shimozono.

Host: Jeff Remmel

### June 24, 2016

### 4:00 PM

### AP&M 7321

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