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arXiv · 1710.07078

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

Adriano Garsia·Jim Haglund·Jeffrey B. Remmel·Meesue Yoo
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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 $t=0$.

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Adriano Garsia, Jim Haglund, Jeffrey B. Remmel, Meesue Yoo. 2017-10-19. A proof of the Delta Conjecture when $q=0$. https://arxiv.org/abs/1710.07078

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