##### Department of Mathematics,

University of California San Diego

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### Math 269 - Combinatorics

## Andrei Negut

#### Department of Mathematics - Columbia University

## New Shuffle Conjectures via Shuffle Algebras

##### Abstract:

I will present the connection between two topics which seem completely unrelated, other than by a fortuitous common name. The original shuffle conjecture equates two symmetric polynomials $Phi_n[X;q,t]$ and $Pi_n[X;q,t]$. The first was conjectured by Garsia-Haiman in the early 1990's to produce the bigraded Frobenius characters of diagonal harmonic polynomials. The second was defined by Haglund et al around 2002 as a weighted enumeration of parking functions in the $n \times n$ lattice square. In a recent paper, Hikita constructs a new polynomial $Pi_{m,n}[X;q,t]$ for any pair of relatively prime natural numbers $m$ and $n$, by extending the notion of parking function to the case of an $m \times n$ lattice rectangle. It follows from Hikita's construction that $Pi_n[X;q,t] = Pi_{n+1,n}[X;q,t]$. The shuffle algebra is a representation-theoretic object constructed by Feigin and Odesskii, which acts on the K-theory of the Hilbert scheme of points in the plane. The K-theory is isomorphic to the ring of symmetric functions in infinitely many variables, so the shuffle algebra action produces many interesting symmetric functions. In particular we are able to construct a new polynomial $Phi_{m,n}[X;q,t]$ which reduces to $Phi_n[X;q,t] for m=n+1$, and conjecture that $Phi_{m,n}[X;q,t] = Pi_{m,n}[X;q,t]$ for general coprime $m,n$. In this talk, we will explain the ample connection that led to this extension of the Shuffle Conjecture. This is joint work with Eugene Gorsky.

Jeff Remmel

### April 16, 2013

### 4:00 PM

### AP&M 7321

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