$2A$ -$A_{12}$的Majorana表示
$2A$-Majorana Representations of $A_{12}$
摘要：
马约拉纳表示已介绍了伊万诺夫，以提供一个公理化的框架，研究行动的格里斯代数的怪物和子群所产生的菲舍尔对合。这个计划的关键步骤是获得$A_{12}$的马约拉纳表示的明确描述，因为这可能最终导致Monster群的新的独立构造。本文证明了$A_{12}$在其$2^2$型和$2^6$型对合（即当$A_{12}$嵌入Monster中时属于Fischer对合类的对合）的集合上有唯一的Majorana表示，并确定了这种表示的阶和不可约分解.由此我们得到了提供$A_{12}$和Harada-Norton散在单群的$2A$ -表示的Majorana代数满足同花顺猜想。作为一个副产品，我们还确定的程度和分解成不可约的Majorana表示诱导的$A_8$子群的$A_{12}$。最后提出了关于交错群$A_n$，$8\leq n\leq 12$的Majorana表示的一个猜想.
英文摘要：
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 $A_n$, $8\leq n\leq 12$.
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