##### Department of Mathematics,

University of California San Diego

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### Special Colloquium

## Sami Assaf

#### University of Pennsylvania

## A combinatorial proof of Macdonald positivity

##### Abstract:

Since they were introduced by Macdonald 20 years ago, Macdonald polynomials have been widely studied and have been found to have applications in such areas as representation theory, algebraic geometry, group theory, statistics, and quantum mechanics. The Macdonald integral form may be defined as the unique function satisfying certain triangularity and orthogonality conditions, from which symmetry follows. The Macdonald Positivity Conjecture (now Theorem) states that the coefficients of the Macdonald integral form expanded into Schur functions are non-negative integers. The original proof, due to Haiman in 2001 building on joint work with Garsia, uses difficult machinery in algebraic geometry and does not provide a combinatorial understanding of the coefficients. In this talk we present a purely combinatorial proof of Macdonald positivity and give a combinatorial interpretation for the Schur coefficients. The proof utilizes an elegant monomial expansion for Macdonald polynomials discovered by Haglund in 2004 and a new combinatorial tool called a dual equivalence graph.

Host: Adriano Garsia

### October 30, 2007

### 4:00 PM

### AP&M 7321

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