2. Cyclic groups in $\textbf{G}_{\textbf{2}}(\mathbb C)$

Elements of finite order in the connected Lie group $G=\mathrm{Aut}\!\left({\mathfrak{g}} \right)^0$ for a simple complex Lie algebra $\mathfrak g$ are classified using the geometry of affine Weyl groups. The very short explanation is that any diagonalisable element of G is conjugate to an element of a Cartan subgroup T, and two elements in a T are conjugate in G if and only if they are conjugate by the Weyl group $N_G(T)/T$ . If an element of G has also finite order, it is a rational point in a (compact) maximal torus in G. A maximal torus is isomorphic, through the exponential map, with a real Cartan subalgebra (CSA) up to translations by the co-weight lattice. Thus, the conjugacy classes of elements of finite order in G are identified with rational linear combinations of simple co-weights in the CSA, modulo the action of the Weyl group and the co-weight lattice. This latter group is known as the extended affine Weyl group.

One of the great insights of Cartan was to work modulo the affine Weyl group instead (the semidirect product of the Weyl group and the co-root lattice, rather than the co-weight lattice), which has a simplex as fundamental domain in the CSA, and handle the remaining symmetry using Dynkin diagrams. Determining the vertices of this simplex then yields

Theorem 2.1 (Cartan) Let $\mathfrak{g}$ be a simple complex Lie algebra and $\{\alpha_1,\ldots,\alpha_\ell\}$ a base for its root system with highest root $\sum_{i=1}^\ell a_i\alpha_i$ . Set $a_0=1$ .

Elements of order n in $\mathrm{Aut}\!\left({\mathfrak g} \right)^0$ , up to conjugation, are in one-to-one correspondence with sequences of nonnegative relative prime integers $\{s_0,\ldots,s_\ell\}$ such that

\begin{equation*}n=\sum_{i=0}^\ell a_i s_i,\end{equation*}

up to symmetry of the affine Dynkin diagram. The conjugacy class associated with $\{s_0,\ldots,s_\ell\}$ is represented by the automorphism sending the Chevalley generator $E_j$ of the Lie algebra to $\zeta^{s_j} E_j$ , where $j=0,\ldots,\ell$ and $\zeta=\exp\!\frac{2\pi i}{n}$ .

The sequence $\{s_0,\ldots,s_\ell\}$ lists the coordinates for the class of automorphisms. For the full story, we refer to the original work of Cartan [Reference Cartan2] and Kac (who extended the result to all automorphisms of finite order) [Reference Kac13, Reference Kac14] and the enlightening treatment of Reeder [Reference Reeder21]. See Bourbaki [Reference Bourbaki1] for a thorough study of (extended) affine Weyl groups.

We present all conjugacy classes of automorphisms of order $\le 5$ in Table 1.

Figure 1. Order 3 elements in $G_2(\mathbb C)$ diagonalised by a Cartan Weyl basis. Thick lines indicate simple roots. The number s at a root indicates an eigenvalue $e^{\frac{2\pi i s}{3}}$ at the root space.

Table 1. Elements of $G_2(\mathbb C)$ of order $\le 5$ (with $\phi^{\pm}=\frac{1\pm\sqrt{5}}{2}$ ).

Two more lemmas are needed in preparation for the next section.

Proof. An element g of order 3 in $\mathrm{PSL}(2,\mathbb C)$ is conjugate to $\pm\mathrm{diag}\big(e^\frac{2\pi i}{6},e^{-\frac{2\pi i}{6}}\big)=\pm\ e^{\frac{2\pi i}{6}H}$ where $H=\mathrm{diag}(1,-1)$ belongs to the Lie algebra $\mathfrak{sl}({2},\mathbb{C})$ of $\mathrm{PSL}(2,\mathbb C)$ . Suppose g is mapped to the conjugacy class of 10—0 in $G_2(\mathbb C)$ . Then, H acts on the simple root spaces with weights 20. By linear extension over the root system, we find all weights of $\mathfrak g_2(\mathbb C)$ as representation of $\mathfrak{sl}({2},\mathbb{C})$ , and see weight 6 has multiplicity two and weight 4 has multiplicity one. Such a representation of $\mathfrak{sl}({2},\mathbb{C})$ does not exist. Hence, g is mapped to the other conjugacy class of order 3 elements in $G_2(\mathbb C)$ .

Lemma 2.4. Let g be a diagonalisable element of $G_2(\mathbb C)$ with trace $\chi_V(g)$ and denote the trace of its action on $\mathfrak g_2(\mathbb C)$ by $\chi_{\mathfrak g_2(\mathbb C)}(g)$ . Then

\begin{equation*}\chi_{\mathfrak g_2(\mathbb C)}(g)=\frac{\chi_V(g)^2-\chi_V(g^2)-2\chi_V(g)}{2}.\end{equation*}

Proof. The Lie group $G_2(\mathbb C)$ is realised as a Lie subgroup of $\mathrm{SO}(V)$ . This turns $\mathfrak{so}(V)$ into a 21-dimensional representation of $\mathfrak g_2(\mathbb C)$ , which has $\mathfrak g_2(\mathbb C)$ as 14-dimensional subrepresentation. By complete reducibility, there must be a 7-dimensional representation U such that $\mathfrak{so}(V)=\mathfrak g_2(\mathbb C)\oplus U$ as $\mathfrak g_2(\mathbb C)$ representation. It follows from the classification of representations of $\mathfrak g_2(\mathbb C)$ that U is either trivial or $U=V$ . If U is trivial, then $\mathfrak g_2(\mathbb C)$ is a nontrivial ideal in $\mathfrak{so}(V)$ , contradicting the simplicity of the latter. Hence, $\mathfrak{so}(V)=\mathfrak g_2(\mathbb C)\oplus V.$

If we consider the trace of g on the left- and right-hand side and observe that the trace of g on $\mathfrak{so}(V)$ is related to $\chi_V(g)$ by $\chi_{\mathfrak{so}(V)}(g)=(\chi_V(g)^2-\chi_V(g^2))/2,$ we obtain the desired result.

3. $\mathsf{TOI}$ groups in $\textbf{G}_{\textbf{2}}(\mathbb C)$

The $\mathsf{T}\mathsf{O}\mathsf{I}$ groups have presentation

\begin{equation*}\Gamma=\langle\gamma_a,\gamma_b,\gamma_c\,|\,\gamma_a^n, \gamma_b^3,\gamma_c^2,\gamma_a\gamma_b\gamma_c\rangle.\end{equation*}

Taking $n=3, 4$ or 5 results in $\mathsf{T}, \mathsf{O}$ or $\mathsf{I},$ respectively. Their character tables are provided in Table 2.

Table 2. Irreducible characters of $\mathsf{T}$ , with $\zeta=e^{\frac{2\pi i}{3}}$ , of $\mathsf{O}$ and $\mathsf{I}$ , with $\phi^{\pm}=\frac{1\pm\sqrt{5}}{2}$ .

Proof. In Table 1, we see that $G_2(\mathbb C)$ only has one class of involutions, which has trace $-1$ , and two classes of elements of order 3, with traces 1 (at the class of 01—1) and $-2$ . The claim follows by observing that the $\mathsf{T}\mathsf{O}\mathsf{I}$ groups do not have a seven dimensional character with values $-1$ and $-2$ at the elements of order 2 and 3, respectively.

In Table 3, we list all 7-dimensional characters of $\mathsf{T}\mathsf{O}\mathsf{I}$ groups with value $-1$ and 1 at elements of order 2 and 3, respectively, and irrational value at elements of order 5. Due to Lemma 3.1 and Table 1, we know that these conditions are necessary for the character of a monomorphism of a $\mathsf{T}\mathsf{O}\mathsf{I}$ group into $G_2(\mathbb{C})$ .

Table 3. Characters of $\mathsf{T}\mathsf{O}\mathsf{I}$ groups in $G_2(\mathbb C)$ .

One way to construct embeddings of polyhedral groups is through a composition

\begin{equation*}\Gamma\hookrightarrow\mathrm{PSL}(2,\mathbb C)\hookrightarrow G_2(\mathbb C).\end{equation*}

There is one embedding $\mathsf{T}\hookrightarrow\mathrm{PSL}(2,\mathbb C)$ , two embeddings $\mathsf{O}\hookrightarrow\mathrm{PSL}(2,\mathbb C)$ and also two embeddings $\mathsf{I}\hookrightarrow\mathrm{PSL}(2,\mathbb C)$ , up to conjugation. Moreover, there are precisely two embeddings $\mathrm{PSL}(2,\mathbb C)\hookrightarrow G_2(\mathbb C)$ up to conjugation [Reference Collingwood and McGovern5]. They are represented by the weighted Dynkin diagrams 22 and 20. The linear extension of these weights to the root system yields the weights of $\mathfrak g_2(\mathbb C)$ as a representation of $\mathfrak{sl}({2},\mathbb{C})$ .

The characters of the various compositions realise all options from Table 3. From character theory, we know that this classifies the embeddings in $G_2(\mathbb C)$ up to conjugation in $\mathrm{GL}(V)$ . Thanks to the work of Larsen and Griess [Reference Larsen18, Reference Griess11], we can conclude that these conjugation classes correspond to the conjugation classes in $G_2(\mathbb C)$ . Thus we arrive at our main results.

The part of this theorem concerning the icosahedral group can also be found in [Reference Griess12, Section 4] where a different proof is given.

There is an automorphism of $\mathsf{I}$ sending $\gamma_a$ to $\gamma_a^2$ . The conjugacy classes of monomorphisms $\mathsf{I}\hookrightarrow G_2(\mathbb C)$ with the first two and last two icosahedral characters of Table 3 are interchanged when precomposed with this automorphism. Thus, we see that there are only two conjugacy classes of images of monomorphisms $\mathsf{I}\hookrightarrow G_2(\mathbb C)$ , and recover [Reference Frey9, Theorem 4.11].

This result provides a practical construction, since monomorphisms $\Gamma\hookrightarrow\mathrm{PSL}(2,\mathbb C)$ and $\mathrm{PSL}(2,\mathbb C)\hookrightarrow G_2(\mathbb C)$ can be found in the literature.

Proof. Using Lemma 2.4 and Theorem 3.2, we can compute all characters of $\Gamma$ -actions on $\mathfrak g_2(\mathbb C)$ and observe that none of them has a trivial component.

We have not classified embeddings of the dihedral groups in $G_2(\mathbb C)$ . The following example shows why this task cannot be completed with the same approach.

Example 3.5 We construct a monomorphism $\mathsf{D}_{3}\hookrightarrow G_2(\mathbb C)$ which shows that Lemma 3.1, Theorems 3.3 and 3.4 all fail for dihedral groups. We do so with a concrete model L of $\mathfrak g_2(\mathbb C)$ in $\mathfrak{gl}(7,\mathbb C)$ . Let $x=(x_1, x_2, x_3)^t$ and let

\begin{equation*}L=\left\{\left(\begin{array}{c@{\quad}c@{\quad}c}0&-\sqrt{2}y^t&-\sqrt{2}x^t\\ \\[-8pt]\sqrt{2}x&a&l_y\\ \\[-8pt]\sqrt{2}y&l_x&-a^t\end{array}\right)\,|\,a\in\mathfrak{sl}({3},\mathbb{C}),\,x,y\in\mathbb{C}^3\right\},\quad l_{x}=\left(\begin{array}{c@{\quad}c@{\quad}c}0&-x_3&x_2\\ \\[-8pt]x_3&0&-x_1\\ \\[-8pt]-x_2&x_1&0\end{array}\right).\end{equation*}

See [Reference Draper Fontanals6] for a proof that (a conjugate of) L is indeed a simple Lie subalgebra of $\mathfrak{gl}(7,\mathbb C)$ of type $G_2$ , so that we can identify L with $\mathfrak g_2(\mathbb C)$ and $\mathrm{Aut}\!\left({L} \right)$ with $G_2(\mathbb C)$ .

Conjugation with the diagonal matrix $\mathrm{diag}(1,\zeta,\zeta,\zeta,\zeta^2,\zeta^2,\zeta^2)$ , $\zeta=e^{\frac{2\pi i}{3}}$ , defines an automorphism of $\mathfrak{gl}(7,\mathbb C)$ which preserves L. Let r be its restriction to L. Then, r is an order 3 automorphism of L. The elements of L fixed by r form a Lie subalgebra isomorphic to $\mathfrak{sl}({3},\mathbb{C})$ ; hence, r belongs to the conjugacy class of 10—0 in $G_2(\mathbb C)$ (cf. Figure 1).

The map $M\mapsto -M^t$ also defines an automorphism of $\mathfrak{gl}(7,\mathbb C)$ which preserves L. Let s be its restriction to L, an automorphism of order 2. Then, $rs=sr^{-1}$ ; hence, r and s generate a dihedral group of order 6 in $G_2(\mathbb C)$ .

Contrary to the case of the $\mathsf{T}\mathsf{O}\mathsf{I}$ groups, the map $\mathsf{D}_{3}\hookrightarrow G_2(\mathbb C)$ we have constructed does not factor through $\mathrm{PSL}(2,\mathbb C)$ because of Lemma 2.3 and the fact that its image has nontrivial intersection with the conjugacy class of 10—0. Moreover, we compute that the elements fixed by $\mathsf{D}_{3}$ form a subalgebra of $\mathfrak g_2(\mathbb C)$ isomorphic to $\mathfrak{sl}({2},\mathbb{C})$ .