$(2B, 3A, 5A)$-subalgebras of the Griess algebra with alternating Miyamoto group

TL;DR

This work classifies subalgebras of the Griess algebra with shape $$(2B,3A,5A)$$ whose Miyamoto groups match alternating groups $A_n$, proving existence only for $n\in\{5,6,8\}$; it consolidates the $n=5$ case, constructs a unique diagonal Majorana representation for $A_6$ of dimension $121$ via a Monster embedding, and shows the $n=8$ case is nonstandard and not realizable from standard representations. The analysis develops diagonal Majorana representations, introduces dormant $4$-axes, and leverages Norton-Sakuma type algebras and maximal subalgebras of $A_6$ to control inner products and subalgebra structure. It culminates in showing $V^{\circ}$ has dimension $121$, decomposes into irreducible ${\mathbb R}[S_6]$-modules, and, crucially, that the overall algebra product is uniquely determined, with $V^{\circ}=V$. These results advance the broader program of classifying Majorana representations of $A_n$ and illuminate the interplay between subalgebra structure and Miyamoto group actions in the Griess algebra framework.

Abstract

We use Majorana representations to study the subalgebras of the Griess algebra that have shape $(2B,3A,5A)$ and whose associated Miyamoto groups are isomorphic to $A_n$. We prove that these subalgebras exist only if $n\in \{5,6,8\}$. The case $n=5$ was already treated by Ivanov, Seress, McInroy, and Shpectorov. In case $n=6$ we prove that these algebras are all isomorphic and provide their precise description. In case $n=8$ we prove that these algebras do not arise from standard Majorana representations.

Theorems & Definitions (98)

• Theorem 1
• Theorem 2
• Remark 2.1
• Remark 2.2
• Lemma 2.3
• proof
• Proposition 2.4
• Lemma 3.1
• proof
• Lemma 3.2