Representation variety

Let $k$ be an algebraically closed field of characteristic zero, $\Gamma$ a finitely presented group, and $G$ an affine algebraic group over $k$ . For a finite presentation

\Gamma=\langle\gamma_1,\ldots,\gamma_r\mid R_1,\ldots,R_s\rangle,

the $G$ -representation scheme $\operatorname{Hom}(\Gamma,G)$ is the closed $k$ -subscheme of $G^r$ defined by

R_j(g_1,\ldots,g_r)=e_G,\qquad 1\leq j\leq s.

For each commutative $k$ -algebra $B$ , its $B$ -points are $\operatorname{Hom}(\Gamma,G(B))$ . In particular, its $k$ -points are the set of group homomorphisms $\rho:\Gamma\to G(k)$ . If the representing coordinate ring is $C$ , the representation variety is the reduced affine scheme $\operatorname{Spec}(C/\sqrt{(0)})$ , with its classical $k$ -points. The full representation scheme can retain nilpotent structure. Here $\sqrt{(0)}$ consists of the nilpotent elements of $C$ .

Coordinates

After choosing generators $\gamma_1,\ldots,\gamma_r$ , a representation is identified with

(\rho(\gamma_1),\ldots,\rho(\gamma_r))\in G^r,

and the relations in $\Gamma$ cut out $\operatorname{Hom}(\Gamma,G)$ by polynomial equations.

Independence of presentation

Changing the finite presentation changes the embedding into a power of the group, but gives an isomorphic representing scheme: its functor sends a commutative $k$ -algebra $B$ to $\operatorname{Hom}(\Gamma,G(B))$ . The reduced variety records the classical reduced geometry; it need not represent the same functor on nonreduced test algebras.

Conjugation and moduli

The group $G$ acts on $\operatorname{Hom}(\Gamma,G)$ by

(g\cdot\rho)(\gamma)=g\rho(\gamma)g^{-1}.

Orbits are isomorphism classes of representations with a chosen target $G$ . The affine quotient of this action, under reductivity hypotheses, is the character variety. The representation variety together with this action records stabilizers and nonclosed orbits that the affine quotient can obscure.

Examples and scope

For the free group $F_r$ , there are no relations, so $\operatorname{Hom}(F_r,G)\cong G^r$ . For $k=\mathbb C$ and $\Gamma=\pi_1(M,x)$ finitely presented, its points include the holonomy representations of flat principal $G(\mathbb C)$ -connections.

References

Alexander Lubotzky and Andy R. Magid, Varieties of Representations of Finitely Generated Groups, Memoirs of the American Mathematical Society 58, no. 336, 1985. AMS record. Relevant: chapter 1, schemes and varieties of representations.
Adam S. Sikora, “Character Varieties,” Transactions of the American Mathematical Society 364 (2012), 5173–5208. DOI record. Relevant: §§5 and 11, representation varieties and conjugation quotients.