Representation variety - Imported Knowlpedia Content

Definition

Let $k$ be an algebraically closed field of characteristic zero, $\Gamma$ a finitely generated group, and $G$ an affine algebraic group over $k$ . The $G$ -representation variety of $\Gamma$ is

\operatorname{Hom}(\Gamma,G),

the set of group homomorphisms $\rho:\Gamma\to G$ , equipped with its natural affine algebraic-set structure. 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.

Presentation model

For a finite presentation

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

the representation variety is the common zero locus of

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

inside $G^r$ . Changing the finite presentation changes this embedding but not the represented affine functor. The foundational construction and its scheme-theoretic refinements are treated by Lubotzky–Magid, chapter 1.

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 itself retains stabilizers, nonclosed orbits, and singularities that a quotient can obscure.

Examples and scope

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