[Paper Review] G-Character varieties for G=SO(n,C) and other not simply connected groups

Adam S. Sikora|arXiv (Cornell University)|Mar 28, 2013

Advanced Algebra and Geometry7 references3 citations

TL;DR

This paper investigates $\mathrm{SO}(n,\mathbb{C})$-character varieties for non-simply connected groups, establishing relations between $G$- and $G/H$-character varieties for finite central subgroups $H$. It provides finite generating sets for $\mathbb{C}[X_{SO(4,\mathbb{C})}(F_2)]$ and proves that the full trace algebra $\mathcal{FT}_{SO(4,\mathbb{C})}(F_2)$ is strictly smaller than the full coordinate ring, resolving a key question about trace function generation in $SO(2n,\mathbb{C})$-varieties.

ABSTRACT

We describe the relation between G-character varieties, $X_G(Γ)$, and $G/H$-character varieties, where $H$ is a finite, central subgroup of $G.$ In particular, we find finite generating sets of coordinate rings $C[X_{G/H}(Γ)]$ for classical groups $G$ and $H$ as above. Using this approach we find an explicit description of $C[X_{SO(4,C)}(F_2)]$ for the free group on two generators, $F_2.$ In the second part of the paper, we prove several properties of SO(2n,C)-character varieties. This is a particularly interesting class of character varieties because unlike for all other classical groups G, the coordinate rings $C[X_{G}(Γ)]$ are generally not generated by trace functions $τ_γ$, for $γ\in Γ$, for G=SO(2n,C). In fact, we prove that the coordinate ring $C[X_{SO(2n,C)}(Γ)]$ is not even generated by "generalized trace functions," $τ_{γ,V},$ for all $γ\in Γ$ and all representations $V$ of $SO(2n,C)$ for $n=2$ and groups $Γ$ of corank $\geq 2$.

Motivation & Objective

• To understand the structure of $G$-character varieties for non-simply connected groups like $SO(n,\mathbb{C})$, which are not fully captured by trace functions.
• To establish a general relation between $G$-character varieties and $G/H$-character varieties when $H$ is a finite central subgroup of $G$.
• To explicitly describe the coordinate ring $\mathbb{C}[X_{SO(4,\mathbb{C})}(F_2)]$ for the free group on two generators.
• To resolve whether the full trace algebra $\mathcal{FT}_{SO(2n,\mathbb{C})}(\Gamma)$ generates the full coordinate ring $\mathbb{C}[X_{SO(2n,\mathbb{C})}(\Gamma)]$ for $n=2$ and $\Gamma$ of corank $\geq 2$.

Proposed method

• Use the isomorphism $SO(4,\mathbb{C}) \cong (SL(2,\mathbb{C}) \times SL(2,\mathbb{C}))/\mathbb{Z}/2$ to lift character varieties from $SL(2,\mathbb{C})$ to $SO(4,\mathbb{C})$ via quotient constructions.
• Construct finite generating sets for $\mathbb{C}[X_{G/H}(\Gamma)]$ using the structure of $G$-representations and invariants under the action of $H$, particularly for classical groups.
• Define generalized trace functions $\tau_{\gamma,V}$ and analyze their role in generating the coordinate ring $\mathbb{C}[X_G(\Gamma)]$.
• Apply a grading argument on the algebra of traces to prove linear independence of key generators in $\mathbb{C}[X_{SO(4,\mathbb{C})}(F_2)]$.
• Use the epimorphism $\Gamma \to F_2$ to lift results from $F_2$ to higher corank groups, showing that the trace algebra extension is proper.
• Leverage the $\mathbb{Z}/2 \times \mathbb{Z}/2$-equivariant structure on $X_{SL(2,\mathbb{C})}(F_2) \times X_{SL(2,\mathbb{C})}(F_2)$ to decompose the coordinate ring and identify generators of the $SO(4,\mathbb{C})$-character variety.

Experimental results

Research questions

• RQ1 Does the full trace algebra $\mathcal{FT}_{SO(2n,\mathbb{C})}(\Gamma)$ generate the full coordinate ring $\mathbb{C}[X_{SO(2n,\mathbb{C})}(\Gamma)]$ for $n=2$ and $\Gamma$ of corank $\geq 2$?
• RQ2 Can the coordinate ring $\mathbb{C}[X_{SO(4,\mathbb{C})}(F_2)]$ be generated by generalized trace functions $\tau_{\gamma,V}$ for all $\gamma \in F_2$ and all $SO(4,\mathbb{C})$-representations $V$?
• RQ3 What is an explicit finite generating set for $\mathbb{C}[X_{SO(4,\mathbb{C})}(F_2)]$?
• RQ4 Are the generators of $\mathbb{C}[X_{SO(4,\mathbb{C})}(F_2)]$ as a $\mathcal{T}_{SO(4,\mathbb{C})}(F_2)$-module linearly independent over the trace algebra?
• RQ5 Is the extension $\mathcal{FT}_{SO(4,\mathbb{C})}(F_2) \subset \mathbb{C}[X_{SO(4,\mathbb{C})}(F_2)]$ proper?

Key findings

• The coordinate ring $\mathbb{C}[X_{SO(4,\mathbb{C})}(F_2)]$ is a $\mathcal{T}_{SO(4,\mathbb{C})}(F_2)$-module generated by $1$ and the functions $Q_4(\gamma_i, \gamma_j)$ for $1 \leq i \leq j \leq 2$, along with four additional $Q_4$-functions involving products and inverses.
• The full trace algebra $\mathcal{FT}_{SO(4,\mathbb{C})}(F_2)$ is strictly smaller than $\mathbb{C}[X_{SO(4,\mathbb{C})}(F_2)]$, as shown by the non-redundancy of the generators $Q_4(\gamma_1, \gamma_2)$, $Q_4(\gamma_1\gamma_2^{-1}, \gamma_2)$, and $Q_4(\gamma_2\gamma_1^{-1}, \gamma_1)$.
• The generators of $\mathbb{C}[X_{SO(4,\mathbb{C})}(F_2)]$ as a $\mathcal{FT}_{SO(4,\mathbb{C})}(F_2)$-module are algebraically independent in degree 3, proving that no generator lies in the $\mathcal{FT}_{SO(4,\mathbb{C})}(F_2)$-span of the others.
• For $\Gamma$ of corank $\geq 2$, the extension $\mathcal{FT}_{SO(4,\mathbb{C})}(\Gamma) \subset \mathbb{C}[X_{SO(4,\mathbb{C})}(\Gamma)]$ is proper, as it is induced from the $F_2$ case via epimorphisms.
• The coordinate ring $\mathbb{C}[X_{SO(4,\mathbb{C})}(F_2)]$ is not generated by trace functions $\tau_\gamma$ alone, nor by generalized trace functions $\tau_{\gamma,V}$, for $SO(4,\mathbb{C})$, unlike for other classical groups.
• The paper provides a finite generating set for $\mathbb{C}[X_{SO(4,\mathbb{C})}(F_2)]$, explicitly listing the generators in terms of $Q_4$-functions and trace invariants.