Perverse sheaves on symmetric products of the plane
Abstract
For any field $k$, we give an algebraic description of the category $\mathrm{Perv}_\mathscr{S}(S^n (\mathbb{C}^2),k)$ of perverse sheaves on the $n$-fold symmetric product of the plane $S^n(\mathbb{C}^2)$ constructible with respect to its natural stratification and with coefficients in $k$. In particular, we show that it is equivalent to the category of modules over a new algebra that is closely related to the Schur algebra. As part of our description we obtain an analogue of modular Springer theory for the Hilbert scheme $\mathrm{Hilb}^n(\mathbb{C}^2)$ of $n$ points in the plane with its Hilbert-Chow morphism.
- Publication:
-
arXiv e-prints
- Pub Date:
- August 2022
- DOI:
- 10.48550/arXiv.2208.14351
- arXiv:
- arXiv:2208.14351
- Bibcode:
- 2022arXiv220814351B
- Keywords:
-
- Mathematics - Algebraic Geometry;
- Mathematics - Representation Theory
- E-Print:
- 47 pages. New introduction with a more concise description of the algebra