arXiv subjects

Mario Mainardis

Publications and source records attributed to Mario Mainardis.

8 recordsLinked to original sources

arXiv2026

The Classification of the 2-generated Primitive Axial Algebras of Monster Type

Axial algebras of Monster type are a class of commutative algebras generated by special idempotents called axes. Some motivating examples of these algebras are the Griess algebra and the Norton-Sakuma algebras, relating to the Monster simple group. A long standing open problem is to classify the 2-generated axial algebras of Monster type. A huge milestone was accomplished by Yabe leading, with additional cases completed by Franchi, Mainardis, and McInroy, to the classification in the symmetric case. In this paper, we complete the classification. To do so, we split the proof into multiple cases: dealing with certain parameters, subalgebras, axets, and axial dimensions. Furthermore, we provide a basis, multiplication and information of the algebras in the classification; consolidating existing results on these algebras into one place.

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arXiv2025

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

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.

math.GR↗

arXiv2022

Quotients of the Highwater algebra and its cover

Axial algebras are a class of non-associative algebra with a strong link to finite (especially simple) groups which have recently received much attention. Of primary interest are the axial algebras of Monster type $(\alpha, \beta)$, of which the Griess algebra (with the Monster as its automorphism group) is an important motivating example. In this paper, we complete the classification of the symmetric $2$-generated primitive axial algebras of Monster type $(\alpha, \beta)$. By previous work of Yabe, and Franchi and Mainardis, any such algebra is either explicitly known, or is a quotient of the infinite-dimensional Highwater algebra $\mathcal{H}$, or its characteristic $5$ cover $\hat{\mathcal{H}}$. In this paper, we classify the ideals of $\mathcal{H}$ and $\hat{\mathcal{H}}$ and thus their quotients. Moreover, we give explicit bases for the ideals. In fact, we proceed in a unified way, by defining a cover $\hat{\mathcal{H}}$ of $\mathcal{H}$ in all characteristics and classifying its ideals. Our new algebra $\hat{\mathcal{H}}$ has a previously unseen fusion law and provides an insight into why the Highwater algebra has a cover which is of Monster type only in characteristic $5$.

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arXiv2021

2-generated axial algebras of Monster type

We provide the basic setup for the project, initiated by Felix Rehren, aiming at classifying all 2-generated axial algebras of Monster type $(\alpha,\beta)$ over a field $\mathbb F$. Using this, we first show that every such algebra has dimension at most 8, except for the case $(\alpha,\beta)=(2,\tfrac{1}{2})$, where the Highwater algebra provides examples of dimension $n$, for all $n\in {\mathbb N}\cup \{\infty\}$. We then classify all 2-generated axial algebras of Monster type $(\alpha,\beta)$ over ${\mathbb Q}(\alpha,\beta)$, for $\alpha$ and $\beta$ algebraically independent over $\mathbb Q$. Finally, we generalise the Norton-Sakuma Theorem to every primitive $2$-generated axial algebra of Monster type $(\frac{1}{4},\frac{1}{32})$ over a field of characteristic zero, dropping the hypothesis on the existence of a Frobenius form.

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arXiv2021

2-generated axial algebras of Monster type $(2\beta, \beta)$

In this paper we prove that $2$-generated primitive axial algebras of Monster type $(2\beta, \beta)$ over a ring $R$ in which $2$ and $\beta$ are invertible can be generated as $R$-module by $8$ vectors. We then completely classify $2$-generated primitive axial algebras of Monster type $(2\beta, \beta)$ over any field of characteristic other than $2$.

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arXiv2021

A note on 2-generated symmetric axial algebras of Monster type

Recently Takahiro Yabe gave an almost complete classification of primitive symmetric $2$-generated axial algebras of Monster type. In this note, we construct a new infinite-dimensional primitive $2$-generated symmetric axial algebra of Monster type $(2, \frac{1}{2})$ over a field of characteristic $5$, and use this algebra to complete the last case left open.

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arXiv2020

An infinite-dimensional 2-generated primitive axial algebra of Monster type

Rehren proved that a primitive 2-generated axial algebra of Monster type $(\alpha,\beta)$ has dimension at most eight if $\alpha\notin\{2\beta,4\beta\}$. In this note we construct an infinite-dimensional 2-generated primitive axial algebra of Monster type $(2,\frac{1}{2})$ over an arbitrary field $F$ with $char(F)\neq 2,3$. This shows that the second special case, $\alpha=4\beta$, is a true exception to Rehren's bound.

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arXiv2020

$2A$-Majorana Representations of $A_{12}$

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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