# OIST Representation Theory Seminar

### Date

Tuesday, November 29, 2022 - 09:30 to 10:30

on Zoom

### Description

Nicolle González, University of California, Berkeley

Title: Higher Rank Rational (q,t)-Catalan Polynomials and a Finite Shuffle Theorem

Abstract: The classical shuffle theorem states that the Frobenius character of the space of diagonal harmonics is given by a certain combinatorial sum indexed by parking functions on square lattice paths. The rational shuffle theorem, conjectured by Gorsky-Negut and proven by Mellit, states that the geometric action on symmetric functions (described by Schiffmmann-Vasserot) of certain elliptic Hall algebra elements $$P_{(m,n)}$$ yield the bigraded Frobenius character of a certain Sn representation. This character is known as the Hikita polynomial. In this talk I will introduce the higher rank rational (q,t)-Catalan polynomials and show these are equal to finite truncations of the Hikita polynomial. By generalizing results of Gorsky-Mazin-Vazirani and constructing an explicit bijection between rational semistandard parking functions and affine compositions, I will derive a finite analog of the rational shuffle theorem in the context of spherical double affine Hecke algebras.

#### Zoom info can be found on the seminar webpage.

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