A proof of the shuffle conjecture

TL;DR

This paper proves the compositional shuffle conjecture, a significant generalization of the shuffle conjecture related to the character of the diagonal coinvariant algebra, using algebraic and combinatorial methods.

Contribution

It provides the first proof of the compositional shuffle conjecture, extending the algebraic framework and operator actions on symmetric functions.

Findings

Confirmed the conjecture for all cases

Established new algebraic operator framework

Connected combinatorial and algebraic structures

Abstract

We present a proof of the compositional shuffle conjecture, which generalizes the famous shuffle conjecture for the character of the diagonal coinvariant algebra. We first formulate the combinatorial side of the conjecture in terms of certain operators on a graded vector space $V_*$ whose degree zero part is the ring of symmetric functions $Sym[X]$ over $\mathbb{Q}(q,t)$. We then extend these operators to an action of an algebra $\tilde{\AA}$ acting on this space, and interpret the right generalization of the $\nabla$ using an involution of the algebra which is antilinear with respect to the conjugation $(q,t)\mapsto (q^{-1},t^{-1})$.