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Stability for hyperplane complements of type B/C and statistics on squarefree polynomials over finite fields
- Publication Year :
- 2017
- Publisher :
- arXiv, 2017.
-
Abstract
- In this paper we explore a relationship between the topology of the complex hyperplane complements $\mathcal{M}_{BC_n} (\mathbb{C})$ in type B/C and the combinatorics of certain spaces of degree-$n$ polynomials over a finite field $\mathbb{F}_q$. This relationship is a consequence of the Grothendieck trace formula and work of Lehrer and Kim. We use it to prove a correspondence between a representation-theoretic convergence result on the cohomology algebras $H^*(\mathcal{M}_{BC_n} (\mathbb{C});\mathbb{C})$, and an asymptotic stability result for certain polynomial statistics on monic squarefree polynomials over $\mathbb{F}_q$ with nonzero constant term. This result is the type B/C analogue of a theorem due to Church, Ellenberg, and Farb in type A, and we include a new proof of their theorem. To establish these convergence results, we realize the sequences of cohomology algebras of the hyperplane complements as FI$_\mathcal{W}$-algebras finitely generated in FI$_\mathcal{W}$- degree $2$, and we investigate the asymptotic behaviour of general families of algebras with this structure. We prove a negative result implying that this structure alone is not sufficient to prove the necessary convergence conditions. Our proof of convergence for the cohomology algebras involves the combinatorics of their relators.<br />Comment: 36 pages. The new version added a citation. Comments welcome!
- Subjects :
- Degree (graph theory)
General Mathematics
010102 general mathematics
Structure (category theory)
Square-free integer
Type (model theory)
01 natural sciences
Cohomology
Mathematics - Algebraic Geometry
Finite field
Hyperplane
0103 physical sciences
Statistics
FOS: Mathematics
Mathematics - Combinatorics
Algebraic Topology (math.AT)
Mathematics - Algebraic Topology
010307 mathematical physics
Combinatorics (math.CO)
0101 mathematics
Representation Theory (math.RT)
Algebraic Geometry (math.AG)
Mathematics - Representation Theory
Monic polynomial
Mathematics
Subjects
Details
- Database :
- OpenAIRE
- Accession number :
- edsair.doi.dedup.....f53d9297a52eb836cf5455686c698a76
- Full Text :
- https://doi.org/10.48550/arxiv.1703.06349