Full automorphism groups of the axial algebra for $M_{11}$ and related algebras

TL;DR

The paper determines the full automorphism groups of key Monster-type axial algebras tied to the smallest Mathieu group and related groups, notably $A_{286}$ with Miyamoto group $M_{11}$, as well as its subalgebras $A_{101}$ and $A_{76}$. It advances the automorphism-program aut by deploying a hybrid method that blends computation with targeted hand proofs to handle large algebras, extending beyond the purely automatic Nuanced algorithm. The main results identify $\mathrm{Aut}(A_{286})\cong M_{11}$, $\mathrm{Aut}(A_{101})\cong \mathrm{PGL}(2,11)$, and $\mathrm{Aut}(A_{76})\cong \mathrm{Aut}(A_{6})$, while also detailing the automorphism structures of several smaller subalgebras arising from maximal subgroups. These findings illuminate how axial algebra structures, shapes, and axet actions constrain and determine automorphism groups, with implications for connections to the Griess algebra and vertex operator algebras.

Abstract

In this paper, in continuation of arXiv:2311.18538, we compute the full automorphism groups of the 286-dimensional algebra for $M_{11}$, its subalgebras and other related algebras. This includes, in particular, the 101-dimensional algebra for $L_2(11)$ and the 76-dimensional algebra for $A_6$. While smaller algebras can be handled by the fully automatic nuanced method from arXiv:2311.18538, the larger algebras, mentioned above, require a hybrid method combining computation with hand-made proofs.

Figures (11)

• Figure 1: The folded shape diagram of $A_{286}$
• Figure 2: The folded shape diagram of $A_{101}$
• Figure 3: The folded shape diagram of $A_{76}$
• Figure 4: The shape diagram of $A_{17}$ on $12$ axes
• Figure 5: Shape diagram of $A_{36}$ on $10+15$ axes

Theorems & Definitions (85)

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