A proof of the Extended Delta Conjecture

Abstract

We prove the Extended Delta Conjecture of Haglund, Remmel, and Wilson, a combinatorial formula for ${\Delta }_{h_l}{\Delta }_{e_k}^{\prime }{e}_n$, where ${\Delta }_{e_k}^{\prime }$ and ${\Delta }_{h_l}$ are Macdonald eigenoperators and $e_n$ is an elementary symmetric function. We actually prove a stronger identity of infinite series of $G{L}_m$ characters expressed in terms of LLT series. This is achieved through new results in the theory of the Schiffmann algebra and its action on the algebra of symmetric functions.

31 pages, 3 figures