论文标题
$ 2A $ -MAJORANA表示$ A_ {12} $
$2A$-Majorana Representations of $A_{12}$
论文作者
论文摘要
伊万诺夫（Ivanov）引入了Majorana表示，以提供一个公理框架，以研究怪物及其由Fischer所产生的怪物代数及其亚组的作用。该程序的关键步骤是获得$ a_ {12} $的Majorana表示形式的明确描述，因为这最终可能会导致怪物组的新独立构造。 在本文中，我们证明$ a_ {12} $在其类型的$ 2^2 $和$ 2^6 $的一组中具有独特的主要代表性（这是当$ a_ {12} $嵌入在Monster中的Fischer范围的参与），并且我们确定了与Irredecibles的学位和分解。结果，我们得到Majoraana代数提供$ 2A $的代表$ a_ {12} $，而Harada-Norton散发性简单组则满足了直阵容的猜想。作为副产品，我们还确定了在$ a_ {12} $的$ a_8 $子组上引起的Majorana代表的不重新分解的程度和分解。我们终于提出了关于交替组$ a_n $，$ 8 \ leq n \ leq 12 $的主要表示的猜想。
Majorana representations have been introduced by Ivanov in order to provide an axiomatic framework for studying the actions on the Griess algebra of the Monster and of its subgroups generated by Fischer involutions. A crucial step in this programme is to obtain an explicit description of the Majorana representations of $A_{12}$, for this might eventually lead to a new and independent construction of the Monster group. In this paper we prove that $A_{12}$ has a unique Majorana representation on the set of its involutions of type $2^2$ and $2^6$ (that is the involutions that fall into the class of Fischer involutions when $A_{12}$ is embedded in the Monster) and we determine the degree and the decomposition into irreducibles of such representation. As a consequence we get that Majorana algebras affording a $2A$-representation of $A_{12}$ and of the Harada-Norton sporadic simple group satisfy the Straight Flush Conjecture. As a by-product we also determine the degree and the decomposition into irreducibles of the Majorana representation induced on the $A_8$ subgroup of $A_{12}$. We finally state a conjecture about Majorana representations of the alternating groups $A_n$, $8\leq n\leq 12$.
