Uniformly Presented Vector Spaces

Abstract

Gaussian elimination answers any question about a finitely presented vector space. However, a "uniform family" of such presentations--given as generic relations among an unspecified number of generators--is susceptible to elimination only once the number of generators is fixed. We develop a theory of "uniformly presented vector spaces" to compute with these uniform families, introducing a formalism of finitely generated functors from the category of finite sets to the category of finite dimensional Q-vector spaces. We show that these representations have finite length and polynomial dimension away from the empty set, and produce finite leftward resolutions by manageable functors.

44 pages

References in corpus (7)

• FI-modules and stability for representations of symmetric groups
• Representation theory and homological stability
• FI-modules over Noetherian rings
• GL-equivariant modules over polynomial rings in infinitely many variables
• Stability patterns in representation theory
• A stability conjecture for the unstable cohomology of SL_n Z, mapping class groups, and Aut(F_n)
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Cited by in corpus (6)

• Homology of FI-modules
• FI-modules and the cohomology of modular representations of symmetric groups
• The representation theory of Brauer categories I: triangular categories
• Algebraic structures on cohomology of configuration spaces of manifolds with flows
• Polynomial representations of the Witt Lie algebra
• Resolving Irreducible $\mathbb{C}{S}_n$-Modules by Modules Restricted from $G{L}_n(\mathbb{C})$