Representations of categories of G-maps

Abstract

We study representations of wreath product analogues of categories of finite sets. This includes the category of finite sets and injections (studied by Church, Ellenberg, and Farb) and the opposite of the category of finite sets and surjections (studied by the authors in previous work). We prove noetherian properties for the injective version when the group in question is polycyclic-by-finite and use it to deduce general twisted homological stability results for such wreath products and indicate some applications to representation stability. We introduce a new class of formal languages (quasi-ordered languages) and use them to deduce strong rationality properties of Hilbert series of representations for the surjective version when the group is finite.

27 pages, split off from arXiv:1409.1670v1; v2: added Section 5.2 on wreath product version of Murnaghan's stability theorem; v3: significant rewrite; v4: corrected some proofs

References in corpus (4)

• Representation stability and finite linear groups
• Introduction to twisted commutative algebras
• Homological stability for automorphism groups
• ${\mathrm{FI}}_G$-modules, orbit configuration spaces, and complex reflection groups

Cited by in corpus (18)

• FI-modules and the cohomology of modular representations of symmetric groups
• FI- and OI-modules with varying coefficients
• Categories of FI type: a unified approach to generalizing representation stability and character polynomials
• Homological stability for automorphism groups
• Codimension and Projective Dimension up to Symmetry
• Homological Invariants of FI-modules and FI_G-modules
• Depth and the Local Cohomology of FI_G-modules
• Homological stability of topological moduli spaces
• On the degree-wise coherence of FI_G-modules
• Koszulity of directed categories in representation stability theory
• ${\mathrm{FI}}_G$-modules, orbit configuration spaces, and complex reflection groups
• A generating function approach to new representation stability phenomena in orbit configuration spaces
• A comparison of twisted coefficient systems
• Asymptotic behavior of symmetric ideals: A brief survey
• A Kruskal-Katona type result and applications
• An application of Nakayama functor in representation stability theory
• ${\mathrm{FI}}_G$-modules and arithmetic statistics
• Asymptotic behaviors of representations of graded categories with inductive functors