Macdonald polynomials and the multispecies zero range process

Algebra Seminar
Monday, March 13, 2023 - 10:20am for 1.5 hours (actually 80 minutes)
Skiles 005
Olya Mandelshtam – University of Waterloo – olya.mandelshtam@uwaterloo.ca
Abeer Al Ahmadieh

Macdonald polynomials are a family of symmetric functions that are known to have remarkable connections to a well-studied particle model called the asymmetric simple exclusion process (ASEP). The modified Macdonald polynomials are obtained from the classical Macdonald polynomials using an operation called plethysm. It is natural to ask whether the modified Macdonald polynomials specialize to the partition function of some other particle system.

We answer this question in the affirmative with a certain multispecies totally asymmetric zero-range process (TAZRP). This link motivated a new tableaux formula for modified Macdonald polynomials. We present a Markov process on those tableaux that projects to the TAZRP and derive formulas for stationary probabilities and certain correlations, proving a remarkable symmetry property. This talk is based on joint work with Arvind Ayyer and James Martin.