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

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 $t=0$.

arXiv comment12 pages, 1 figure

References in corpus (4)

• Vanishing theorems and character formulas for the Hilbert scheme of points in the plane
• Ordered set partitions, generalized coinvariant algebras, and the Delta Conjecture
• Hall-Littlewood expansions of Schur delta operators at $t=0$
• Decorated Dyck paths, the Delta conjecture, and a new q,t-square

Cited by in corpus (11)

• A module for the Delta conjecture
• Ordered set partitions, generalized coinvariant algebras, and the Delta Conjecture
• Hall-Littlewood expansions of Schur delta operators at $t=0$
• The Schröder case of the generalized Delta conjecture
• The Delta square conjecture
• $e$-Positivity Results and Conjectures
• The generalized Delta conjecture at t=0
• A valley version of the Delta square conjecture
• (GL_k x S_n)-modules and nabla of hook-indexed Schur functions
• A complete multipartite basis for the chromatic symmetric function
• The valley version of the Extended Delta Conjecture