Shuffle formula in science fiction for Macdonald polynomials

TL;DR

This paper introduces Macdonald intersection polynomials, explores their properties, and connects them to the shuffle formula, providing new proofs and combinatorial tools in algebraic combinatorics.

Contribution

It establishes key identities and connections for Macdonald intersection polynomials, including their relation to the shuffle formula and diagonal coinvariant algebra.

Findings

Proved vanishing and shape independence of Macdonald intersection polynomials.

Connected Macdonald intersection polynomials to the character bla e_{k-1} of diagonal coinvariant algebra.

Provided a new proof of the shuffle theorem.

Abstract

We initiate the study of the Macdonald intersection polynomials $\operatorname{I}_{\mu^{(1)},\dots,\mu^{(k)}}[X;q,t]$, which are indexed by $k$-tuples of partitions $\mu^{(1)},\dots,\mu^{(k)}$. These polynomials are conjectured to be equal to the bigraded Frobenius characteristic of the intersection of Garsia-Haiman modules, as proposed by the science fiction conjecture of Bergeron and Garsia. In this work, we establish the vanishing identity and the shape independence of the Macdonald intersection polynomials. Additionally, we unveil a remarkable connection between $\operatorname{I}_{\mu^{(1)},\dots,\mu^{(k)}}$ and the character $\nabla e_{k-1}$ of diagonal coinvariant algebra by employing the plethystic formula for the Macdonald polynomials of Garsia--Haiman--Tesler. Furthermore, we establish a connection between $\operatorname{I}_{\mu^{(1)},\dots,\mu^{(k)}}$ and the shuffle formula $D_{k-1}[X;q,t]$, utilizing novel combinatorial tools such as the column exchange rule, a new fermionic formula for the shuffle formula, and the lightning bolt formula for Macdonald intersection polynomials. Notably, our findings provide a new proof for the shuffle theorem.