##### Department of Mathematics,

University of California San Diego

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### Food For Thought Seminar

## Angela Hicks

#### UCSD

## Combinatorics of the Diagonal Harmonics

##### Abstract:

The space of diagonal harmonics has emerged as one of the key ingredients in a program initiated by Garsia and Haiman to give a representation-theoretical proof of some conjectures in the theory of Macdonald polynomials. The study of this particular space has provided a remarkable display of connections between several areas, including representation theory, symmetric function theory, and combinatorics. Over two decades since the introduction of the diagonal harmonics, the bivariate Hilbert series of the diagonal harmonics has been the object of a variety of algebraic and combinatorial conjectures. In the following lecture, we will define the diagonal harmonics and explore some of the combinatorial objects related to this space. We assume only a basic understanding of undergraduate algebra and a passing appreciation for either free food or beautiful mathematical pictures.

### March 12, 2009

### 11:00 AM

### AP&M 7321

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