Regularity and stable ranges of FI-modules. (English) Zbl 1535.18001
Let \(FI\) denote the category whose objects are finite sets and whose morphisms are all injective maps. T. Church et al. [Duke Math. J. 164, No. 9, 1833–1910 (2015; Zbl 1339.55004)] used the notation \(FI\) for this category as an acronym for Finite sets and Injective maps. The category has appeared in other contexts in algebraic topology, algebraic geometry, and computer science under various names.
The author gives refined bounds for the regularity of \(FI\)-modules and the stable ranges of \(FI\)-modules for various forms of their stabilization studied in the representation stability literature. The first bound, is in terms of the generation and presentation degrees of an \(FI\)-module \(V\). The second bound, is in terms of the local and stable degrees of \(V\) (in the sense of [T. Church et al., Adv. Math. 333, 1–40 (2018; Zbl 1392.15030)]) which is also often sharp. Presented results are applied to get explicit stable ranges for diagonal coinvariant algebras and improve those for ordered configuration spaces of manifolds and congruence subgroups of general linear groups.
The author gives refined bounds for the regularity of \(FI\)-modules and the stable ranges of \(FI\)-modules for various forms of their stabilization studied in the representation stability literature. The first bound, is in terms of the generation and presentation degrees of an \(FI\)-module \(V\). The second bound, is in terms of the local and stable degrees of \(V\) (in the sense of [T. Church et al., Adv. Math. 333, 1–40 (2018; Zbl 1392.15030)]) which is also often sharp. Presented results are applied to get explicit stable ranges for diagonal coinvariant algebras and improve those for ordered configuration spaces of manifolds and congruence subgroups of general linear groups.
Reviewer: Marek Golasiński (Olsztyn)
MSC:
| 18A25 | Functor categories, comma categories |
| 05E10 | Combinatorial aspects of representation theory |
| 55R80 | Discriminantal varieties and configuration spaces in algebraic topology |
| 11F75 | Cohomology of arithmetic groups |
Keywords:
configuration spaces; congruence subgroups; diagonal coinvariant algebras; \(FI\)-modules;representation stabilityReferences:
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