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Motivic random variables and representation stability, I: Configuration spaces

Sean Howe

Algebraic & Geometric Topology 20 (2020) 3013–3045
DOI: 10.2140/agt.2020.20.3013

We prove a motivic stabilization result for the cohomology of the local systems on configuration spaces of varieties over attached to character polynomials. Our approach interprets the stabilization as a probabilistic phenomenon based on the asymptotic independence of certain motivic random variables, and gives explicit universal formulas for the limits in terms of the exponents of a motivic Euler product for the Kapranov zeta function. The result can be thought of as a weak but explicit version of representation stability for the cohomology of ordered configuration spaces. In the sequel, we find similar stability results in spaces of smooth hypersurface sections, providing new examples to be investigated through the lens of representation stability for symmetric, symplectic and orthogonal groups.

representation stability, motivic stabilization, arithmetic statistics, configuration spaces, cohomological stability.
Mathematical Subject Classification 2010
Primary: 14G10, 18F30, 55R80
Received: 14 April 2019
Revised: 27 November 2019
Accepted: 10 March 2020
Published: 8 December 2020
Sean Howe
Department of Mathematics
University of Utah
Salt Lake City, UT
United States