A combinatorial formula for the nabla operator
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866908552904310784 |
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| author | Carlsson, Erik Mellit, Anton |
| author_facet | Carlsson, Erik Mellit, Anton |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2012_01627 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | A combinatorial formula for the nabla operator Carlsson, Erik Mellit, Anton Combinatorics Representation Theory 05E10, 20C30, 33D52, 05A30, 14M15, 14C05 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. |
| title | A combinatorial formula for the nabla operator |
| topic | Combinatorics Representation Theory 05E10, 20C30, 33D52, 05A30, 14M15, 14C05 |
| url | https://arxiv.org/abs/2012.01627 |