##### Department of Mathematics,

University of California San Diego

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### AWM Colloquium

## Sami Assaf

#### MIT

## A bijective approach to Macdonald polynomials

##### Abstract:

In 1913, Major Percy MacMahon showed that the major index and inversion number statistics are equidistributed over permutations. A bijective proof of this fact was first given in 1968 by Dominique Foata who constructed a recursive bijection on permutations such that the major index of the source is the inversion number of image. In 2004, Jim Haglund made a major breakthrough in the theory of Macdonald polynomials by conjecturing a formula for Macdonald polynomials, proved shortly thereafter with Haiman and Loehr, that involved relatively simple generalizations of the major index and the inversion number. In this talk, we will show how a filtration of Foata's bijection can be used to give a simple proof of Macdonald positivity for certain cases and outline how this approach might be generalized. Time permitting, we will outline how similar techniques might be useful in giving a bijective proof of the so-called q,t symmetry of Macdonald polynomials. This talk will be accessible to first year graduate students and contains several open problems.

### January 21, 2011

### 12:00 PM

### AP&M 6402

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