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
Abstract:
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: clara.franchi@unicatt.it
- 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: a.ivanov@imperial.ac.uk
- 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: mario.mainardis@uniud.it
- 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: https://doi.org/10.1090/tran/8669
- MathSciNet review: 4469236