Leaving the Hall: explicit formulas for Negut operators
Abstract
Recent major breakthroughs in q,t-combinatorics include the introduction of the Dyck path algebra ๐ธq,t by Carlsson and Mellit and of the Catalanimals by Blasiak et al., both of which led, among other things, to independent proofs of different extensions of the rational shuffle conjecture of Bergeron et al. The first main contribution of this paper is a simple, explicit formula inside the algebra ๐ธq,t for the Negut operators, yielding a direct, elementary connection between the original operators of the rational shuffle conjecture and the corresponding Catalanimals. Our formula bypasses the elliptic Hall algebra, turning these operators into transparent, workable tools whose action we can compute exactly and efficiently on any symmetric function, not just constants. Our second main contribution consists of a series of explicit formulas relating the Negut operators to the Theta operators introduced by D'Adderio et al. To prove these formulas, we provide an extension of the aforementioned Theta operators to the entire algebra ๐ธq,t, allowing us to obtain a series of new combinatorial results. The algebraic computations underlying this extension have been formalized in Lean. To showcase the power of our results, we give a proof, also partially formalized in Lean, of the Theta conjecture of D'Adderio et al., first stated in 2019.
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BibTeX
@misc{dadderio2026leaving,
title = {{Leaving the Hall: explicit formulas for Negut operators}},
author = {Michele D'Adderio and Giovanni Interdonato and Alessandro Iraci and Roberto Pagaria},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.14836},
eprint = {2608.14836},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.14836},
}
RIS
TY - UNPB
TI - Leaving the Hall: explicit formulas for Negut operators
AU - Michele D'Adderio
AU - Giovanni Interdonato
AU - Alessandro Iraci
AU - Roberto Pagaria
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2608.14836
UR - https://arxiv.org/abs/2608.14836
AB - Recent major breakthroughs in q,t-combinatorics include the introduction of the Dyck path algebra ๐ธ_q,t by Carlsson and Mellit and of the Catalanimals by Blasiak et al., both of which led, among other things, to independent proofs of different extensions of the rational shuffle conjecture of Bergeron et al. The first main contribution of this paper is a simple, explicit formula inside the algebra ๐ธ_q,t for the Negut operators, yielding a direct, elementary connection between the original operators of the rational shuffle conjecture and the corresponding Catalanimals. Our formula bypasses the elliptic Hall algebra, turning these operators into transparent, workable tools whose action we can compute exactly and efficiently on any symmetric function, not just constants. Our second main contribution consists of a series of explicit formulas relating the Negut operators to the Theta operators introduced by D'Adderio et al. To prove these formulas, we provide an extension of the aforementioned Theta operators to the entire algebra ๐ธ_q,t, allowing us to obtain a series of new combinatorial results. The algebraic computations underlying this extension have been formalized in Lean. To showcase the power of our results, we give a proof, also partially formalized in Lean, of the Theta conjecture of D'Adderio et al., first stated in 2019.
ER -