A proof of the shuffle conjecture
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 $\tildeÅ$ 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 $(q,t)\mapsto (q^{1},t^{1})$.
 Publication:

arXiv eprints
 Pub Date:
 August 2015
 DOI:
 10.48550/arXiv.1508.06239
 arXiv:
 arXiv:1508.06239
 Bibcode:
 2015arXiv150806239C
 Keywords:

 Mathematics  Representation Theory;
 Mathematics  Combinatorics;
 05E05;
 05E10;
 05A30
 EPrint:
 some proofs are expanded. Accepted in JAMS