A Proof of the Extended Delta Conjecture. (22nd February 2023)

• Record Type: Journal Article
• Authors: Blasiak, Jonah; Haiman, Mark; Morse, Jennifer; Pun, Anna; Seelinger, George H.
• Is Part Of: Forum of mathematics. Volume 11(2023)
• Publication Date: 2023-02-22
• Subjects: 05E05 -- 16T30; Mathematics -- Periodicals; 510
• DOI: 10.1017/fmp.2023.3
• Languages: English
• ISSNs: 2050-5086

Abstract

Abstract: We prove the extended delta conjecture of Haglund, Remmel and Wilson, a combinatorial formula for $\Delta _{h_l}\Delta ' _{e_k} e_{n}$, where $\Delta ' _{e_k}$ 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 $\operatorname {\mathrm {GL}}_m$ 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.