Duke Mathematical Journal· 2015Q1
FI-modules and stability for representations of symmetric groups
- 349citations
- Q1SCImago
- 2015year
Short summary
A new theory of FI-modules is introduced, proving that the character of symmetric group representations on various mathematical objects (like configuration spaces and moduli spaces) is a polynomial in cycle-counting functions for large n, meaning their dimensions are also polynomial in n.
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Key points
- Introduces and develops the theory of FI-modules.
- Proves that the character of symmetric group representations is polynomial in cycle-counting functions for large n.
- Demonstrates that the dimension of these representations is eventually a polynomial in n.
- Connects representation stability to finite generation of FI-modules.
AI-generated from the title and abstract; the full text is not read.
Abstract
In this paper we introduce and develop the theory of FI-modules. We apply this theory to obtain new theorems about: • the cohomology of the configuration space of n distinct ordered points on an arbitrary (connected, oriented) manifold; • the diagonal coinvariant algebra on r sets of n variables; • the cohomology and tautological ring of the moduli space of n -pointed curves; • the space of polynomials on rank varieties of n × n matrices; • the subalgebra of the cohomology of the genus n Torelli group generated by H 1 ; and more. The symmetric group S n acts on each of these vector spaces. In most cases almost nothing is known about the characters of these representations, or even their dimensions. We prove that in each fixed degree the character is given, for n large enough, by a polynomial in the cycle-counting functions that is independent of n . In particular, the dimension is eventually a polynomial in n . In this framework, representation stability (in the sense of Church–Farb) for a sequence of S n -representations is converted to a finite generation property for a single FI-module.
The authors' abstract, as published at the source. Duke Mathematical Journal, 2015 · DOI ↗
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Field: Geometry and Topology
Geometry and TopologyMathematics