A proof of the Delta Conjecture when $q=0$

TL;DR

This paper proves the Delta Conjecture for the case when either parameter q or t equals zero, confirming a key combinatorial prediction related to symmetric functions and Macdonald polynomials.

Contribution

It provides the first proof of the Delta Conjecture in the special case when q=0 or t=0, advancing understanding of the conjecture's validity.

Findings

Confirmed the conjecture for q=0

Confirmed the conjecture for t=0

Established a foundation for future proofs

Abstract

In [The Delta Conjecture, Trans. Amer. Math. Soc., to appear] Haglund, Remmel, Wilson introduce a conjecture which gives a combinatorial prediction for the result of applying a certain operator to an elementary symmetric function. This operator, defined in terms of its action on the modified Macdonald basis, has played a role in work of Garsia and Haiman on diagonal harmonics, the Hilbert scheme, and Macdonald polynomials [A. M. Garsia and M. Haiman. A remarkable $q,t$-Catalan sequence and $q$-Lagrange inversion, J. Algebraic Combin. 5 (1996), 191--244], [M. Haiman, Vanishing theorems and character formulas for the Hilbert scheme of points in the plane, Invent. Math. 149 (2002), 371-407]. The Delta Conjecture involves two parameters $q,t$; in this article we give the first proof that the Delta Conjecture is true when $q=0$ or $t=0$.