## Saturated Majorana representations of $A_{12}$

HTML articles powered by AMS MathViewer

- 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 $A_n$, $8\leq n\leq 12$.

## References

- Eiichi Bannai and Tatsuro Ito,
*Algebraic combinatorics. I*, The Benjamin/Cummings Publishing Co., Inc., Menlo Park, CA, 1984. Association schemes. MR**882540** - A. Castillo-Ramirez and A. A. Ivanov,
*The axes of a Majorana representation of $A_{12}$*, Groups of exceptional type, Coxeter groups and related geometries, Springer Proc. Math. Stat., vol. 82, Springer, New Delhi, 2014, pp. 159–188. MR**3207276**, DOI 10.1007/978-81-322-1814-2_{9} - J. H. Conway,
*A simple construction for the Fischer-Griess monster group*, Invent. Math.**79**(1985), no. 3, 513–540. MR**782233**, DOI 10.1007/BF01388521 - J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson,
*$\Bbb {ATLAS}$ of finite groups*, Oxford University Press, Eynsham, 1985. Maximal subgroups and ordinary characters for simple groups; With computational assistance from J. G. Thackray. MR**827219** - Clara Franchi, Alexander A. Ivanov, and Mario Mainardis,
*The $2A$-Majorana representations of the Harada-Norton group*, Ars Math. Contemp.**11**(2016), no. 1, 175–187. MR**3546657**, DOI 10.26493/1855-3974.859.0c3 - Clara Franchi, Alexander A. Ivanov, and Mario Mainardis,
*Standard Majorana representations of the symmetric groups*, J. Algebraic Combin.**44**(2016), no. 2, 265–292. MR**3533555**, DOI 10.1007/s10801-016-0668-8 - Clara Franchi, Alexander A. Ivanov, and Mario Mainardis,
*Permutation modules for the symmetric group*, Proc. Amer. Math. Soc.**145**(2017), no. 8, 3249–3262. MR**3652780**, DOI 10.1090/proc/13474 - Clara Franchi, Alexander A. Ivanov, and Mario Mainardis,
*Radicals of $S_n$-invariant positive semidefinite hermitian forms*, Algebr. Comb.**1**(2018), no. 4, 425–440. MR**3875072**, DOI 10.5802/alco - Clara Franchi, Alexander A. Ivanov, and Mario Mainardis,
*Majorana representations of finite groups*, Algebra Colloq.**27**(2020), no. 1, 31–50. MR**4069349**, DOI 10.1142/S1005386720000048 - The GAP Group,
*GAP – Groups, Algorithms, and Programming*, Version 4.10.2, 2019 https://www.gap-system.org. - Robert L. Griess Jr.,
*The friendly giant*, Invent. Math.**69**(1982), no. 1, 1–102. MR**671653**, DOI 10.1007/BF01389186 - D. G. Higman,
*Coherent configurations. I. Ordinary representation theory*, Geometriae Dedicata**4**(1975), no. 1, 1–32. MR**398868**, DOI 10.1007/BF00147398 - J. I. Hall, F. Rehren, and S. Shpectorov,
*Universal axial algebras and a theorem of Sakuma*, J. Algebra**421**(2015), 394–424. MR**3272388**, DOI 10.1016/j.jalgebra.2014.08.035 - J. I. Hall, F. Rehren, and S. Shpectorov,
*Primitive axial algebras of Jordan type*, J. Algebra**437**(2015), 79–115. MR**3351958**, DOI 10.1016/j.jalgebra.2015.03.026 - I. Martin Isaacs,
*Character theory of finite groups*, Dover Publications, Inc., New York, 1994. Corrected reprint of the 1976 original [Academic Press, New York; MR0460423 (57 #417)]. MR**1280461** - A. A. Ivanov,
*The Monster group and Majorana involutions*, Cambridge Tracts in Mathematics, vol. 176, Cambridge University Press, Cambridge, 2009. MR**2503090**, DOI 10.1017/CBO9780511576812 - A. A. Ivanov and S. Shpectorov,
*Majorana representations of $L_3(2)$*, Adv. Geom.**12**(2012), no. 4, 717–738. MR**3005109**, DOI 10.1515/advgeom-2012-0018 - A. A. Ivanov,
*Majorana representation of the Monster group*, Finite simple groups: thirty years of the atlas and beyond, Contemp. Math., vol. 694, Amer. Math. Soc., Providence, RI, 2017, pp. 11–17. MR**3682586**, DOI 10.1090/conm/694 - Alexander A. Ivanov,
*The future of Majorana theory*, Group theory and computation, Indian Stat. Inst. Ser., Springer, Singapore, 2018, pp. 107–118. MR**3839690** - A. A. Ivanov, D. V. Pasechnik, Á. Seress, and S. Shpectorov,
*Majorana representations of the symmetric group of degree 4*, J. Algebra**324**(2010), no. 9, 2432–2463. MR**2684148**, DOI 10.1016/j.jalgebra.2010.07.015 - A. A. Ivanov,
*On Majorana representations of $A_6$ and $A_7$*, Comm. Math. Phys.**307**(2011), no. 1, 1–16. MR**2835871**, DOI 10.1007/s00220-011-1298-6 - A. A. Ivanov,
*Majorana representation of $A_6$ involving $3C$-algebras*, Bull. Math. Sci.**1**(2011), no. 2, 365–378. MR**2901004**, DOI 10.1007/s13373-011-0010-7 - A. A. Ivanov and Á. Seress,
*Majorana representations of $A_5$*, Math. Z.**272**(2012), no. 1-2, 269–295. MR**2968225**, DOI 10.1007/s00209-011-0933-4 - G. D. James,
*The representation theory of the symmetric groups*, Lecture Notes in Mathematics, vol. 682, Springer, Berlin, 1978. MR**513828** - S. M. S. Khasraw, J. McInroy, and S. Shpectorov,
*On the structure of axial algebras*, Trans. Amer. Math. Soc.**373**(2020), no. 3, 2135–2156. MR**4068292**, DOI 10.1090/tran/7979 - S. M. S. Khasraw, J. McInroy, and S. Shpectorov,
*Enumerating 3-generated axial algebras of Monster type*, J. Pure Appl. Algebra**226**(2022), no. 2, Paper No. 106816, 21. MR**4276486**, DOI 10.1016/j.jpaa.2021.106816 - Serge Lang,
*Algebra*, 3rd ed., Graduate Texts in Mathematics, vol. 211, Springer-Verlag, New York, 2002. MR**1878556**, DOI 10.1007/978-1-4613-0041-0 - C. S. Lim,
*From the Monster to Majorana. A study of the $3A$-axes*, Imperial College London, 2017. - Justin McInroy and Sergey Shpectorov,
*An expansion algorithm for constructing axial algebras*, J. Algebra**550**(2020), 379–409. MR**4065996**, DOI 10.1016/j.jalgebra.2019.12.018 - Masahiko Miyamoto,
*Griess algebras and conformal vectors in vertex operator algebras*, J. Algebra**179**(1996), no. 2, 523–548. MR**1367861**, DOI 10.1006/jabr.1996.0023 - S. Norton,
*The Monster algebra: some new formulae*, Moonshine, the Monster, and related topics (South Hadley, MA, 1994) Contemp. Math., vol. 193, Amer. Math. Soc., Providence, RI, 1996, pp. 297–306. MR**1372728**, DOI 10.1090/conm/193/02377 - M. Pfeiffer and M. Whybrow,
*MajoranaAlgebras, a package for constructing Majorana algebras and representations*, Version 1.4, 2018, GAP package, https://MWhybrow92.github.io/MajoranaAlgebras/. - Shinya Sakuma,
*6-transposition property of $\tau$-involutions of vertex operator algebras*, Int. Math. Res. Not. IMRN**9**(2007), Art. ID rnm 030, 19. MR**2347298**, DOI 10.1093/imrn/rnm030 - J. P. Serre,
*Letter to Donna Testerman*, Finite simple groups: thirty years of the atlas and beyond, Contemp. Math., vol. 694, Amer. Math. Soc., Providence, RI, 2017, pp. 19–20. - R. M. Thrall,
*On symmetrized Kronecker powers and the structure of the free Lie ring*, Amer. J. Math.**64**(1942), 371–388. MR**6149**, DOI 10.2307/2371691 - Madeleine L. Whybrow,
*Majorana algebras generated by a $2A$ algebra and one further axis*, J. Group Theory**21**(2018), no. 3, 417–437. MR**3794923**, DOI 10.1515/jgth-2017-0047 - Madeleine Whybrow,
*An infinite family of axial algebras*, J. Algebra**577**(2021), 1–31. MR**4232631**, DOI 10.1016/j.jalgebra.2020.07.026

## 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