Affine Schubert calculus and double coinvariants
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2018
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| _version_ | 1866914568047951872 |
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| author | Carlsson, Erik Oblomkov, Alexei |
| author_facet | Carlsson, Erik Oblomkov, Alexei |
| contents | We define an action of the double coinvariant algebra
$DR_n$ on the equivariant Borel-Moore homology of the affine flag variety
$\widetilde{Fl}_n$ in type $A$, which has an explicit form in terms of
the left and right action of the (extended) affine Weyl group and multiplication by Chern classes.
Up to first order in the augmentation ideal, we show that it coincides with the
action of the Cherednik algebra on the equivariant homology of
the homogeneous affine Springer fiber $\widetilde{S}_{n,n+1} \subset \widetilde{Fl}_n$
due to Yun and the second author, and therefore
preserves the non-equivariant Borel-Moore homology groups $H_*(\widetilde{S}_{n,n+1})\hookrightarrow H_*(\widetilde{Fl}_n)$.
We then define a geometric filtration
$F_{a} H_*(\widetilde{S}_{n,n+1})=H_*(\widetilde{S}(a))$
by closed subspaces $\widetilde{S}(a)\subset \widetilde{S}_{n,n+1}$,
which we prove recovers the Garsia-Stanton descent order on $DR_n$. We use this to deduce an explicit monomial basis of $DR_n$, as well as an independent proof of the (non-compositional) Shuffle Theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1801_09033 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Affine Schubert calculus and double coinvariants Carlsson, Erik Oblomkov, Alexei Combinatorics Algebraic Geometry We define an action of the double coinvariant algebra $DR_n$ on the equivariant Borel-Moore homology of the affine flag variety $\widetilde{Fl}_n$ in type $A$, which has an explicit form in terms of the left and right action of the (extended) affine Weyl group and multiplication by Chern classes. Up to first order in the augmentation ideal, we show that it coincides with the action of the Cherednik algebra on the equivariant homology of the homogeneous affine Springer fiber $\widetilde{S}_{n,n+1} \subset \widetilde{Fl}_n$ due to Yun and the second author, and therefore preserves the non-equivariant Borel-Moore homology groups $H_*(\widetilde{S}_{n,n+1})\hookrightarrow H_*(\widetilde{Fl}_n)$. We then define a geometric filtration $F_{a} H_*(\widetilde{S}_{n,n+1})=H_*(\widetilde{S}(a))$ by closed subspaces $\widetilde{S}(a)\subset \widetilde{S}_{n,n+1}$, which we prove recovers the Garsia-Stanton descent order on $DR_n$. We use this to deduce an explicit monomial basis of $DR_n$, as well as an independent proof of the (non-compositional) Shuffle Theorem. |
| title | Affine Schubert calculus and double coinvariants |
| topic | Combinatorics Algebraic Geometry |
| url | https://arxiv.org/abs/1801.09033 |