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arXiv · 2411.17222

Irreducible components of two-column $\Delta$-Springer fibers

Joshua P. Connor·Sean T. Griffin·Kavish A. Purohit
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Abstract

The $\Delta$-Springer fibers $Y_{n,\lambda,s}$, introduced by Levinson, Woo, and the second author, generalize Springer fibers for $\mathrm{GL}_n(\mathbb{C})$ and give a geometric interpretation of the of the Delta Conjecture from algebraic combinatorics (at $t=0$). We prove that all irreducible components of the $\Delta$-Springer fiber $Y_{n,n-1}=Y_{n,(1^{n-1}),n-1}$ are smooth. In fact, we prove that any intersection of irreducible components of $Y_{n,n-1}$ is a smooth Hessenberg variety which has the structure of an iterated Grassmannian fiber bundle. We then give a presentation of the singular cohomology ring of each irreducible component of $Y_{n,n-1}$ and a combinatorial formula for the Poincar\'e polynomial of an arbitrary union of intersections of irreducible components in terms of arm and leg statistics on Dyck paths.

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Joshua P. Connor, Sean T. Griffin, Kavish A. Purohit. 2024-11-26. Irreducible components of two-column $\Delta$-Springer fibers. https://arxiv.org/abs/2411.17222

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