A combinatorial formula for the nabla operator

Abstract

We present an LLT-type formula for a general power of the nabla operator applied to the Cauchy product for the modified Macdonald polynomials, and use it to deduce a new proof of the generalized shuffle theorem describing ${\nabla }^k{e}_n$, and the Elias-Hogancamp formula for $({\nabla }^k{p}_1^n,e_n)$ as corollaries. We give a direct proof of the theorem by verifying that the LLT expansion satisfies the defining properties of ${\nabla }^k$, such as triangularity in the dominance order, as well as a geometric proof based on a method for counting bundles on ${\mathbb{P}}^1$ due to the second author. These formulas are related to an affine paving of the type A unramified affine Springer fiber studied by Goresky, Kottwitz, and MacPherson, and also to Stanley's chromatic symmetric functions.

35 Pages

References in corpus (4)

• Toric braids and $(m,n)$-parking functions
• Hilbert schemes and $y$-ification of Khovanov-Rozansky homology
• On the computation of torus link homology
• Sweeping up Zeta