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arXiv · 2501.00197

A geometric interpretation of the Delta Conjecture

Maria Gillespie·Eugene Gorsky·Sean T. Griffin
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Abstract

We introduce a variety $Y_{n,k}$, which we call the \textit{affine $\Delta$-Springer fiber}, generalizing the affine Springer fiber studied by Hikita, whose Borel-Moore homology has an $S_n$ action and a bigrading that corresponds to the Delta Conjecture symmetric function $\mathrm{rev}_q\,\omega \Delta'_{e_{k-1}}e_n$ under the Frobenius character map. We similarly provide a geometric interpretation for the Rational Shuffle Theorem in the integer slope case $(km,k)$. The variety $Y_{n,k}$ has a map to the affine Grassmannian whose fibers are the $\Delta$-Springer fibers introduced by Levinson, Woo, and the third author. Part of our proof of our geometric realization relies on our previous work on a Schur skewing operator formula relating the Rational Shuffle Theorem to the Delta Conjecture.

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Maria Gillespie, Eugene Gorsky, Sean T. Griffin. 2024-12-31. A geometric interpretation of the Delta Conjecture. https://arxiv.org/abs/2501.00197

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