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

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.

💡 Research Summary

This paper continues the line of investigation started in arXiv:2311.18538, focusing on the determination of full automorphism groups of several high‑dimensional axial algebras of Monster type. The authors study the 286‑dimensional algebra A₍₂₈₆₎ constructed from the smallest Mathieu sporadic simple group M₁₁, together with a collection of its subalgebras: a 101‑dimensional algebra A₍₁₀₁₎ arising from the maximal subgroup L₂(11), a 76‑dimensional algebra A₍₇₆₎ associated with M₁₀≅A₆, and several smaller algebras (dimensions 17, 24, 18, 12) corresponding to other maximal subgroups of M₁₁.

The paper begins with a concise review of axial algebras, the notion of an “axis”, the fusion law M(α,β) that defines the Monster type, and the associated Miyamoto group (the subgroup generated by the involutive Miyamoto automorphisms attached to each axis). A key combinatorial invariant, called the “shape”, records the orbit structure of the axet (the closed set of axes) under the Miyamoto group and is essential for organizing the computational work.

Two complementary computational strategies are employed. For algebras of dimension up to roughly 150, the authors rely on the fully automated “nuanced” algorithm introduced in