The dimension of solution sets to systems of equations in algebraic groups
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THE DIMENSION OF SOLUTION SETS TO SYSTEMS OF EQUATIONS IN ALGEBRAIC GROUPS
BY
Anton A. Klyachko∗ and Maria A. Ryabtseva Faculty of Mechanics and Mathematics Moscow State University Leninskie gory, Moscow 119991, Russia e-mail: [email protected], [email protected]
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 dim G.
1. Introduction Solomon’s theorem ([Solo69]): In any group, the number of solutions to a system of coefficient-free equations is divisible by the order of the group if the system has less equations than unknowns. Here, as usual, an equation over a group G is an expression of the form v(x1 , . . . , xm ) = 1, ∗ The work of the first author was supported by the Russian Foundation for Basic
Research, project no. 19-01-00591. Received April 4, 2019 and in revised form April 29, 2019
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A. A. KLYACHKO AND M. A. RYABTSEVA
Isr. J. Math.
where v is a word whose letters are unknowns, their inverses, and elements of G called coefficients (there are no coefficients though by the condition of Solomon’s theorem). In other words, the left-hand side of an equation is an element of the free product G∗F (x1 , . . . , xm ) of G and the free group F (x1 , . . . , xm ) of rank m (where m is the number of unknowns). Solomon’s theorem was generalized in different directions; see [Isaa70, Stru95, AmV11, GRV12, KM14, KM17, BKV18], and literature cited therein. For example, the following fact was proved in [KM14]. Theorem KM ([KM14]): The number of solutions to a system of equations over a group is divisible by the order of the centralizer of the set of coefficients 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. For coefficient-free equations, this theorem was obtained earlier by Gordon and Rodriguez-Villegas [GRV12] (and, in this case, the number of solutions is divisible by the order of the whole group). For example, the number of solutions to the system of equations {x100 y 100 [x, y]777 = 1, (xy)2020 = 1} is always divisible by the order of the group because this system (although not covered by Solomon’s theorem) satisfies the conditions of the Gordon– Rodriguez-Villegas theorem: the matrix composed of the exponent sums (called 100 100 ), and its the matrix of the system of equations) has the form ( 2020 2020 rank is one (while the number of unknowns is two). Note that, in these divisibility theorems, the group is not assumed to be necessarily finite; the divisibility is always understood
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