Algebraic independence in \(\mathrm{SL}(3, \mathbb C)\) character varieties of free groups. (English) Zbl 1209.14036
In this work are described local parameters (maximal independent coordinates) for the moduli spaces of \(\mathrm{SL}(3,\mathbb{C})\)-representations of free groups of arbitrary rank.
Let \(F_r\) be a rank \(r\) (non-abelian) free group. The author defines a complex \(m\)-dimensional representation as a homomorphism of \(F_r\) into \(\mathrm{SL}(m,\mathbb{C})\). The set of these representations \(R_r\), called the \(G=\mathrm{SL}(m,\mathbb{C})\)-representation variety of \(F_r\), is in bijection with \(\mathrm{SL}(m,\mathbb{C})^r\), so it is a smooth affine variety. The \(\mathrm{SL}(m,\mathbb{C})\)-character variety of \(F_r\) is the categorical quotient \(\mathfrak{X}_r=Spec(\mathbb{C}[R_r]^{\mathrm{SL}(m,\mathbb{C})})\) of \(R_r\) by \(\mathrm{SL}(m,\mathbb{C})\). In [J. Algebra 313, No. 2, 782–801 (2007; Zbl 1119.13004)], the author completely described \(\mathfrak{X}_2\) (with \(m=3\)). In this work there are some generalizations of such results to \(\mathfrak{X}_3\) and \(\mathfrak{X}_r\) (again with \(m=3\)).
Let \(\mathfrak{gl}(m,\mathbb{C})\) be the Lie algebra of \(GL(m,\mathbb{C})\), then \(R_r\equiv \mathrm{SL}(m,\mathbb{C})^r\subset \mathfrak{gl}(m,\mathbb{C})^r\) is an affine lift of \(\mathfrak{X}_r\), namely \(\Pi: \mathbb{C}[\mathfrak{gl}(m,\mathbb{C})^r]\rightarrow \mathbb{C}[R_r]\) and \(\Pi_G: \mathbb{C}[\mathfrak{gl}(m,\mathbb{C})^r//G]\rightarrow \mathbb{C}[R_r//G]\) are surjective.
Procesi described a set of generators \(I\) for \(\mathbb{C}[\mathfrak{gl}(m,\mathbb{C})^r//G]\). The number of these generators depends by an integer, called the degree of nilpotency, which is known only for \(m\leq4\). This result gives, in particular, a set of generators for \(\mathfrak{X}_r\). The main results of his paper [J. Algebra 320, No. 10, 3773–3810 (2008; Zbl 1157.14030)] is a description, when \(m=3\), of certain minimal sets of generators for \(\mathfrak{X}_r\) which are contained in \(I\).
A set of local parameters of \(\mathfrak{X}_r\) is a set of \(\dim\,\mathfrak{X}_r=m^2-1\) functions whose differentials give a basis of the cotangent space at a fixed point. Any set of \(m^2-1\) functions in the local coordinate ring of the point can be written in their terms.
The main result of this work describes local parameters (when \(m=3\)), which are contained in the set of minimal generator of the previous result in [Zbl 1157.14030]. This result is established by induction on \(r\), and the base of induction is proved in [Zbl 1119.13004].
There is another interesting result. The author says that a set of generators of \(\mathfrak{X}_r\) or of \(\mathfrak{gl}(m,\mathbb{C})^r//G\) is a set of Procesi generators if they are in set \(I\) defined by Procesi plus another condition. In [Zbl 1157.14030] the author shows that minimal Procesi generators of \(\mathfrak{gl}(m,\mathbb{C})^r//G\) projects to minimal Procesi generators of \(\mathfrak{X}_r\). In this work he shows that a maximal set of algebraically independent Procesi generators of \(\mathfrak{X}_r\) can be lifted to a set of algebraically independent Procesi generators of \(\mathfrak{gl}(m,\mathbb{C})^r//G\) which becomes maximal by adding \(m\) fixed (non-Procesi) generators.
Let \(F_r\) be a rank \(r\) (non-abelian) free group. The author defines a complex \(m\)-dimensional representation as a homomorphism of \(F_r\) into \(\mathrm{SL}(m,\mathbb{C})\). The set of these representations \(R_r\), called the \(G=\mathrm{SL}(m,\mathbb{C})\)-representation variety of \(F_r\), is in bijection with \(\mathrm{SL}(m,\mathbb{C})^r\), so it is a smooth affine variety. The \(\mathrm{SL}(m,\mathbb{C})\)-character variety of \(F_r\) is the categorical quotient \(\mathfrak{X}_r=Spec(\mathbb{C}[R_r]^{\mathrm{SL}(m,\mathbb{C})})\) of \(R_r\) by \(\mathrm{SL}(m,\mathbb{C})\). In [J. Algebra 313, No. 2, 782–801 (2007; Zbl 1119.13004)], the author completely described \(\mathfrak{X}_2\) (with \(m=3\)). In this work there are some generalizations of such results to \(\mathfrak{X}_3\) and \(\mathfrak{X}_r\) (again with \(m=3\)).
Let \(\mathfrak{gl}(m,\mathbb{C})\) be the Lie algebra of \(GL(m,\mathbb{C})\), then \(R_r\equiv \mathrm{SL}(m,\mathbb{C})^r\subset \mathfrak{gl}(m,\mathbb{C})^r\) is an affine lift of \(\mathfrak{X}_r\), namely \(\Pi: \mathbb{C}[\mathfrak{gl}(m,\mathbb{C})^r]\rightarrow \mathbb{C}[R_r]\) and \(\Pi_G: \mathbb{C}[\mathfrak{gl}(m,\mathbb{C})^r//G]\rightarrow \mathbb{C}[R_r//G]\) are surjective.
Procesi described a set of generators \(I\) for \(\mathbb{C}[\mathfrak{gl}(m,\mathbb{C})^r//G]\). The number of these generators depends by an integer, called the degree of nilpotency, which is known only for \(m\leq4\). This result gives, in particular, a set of generators for \(\mathfrak{X}_r\). The main results of his paper [J. Algebra 320, No. 10, 3773–3810 (2008; Zbl 1157.14030)] is a description, when \(m=3\), of certain minimal sets of generators for \(\mathfrak{X}_r\) which are contained in \(I\).
A set of local parameters of \(\mathfrak{X}_r\) is a set of \(\dim\,\mathfrak{X}_r=m^2-1\) functions whose differentials give a basis of the cotangent space at a fixed point. Any set of \(m^2-1\) functions in the local coordinate ring of the point can be written in their terms.
The main result of this work describes local parameters (when \(m=3\)), which are contained in the set of minimal generator of the previous result in [Zbl 1157.14030]. This result is established by induction on \(r\), and the base of induction is proved in [Zbl 1119.13004].
There is another interesting result. The author says that a set of generators of \(\mathfrak{X}_r\) or of \(\mathfrak{gl}(m,\mathbb{C})^r//G\) is a set of Procesi generators if they are in set \(I\) defined by Procesi plus another condition. In [Zbl 1157.14030] the author shows that minimal Procesi generators of \(\mathfrak{gl}(m,\mathbb{C})^r//G\) projects to minimal Procesi generators of \(\mathfrak{X}_r\). In this work he shows that a maximal set of algebraically independent Procesi generators of \(\mathfrak{X}_r\) can be lifted to a set of algebraically independent Procesi generators of \(\mathfrak{gl}(m,\mathbb{C})^r//G\) which becomes maximal by adding \(m\) fixed (non-Procesi) generators.
Reviewer: Alessandro Ruzzi (Roma)
MSC:
| 14L24 | Geometric invariant theory |
| 20G05 | Representation theory for linear algebraic groups |
| 14L30 | Group actions on varieties or schemes (quotients) |
| 13A50 | Actions of groups on commutative rings; invariant theory |
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