The Axes of a Majorana Representation of $A_{12}$

Abstract

This paper contributes to an axiomatic approach (which goes under the name of Majorana theory) to the largest sporadic simple group ${\mathbf {M}}$, known as the Monster, and to its $196,884$-dimensional algebra $V_{\mathbf {M}}$. We study the axiomatic version $V$ of the subalgebra $V_A$ of $V_{\mathbf {M}}$ generated by the Majorana axes corresponding to the $2A$-involutions of an alternating subgroup $A_{12}$ of $\mathbf {M}$ whose centraliser is isomorphic to $A_5$. Although the dimension of $V^{(2A)}_A$, the linear span of the generating Majorana axes of $V_A$, has been known for a while to be $3498$, the dimension of $V_A$ itself remains unknown. For each $3 \le N \le 5$, let $V^{(NA)}$ be the linear span of the $NA$-axes of $V$. In this paper, we examine the space

$$\begin{aligned} V^{\circ } = \left\langle V^{(NA)} : 2 \le N \le 5 \right\rangle . \end{aligned}$$

We prove that $V^{(2A)}$ contains the $3A$-axes corresponding to $3$-cycles in $A_{12}$, but it does not contain any of those corresponding to products of two $3$-cycles. By considering a $21$-dimensional space determined by a $2B$-involution, we also show that no $4A$-axis of $V$ belongs to $V^{(2A)}$. An argument due to Á. Seress enables us to show further that any $5A$-axis of $V$ is linearly expressible in terms of Majorana axes and $3A$-axes of $V$. When $V=V_A$, our results, enhanced by information about the characters of ${\mathbf {M}}$, allow us to deduce that

$$\begin{aligned} \left\langle V^{(2A)}_A, V^{(3A)}_A \right\rangle = \left\langle V^{(2A)}_A, V^{(4A)}_A \right\rangle = V^{\circ }_A, \end{aligned}$$

and that $V^{\circ }_A$ is the the direct sum of $V_A^{(2A)}$ and a $462$-dimensional irreducible $\mathbb {R}A$-module; hence, we conclude that $\dim (V_A^{\circ }) = 3960$. The whole algebra $V_A$ may still contain $V_A^{\circ }$ properly, but its codimension is bounded by $1191$.

A Castillo-Ramirez—Funded by the Universidad de Guadalajara and an Imperial College International Scholarship.

Appendix: Orthogonality Relations

Let $\left( G,T,V,\left( ,\right) ,\cdot ,\varphi ,\psi \right)$ be a Majorana representation. It follows directly from axiom M1 that eigenvectors of the adjoint transformation of a Majorana axis corresponding to distinct eigenvalues are orthogonal. This fact may be used to obtain several inner product relations involving distinct $3A$- and $4A$-axes. This appendix is devoted to state some of these relations.

Let $t,g,q,s,h\in T$ and suppose that the Norton-Sakuma algebras $\left\langle \left\langle a_{t},a_{g}\right\rangle \right\rangle$, $\left\langle \left\langle a_{t},a_{q}\right\rangle \right\rangle$, $\left\langle \left\langle a_{t},a_{s}\right\rangle \right\rangle$ and $\left\langle \left\langle a_{t},a_{h}\right\rangle \right\rangle$ have types $2A$, $2B$, $3A$ and $4A$, respectively. Let $\rho _{1}:=ts$ and $\rho _{2}:=th$, and define $s_{i}:=t\rho _{1}^{i}$ and $h_{i}:=t\rho _{2}^{i}$, for $i\in \mathbb {Z}$. Table 2 describes some of the eigenvectors of the adjoint transformation of $a_{t}$ contained in the previous algebras. These eigenvectors were obtained in [5].

Table 2 Eigenvectors of Norton-Sakuma algebras

We summarise in Table 3 some of the relations obtained using the orthogonality between the eigenvectors of Table 2. In Table 3, the expression of each entry equals to the inner product between the axes labeling the row and column.

Table 3 Inner product relations using Norton-Sakuma algebras

Table 4 Eigenvectors of $a_{(ij)}$ in the $S_{4}$-algebra of shape $(2B,3A)$

The following lemmas state further useful relations.

Lemma A.1

Consider the algebras $\left\langle \left\langle a_{t},a_{s}\right\rangle \right\rangle$ and $\left\langle \left\langle a_{t},a_{h}\right\rangle \right\rangle$ of types $3A$ and $4A$, respectively, as defined above. Then,

$$\begin{aligned} \left( a_{h_{2}},u_{\rho _{1}}\right)&=\frac{2^{6}}{3^{2}5}\left( a_{h_{2}},a_{s}\right) , \\ \left( v_{\rho _{2}},u_{\rho _{1}}\right)&=\frac{11}{2^{2}3^{3}}-2\left( a_{h},u_{\rho _{1}}\right) +\frac{2^{6}}{3^{3}}\left[ \left( a_{h},a_{s}+a_{s_{-1}}\right) +\frac{1}{5}\left( a_{h_{2}},a_{s}\right) \right] \!, \\ \left( v_{\rho _{2}},a_{s}\right)&=\frac{7}{2^{8}3}+\frac{3^{2}5}{2^{5}}\left( a_{h},u_{\rho _{1}}\right) +\frac{1}{3}\left[ \left( a_{h_{2}},a_{s}\right) -\left( a_{h},a_{s}+a_{s_{-1}}\right) \right] \!. \end{aligned}$$

Lemma A.2

Let $\left\langle \left\langle a_{t},a_{h}\right\rangle \right\rangle$ be the algebra of type $4A$ defined above, and suppose that $\left\langle \left\langle a_{t},a_{h^{\prime }}\right\rangle \right\rangle$ is also an algebra of type $4A$ with $\rho _{2}^{\prime }:=th^{\prime }$ and $h_{i}^{\prime }:=t\left( \rho _{2}^{\prime }\right) ^{i}$, $i\in \mathbb {Z}$. Suppose that $h_{2}=h_{2}^{\prime }$. Then

$$\begin{aligned} \left( v_{\rho _{2}},v_{\rho _{2}^{\prime }}\right) =\frac{1}{3}+\frac{2^{2}}{3}\left( v_{\rho _{2}},a_{h^{\prime }}\right) -2^{2}\left( a_{h},v_{\rho _{2}^{\prime }}\right) +\frac{2^{3}}{3}\left( a_{h}+a_{h_{-1}},a_{h^{\prime }}\right) \!. \end{aligned}$$

For future reference, Table 4 contains some eigenvectors of $a_{(i,j)}$ in the $S_4$-algebra of shape $(2B,3A)$, using the notation of the basis introduced in Sect. 5 of [5].

Lemma A.3

Let $t , (ij), (kl), (ik) \in T$. Suppose that

$$\begin{aligned} \left\langle \left\langle a_{t},a_{(ij)}\right\rangle \right\rangle \textit{ and } \left\langle \left\langle a_{(ij)},a_{(kl)},a_{(ik)}\right\rangle \right\rangle \end{aligned}$$

are a Norton-Sakuma algebra of type $2A$ and an $S_{4}$-algebra of shape $( 2B,3A )$, respectively. Then, we have that

$$\begin{aligned} ( a_{t}-a_{t \cdot (ij)},v_j ) = - \frac{2}{3}\left( a_{t} - a_{t \cdot (ij)},a_{(ik)}+a_{(il)}\right) +\frac{3\cdot 5}{2^{4}}\left( a_{t}-a_{t \cdot (ij)},u_i\right) , \end{aligned}$$

where $v_j$ is the $4A$-axis corresponding to $(i,k,j,l)$, and $u_i$ is the $3A$-axis corresponding to $(j,k,l)$.

Proof

The result follows by the orthogonality between the first $0$-eigenvector of $a_{(i,j)}$ given in Table 4 and the $\frac{1}{4}$-eigenvector of $a_{(ij)}$ in $\left\langle \left\langle a_{(ij)},a_{(kl)},a_{(ik)}\right\rangle \right\rangle$.