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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Saturated Majorana representations of $A_{12}$
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by Clara Franchi, Alexander A. Ivanov and Mario Mainardis PDF
Trans. Amer. Math. Soc. 375 (2022), 5753-5801 Request permission


Majorana representations have been introduced by Ivanov [Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2009] 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 $A_{12}$ (by Franchi, Ivanov, and Mainardis [J. Algebraic Combin. 44 (2016), pp. 265-292], the largest alternating group admitting a Majorana representation) for this might eventually lead to a new and independent construction of the Monster group (see A.A Ivanov [Group theory and computation, Indian Stat. Inst. Ser., Springer, Singapore, 2018, Section 4, page 115]).

In this paper we prove that $A_{12}$ has two possible Majorana sets, one of which is the set $\mathcal X_b$ of involutions of cycle type $2^2$, the other is the union of $\mathcal X_b$ with the set $\mathcal X_s$ of involutions of cycle type $2^6$. The latter case (the saturated case) is most interesting, since the Majorana set is precisely the set of involutions of $A_{12}$ that fall into the class of Fischer involutions when $A_{12}$ is embedded in the Monster. We prove that $A_{12}$ has a unique saturated Majorana representation and we determine its degree and decomposition into irreducibles. As consequences we get that the Harada-Norton group has, up to equivalence, a unique Majorana representation and every Majorana algebra, affording either a Majorana representation of the Harada-Norton group or a saturated Majorana representation of $A_{12}$, satisfies the Straight Flush Conjecture (see A. A. Ivanov [Contemp. Math., Amer. Math. Soc., Providence, RI, 2017, pp. 11-17] and A. A. Ivanov [Group theory and computation, Indian Stat. Inst. Ser., Springer, Singapore, 2018, pp. 107-118]). As a by-product we also determine the degree and the decomposition into irreducibles of the Majorana representation induced on $A_8$, the four point stabilizer 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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Additional Information
  • Clara Franchi
  • Affiliation: Dipartimento di Matematica e Fisica, Università Cattolica del Sacro Cuore, Via Garzetta 48, I-25133 Brescia, Italy
  • MR Author ID: 659086
  • ORCID: 0000-0002-2830-2540
  • Email:
  • Alexander A. Ivanov
  • Affiliation: Three Gorges Mathematical Research Center, Yichang, Hubei, China; Department of Mathematics, Imperial College, 180 Queen’s Gt., London SW7 2AZ, United Kingdom; Institute for System Analysis ERC, CSC RAS, Moscow, Russia
  • MR Author ID: 191708
  • Email:
  • Mario Mainardis
  • Affiliation: Dipartimento di Scienze Matematiche, Informatiche e Fisiche, Università degli Studi di Udine, via delle Scienze 206, I-33100 Udine, Italy
  • MR Author ID: 313708
  • ORCID: 0000-0002-8567-3014
  • Email:
  • Received by editor(s): April 16, 2020
  • Received by editor(s) in revised form: April 16, 2020, and January 24, 2022
  • Published electronically: June 3, 2022
  • Additional Notes: This work was partially supported by PRID Marfap - Università di Udine
  • © Copyright 2022 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 375 (2022), 5753-5801
  • MSC (2020): Primary 20C30, 20C15, 17B69, 05B99
  • DOI:
  • MathSciNet review: 4469236