Character variety

In mathematics , character varieties are an important tool in group theory , topology, and geometry .

definition

Let it be a finitely generated group , a Lie group and ${\displaystyle \Gamma }$ ${\displaystyle G}$

${\displaystyle Hom(\Gamma ,G)}$

the representation variety . The group acts on by conjugation , d. H. for and is ${\displaystyle G}$ ${\displaystyle Hom(\Gamma ,G)}$ ${\displaystyle g\in G,\rho \in Hom(\Gamma ,G)}$ ${\displaystyle \gamma \in \Gamma }$

${\displaystyle g.\rho (\gamma ):=g\rho (\gamma )g^{-1}}$.

The quotient space is generally not an algebraic set . One therefore uses the geometric invariant theory and considers the GIT quotient ${\displaystyle Hom(\Gamma ,G)/G}$

${\displaystyle X(\Gamma ,G):=Hom(\Gamma ,G)//G}$.

Its coordinate ring is isomorphic to by the definition of the GIT quotient

${\displaystyle \mathbb {C} \left[Hom(\Gamma ,G)\right]^{G}\subset \mathbb {C} \left[Hom(\Gamma ,G)\right]}$,

the subring of the functions invariant under conjugation with elements from the coordinate ring . ${\displaystyle G}$ ${\displaystyle \mathbb {C} \left[Hom(\Gamma ,G)\right]}$

Coordinate ring

If there is a reductive group , then the coordinate ring is finitely generated ( Nagata theorem ), so the GIT quotient is a (not necessarily irreducible ) algebraic variety . ${\displaystyle G}$ ${\displaystyle \mathbb {C} \left[Hom(\Gamma ,G)\right]^{G}}$ ${\displaystyle X(\Gamma ,G)}$

For becomes the coordinate ring ${\displaystyle G=SL(n,\mathbb {C} )}$

${\displaystyle \mathbb {C} \left[Hom(\Gamma ,SL(n,\mathbb {C} ))\right]^{SL(n,\mathbb {C} )}}$

of the track functions

${\displaystyle I_{\rho }\colon \rho \to Spur(\rho (\gamma )}$

for generated, the points of the character variety correspond to the characters of , which also explains the naming. ${\displaystyle \gamma \in \Gamma }$ ${\displaystyle \Gamma }$

Explicit description

Mark with

${\displaystyle Hom(\Gamma ,G)^{*}\subset Hom(\Gamma ,G)}$

the union of all closed orbits of the effect. This is a closed subset and the quotient space ${\displaystyle G}$

${\displaystyle X(\Gamma ,G):=Hom(\Gamma ,G)^{*}/G}$

is a Hausdorff room . It is referred to as a character variety , although generally it does not need to be an algebraic variety . In the case of complex reductive groups, this definition agrees with the above definition as a GIT quotient.

For an orbit of the effect is completed if and only if the corresponding representations are semi-simple . It is well known that semi-simple representations are conjugated if and only if they have identical characters. ${\displaystyle G=SL(n,\mathbb {C} )}$ ${\displaystyle G}$

Basic properties

• If is compact , then is and . ${\displaystyle G}$ ${\displaystyle Hom^{*}(\Gamma ,G)=Hom(\Gamma ,G)}$ ${\displaystyle X(\Gamma ,G)=Hom(\Gamma ,G)/G}$
• If is a real algebraic group , then is a semi- algebraic set . ${\displaystyle G}$ ${\displaystyle X(\Gamma ,G)}$
• If is a complex reductive group , then is a (not necessarily irreducible) algebraic variety. ${\displaystyle G}$ ${\displaystyle X(\Gamma ,G)}$

Examples

• There is no variety for the group of integers . ${\displaystyle \mathbb {Z} }$ ${\displaystyle X(\mathbb {Z} ,SU_{2})\cong \left[-2,2\right]}$
• Fricke-Vogt's theorem : For the free group with two producers is ${\displaystyle F_{2}}$ ${\displaystyle X,Y}$

${\displaystyle X(F_{2},SL_{2}\mathbb {C} )\cong \mathbb {C} ^{3}}$

parameterized by the tracks . ${\displaystyle Tr(\rho (X)),Tr(\rho (Y)),Tr(\rho (XY))}$

• ${\displaystyle X(\mathbb {Z} ,SL_{3}\mathbb {C} )}$is isomorphic to , the isomorphism maps the equivalence class of a representation to . ${\displaystyle \mathbb {C} ^{2}}$ ${\displaystyle \rho }$ ${\displaystyle (Tr(\rho (1)),Tr(\rho (1)^{-1})\in \mathbb {C} ^{2}}$
• ${\displaystyle X(F_{2},SL_{3}\mathbb {C} )}$is a branched 2-fold consideration of , it is parameterized by the tracks and , whereby it is related to the eight other parameters by a quadratic polynomial. ${\displaystyle \mathbb {C} ^{8}}$ ${\displaystyle Tr(\rho (X^{\pm 1})),Tr(\rho (Y^{\pm 1})),Tr(\rho (X^{\pm 1}Y^{\pm 1}))}$ ${\displaystyle Tr(\rho (\left[X,Y\right]))}$ ${\displaystyle Tr(\rho (\left[X,Y\right]))}$
• For the node group of a hyperbolic node , the component of the hyperbolic monodromy containing the character is a complex curve , i.e. H. complex 1-dimensional. ${\displaystyle \Gamma }$ ${\displaystyle X(\Gamma ,SL_{2}\mathbb {C} )}$
• The node group of the Achknot consists of two components: one contains the hyperbolic monodromy, the other only consists of reducible representations . ${\displaystyle X(\Gamma ,SL_{2}\mathbb {C} )}$

literature

• Alexander Lubotzky , Andy Magid : Varieties of representations of finitely generated groups. Mem. Amer. Math. Soc. 58 (1985), no.336
• Igor Dolgachev : Lectures on invariant theory. London Mathematical Society Lecture Note Series, 296. Cambridge University Press, Cambridge, 2003. ISBN 0-521-52548-9
• Adam Sikora: SL _n -character varieties as spaces of graphs. Trans. Amer. Math. Soc. 353 (2001), no. 7, 2773-2804. online (pdf)
• Adam Sikora: Character varieties. Trans. Amer. Math. Soc. 364 (2012), no. 10, 5173-5208. online (pdf)

Individual evidence

1. ^ Claudio Procesi : The invariant theory of n × n matrices. Advances in Math. 19 (1976) no. 3, 306-381. online (PDF)
2. ^ Richardson, Slodowy: Minimum vectors for real reductive algebraic groups. J. London Math. Soc. (2) 42 (1990) no. 3, 409-429. online (PDF)
3. ↑ Sean Lawton: Generators, relations and symmetries in pairs of 3 × 3 unimodular matrices. J. Algebra 313 (2007), no. 2, 782-801. online (pdf)