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The axes of a Majorana representation of \(A_{12}\). (English) Zbl 1341.20010
Narasimha Sastry, N.S. (ed.), Groups of exceptional type, Coxeter groups and related geometries. Invited articles based on the presentations at the international conference on “Groups and geometries”, Bangalore, India, December 10–21, 2012. New Delhi: Springer (ISBN 978-81-322-1813-5/hbk; 978-81-322-1814-2/ebook). Springer Proceedings in Mathematics & Statistics 82, 159-188 (2014).
Summary: 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 \(3\leq N\leq 5\), let \(V^{(NA)}\) be the linear span of the \(NA\)-axes of \(V\). In this paper, we examine the space \[ V^\circ=\left\langle V^{(NA)}:2\leq N\leq 5\right\rangle. \] 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 \[ \left\langle V^{(2A)}_A,V^{(3A)}_A\right\rangle=\left\langle V^{(2A)}_A,V^{(4A)}_A\right\rangle=V^\circ_A, \] and that \(V^\circ_A\) is the direct sum of \(V_A^{(2A)}\) and a \(462\)-dimensional irreducible \(\mathbb RA\)-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.
For the entire collection see [Zbl 1297.20001].

20D08 Simple groups: sporadic groups
17B69 Vertex operators; vertex operator algebras and related structures
20C34 Representations of sporadic groups
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