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The dimension of solution sets to systems of equations in algebraic groups

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Abstract

The Gordon-Rodriguez-Villegas theorem says that, in a finite group, the number of solutions to a system of coefficient-free equations is divisible by the order of the group if the rank of the matrix composed of the exponent sums of the j-th unknown in the i-th equation is less than the number of unknowns. We obtain analogues of this and similar facts for algebraic groups. In particular, our results imply that the dimension of each irreducible component of the variety of homomorphisms from a finitely generated group with infinite abelianization into an algebraic group G is at least dimG.

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Authors and Affiliations

  1. Faculty of Mechanics and Mathematics, Moscow State University, Leninskie gory, Moscow, 119991, Russia

    Anton A. Klyachko & Maria A. Ryabtseva

Authors
  1. Anton A. Klyachko
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  2. Maria A. Ryabtseva
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Correspondence to Anton A. Klyachko.

Additional information

The work of the first author was supported by the Russian Foundation for Basic Research, project no. 19-01-00591.

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Klyachko, A.A., Ryabtseva, M.A. The dimension of solution sets to systems of equations in algebraic groups. Isr. J. Math. 237, 141–154 (2020). https://doi.org/10.1007/s11856-020-2002-3

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  • DOI: https://doi.org/10.1007/s11856-020-2002-3

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