Algebraic Description of Character Varieties

We find finite, reasonably small, explicit generator sets of the coordinate rings of G-character varieties of finitely generated groups for all classical groups G. This result together with the method of Gröbner basis gives an algorithm for describing character varieties by explicit polynomial equations. For a reductive group G over a field K, denote the G-character variety of a finitely generated group Γ by XG(Γ). For every γ ∈ Γ and for every representation φ : G → GL(n,K), there is a regular map τφ,γ : XG → K sending the equivalence class of ρ : Γ → G to tr(φρ(γ)). We prove that functions of this form do not generate the coordinate rings of SO(2n,K)-character varieties. We discuss the generating sets over fields of arbitrary characteristics. 1. Character Varieties We will assume throughout the paper that G is an affine reductive group over an algebraically closed field K of characteristic zero and that Γ is a (discrete) group generated by γ1, ..., γN . The space of all G-representations of Γ forms an algebraic subset, Hom(Γ, G), of G , called the G-representation variety of Γ. The group G acts on this set by conjugating representations and the categorical quotient of that action XG(Γ) = Hom(Γ, G)//G is the G-character variety of Γ, c.f. [S2] and the references within. With Hom(Γ, G) and XG(Γ), there are naturally associated algebraic schemes Hom(Γ, G) and XG(Γ) = Hom(Γ, G)//G such that the coordinate rings, K[Hom(Γ, G)] and K[XG(Γ)], are nil-radical quotients of the algebras of global sections K[Hom(Γ, G)] and K[XG(Γ)], c.f. [S2]. Due to ubiquitous applications of character varieties in low-dimensional topology, geometry, gauge theory, and quantum field theories one is interested in an explicit description of them by polynomial equations, or, equivalently a description of K[X(Γ, G)] and of K[X (Γ, G)] by generators and relations. In this paper we describe generating sets of these rings for all classical G and all Γ. We do not discuss here the second part of the problem: finding complete sets of relations between generator sets. An algorithmic solution to this problem is given by the theory of Gröbner basis. (However, due to its computational complexity, this method may be difficult to be employed in practice.) Let T be a maximal torus in G.