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The dimension of solution sets to systems of equations in algebraic groups
- Source :
- Israel Journal of Mathematics. 237:141-154
- Publication Year :
- 2020
- Publisher :
- Springer Science and Business Media LLC, 2020.
-
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 $j$-th unknown in $i$-th equation is less than the number 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 abelianisation into an algebraic group $G$ is at least $\dim G$.<br />Comment: 6 pages. A Russian version of this paper is at http://halgebra.math.msu.su/staff/klyachko/papers.htm . V.2: An open question added, misprints corrected. V.3: misprints corrected
- Subjects :
- Finite group
Pure mathematics
Rank (linear algebra)
Group (mathematics)
General Mathematics
010102 general mathematics
Group Theory (math.GR)
0102 computer and information sciences
01 natural sciences
Mathematics - Algebraic Geometry
010201 computation theory & mathematics
Algebraic group
FOS: Mathematics
Order (group theory)
Finitely generated group
0101 mathematics
Algebraic number
Mathematics - Group Theory
Algebraic Geometry (math.AG)
Irreducible component
Mathematics
Subjects
Details
- ISSN :
- 15658511 and 00212172
- Volume :
- 237
- Database :
- OpenAIRE
- Journal :
- Israel Journal of Mathematics
- Accession number :
- edsair.doi.dedup.....96e003ef8af36da10cfd710d9689ca6c