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A combinatorial formula for the nabla operator
- Part of
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- Journal:
- Compositio Mathematica / Volume 161 / Issue 4 / April 2025
- Published online by Cambridge University Press:
- 17 September 2025, pp. 800-830
- Print publication:
- April 2025
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- Article
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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
$\nabla^k e_n$ [HHL+05a, CM18, Mel21], and the formula for 
$(\nabla^k p_1^n,e_n)$ from [EH16, GH22] 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 [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.
