A combinatorial formula for the nabla operator

We find an Lascoux–Leclerc–Thibon (LLT)-type formula for a general power of the nabla operator of [BG99] applied to the Cauchy product for the modified Macdonald polynomials, and use it to deduce a new proof of the generalized shuffle theorem describing ∇ᵏ eₙ [HHL + 05a, CM18, Mel21], and the formula for (∇ᵏ p₁ⁿ,eₙ) from [EH16, GH22] as corollaries. We give a direct proof of the theorem by verifying that the LLT expansion satisfies the defining properties of ∇ᵏ , such as triangularity in the dominance order, as well as a geometric proof based on a method for counting bundles on P¹ due to the second author [Mel20]. These formulas are related to an affine paving of the type A unramified affine Springer fiber studied by Goresky, Kottwitz, and MacPherson in [GKM04], and also to Stanley’s chromatic symmetric functions.