Tits endomorphisms and buildings of type F 4
[Endomorphismes de Tits et immeubles de type F 4 ]
Annales de l'Institut Fourier, Tome 67 (2017) no. 6, pp. 2349-2421.

L’immeuble de points fixes d’une polarité d’un quadrangle de Moufang de type F 4 est un ensemble de Moufang. Il en va de même pour l’immeuble de points fixes d’un automorphisme semi-linéaire d’ordre 2 d’un octogone de Moufang qui stabilise au moins deux cloisons d’un type mais aucun de l’autre. Nous montrons que ces deux classes d’ensembles de Moufang sont en fait identiques, que chaque membre de cette classe peut être construit comme l’immeuble de points fixes d’un groupe d’ordre 4 agissant sur un immeuble de type F 4 , et que pour chacun de ces ensembles de Moufang, le groupe engendré par tous les sous-groupes radiciels est un groupe simple.

The fixed point building of a polarity of a Moufang quadrangle of type F 4 is a Moufang set, as is the fixed point building of a semi-linear automorphism of order 2 of a Moufang octagon that stabilizes at least two panels of one type but none of the other. We show that these two classes of Moufang sets are, in fact, the same, that each member of this class can be constructed as the fixed point building of a group of order 4 acting on a building of type F 4 and that the group generated by all the root groups of any one of these Moufang sets is simple.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : https://doi.org/10.5802/aif.3138
Classification : 20E42,  51E12,  51E24
Mots clés : immeubles, descent, polarité, ensemble de Moufang, quadrangle de Moufang, octogone de Moufang
@article{AIF_2017__67_6_2349_0,
     author = {De Medts, Tom and Segev, Yoav and Weiss, Richard M.},
     title = {Tits endomorphisms and buildings of type~$F_4$},
     journal = {Annales de l'Institut Fourier},
     pages = {2349--2421},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {67},
     number = {6},
     year = {2017},
     doi = {10.5802/aif.3138},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3138/}
}
TY  - JOUR
AU  - De Medts, Tom
AU  - Segev, Yoav
AU  - Weiss, Richard M.
TI  - Tits endomorphisms and buildings of type $F_4$
JO  - Annales de l'Institut Fourier
PY  - 2017
DA  - 2017///
SP  - 2349
EP  - 2421
VL  - 67
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3138/
UR  - https://doi.org/10.5802/aif.3138
DO  - 10.5802/aif.3138
LA  - en
ID  - AIF_2017__67_6_2349_0
ER  - 
%0 Journal Article
%A De Medts, Tom
%A Segev, Yoav
%A Weiss, Richard M.
%T Tits endomorphisms and buildings of type $F_4$
%J Annales de l'Institut Fourier
%D 2017
%P 2349-2421
%V 67
%N 6
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.3138
%R 10.5802/aif.3138
%G en
%F AIF_2017__67_6_2349_0
De Medts, Tom; Segev, Yoav; Weiss, Richard M. Tits endomorphisms and buildings of type $F_4$. Annales de l'Institut Fourier, Tome 67 (2017) no. 6, pp. 2349-2421. doi : 10.5802/aif.3138. https://aif.centre-mersenne.org/articles/10.5802/aif.3138/

[1] Carter, Roger W. Simple groups of Lie type Tome 28, John Wiley & Sons, 1972, viii+331 pages (Pure and Applied Mathematics) | MR 0407163 | Zbl 0248.20015

[2] De Medts, Tom Automorphisms of F 4 quadrangles, Math. Ann., Tome 328 (2004) no. 3, pp. 399-413 | Article | MR 2036328 | Zbl 1056.51003

[3] De Medts, Tom An algebraic structure for Moufang quadrangles, Mem. Am. Math. Soc., Tome 173 (2005) no. 818, vi+99 pages | Article | MR 2109785 | Zbl 1065.51003

[4] De Medts, Tom; Segev, Yoav A course on Moufang sets, Innov. Incidence Geom., Tome 9 (2009), pp. 79-122 | MR 2658895 | Zbl 1233.20028

[5] De Medts, Tom; Van Maldeghem, Hendrik Moufang sets of type F 4 , Math. Z., Tome 265 (2010) no. 3, pp. 511-527 | Article | MR 2644307 | Zbl 1198.20028

[6] De Medts, Tom; Weiss, Richard M. Moufang sets and Jordan division algebras, Math. Ann., Tome 335 (2006) no. 2, pp. 415-433 | Article | MR 2221120 | Zbl 1163.17031

[7] Elduque, Alberto; Pérez, José María Infinite-dimensional quadratic forms admitting composition, Proc. Am. Math. Soc., Tome 125 (1997) no. 8, pp. 2207-2216 | Article | MR 1376759 | Zbl 0938.17003

[8] Engler, Antonio J.; Prestel, Alexander Valued fields, Springer Monographs in Mathematics, Springer, 2005, x+205 pages | MR 2183496 | Zbl 1128.12009

[9] Humphreys, James E. Introduction to Lie algebras and representation theory Tome 9, Springer, 1972, xii+169 pages (Graduate Texts in Mathematics) | MR 0323842 | Zbl 0254.17004

[10] Iwasawa, Kenkiti Über die Einfachheit der speziellen projektiven Gruppen, Proc. Imp. Acad. Tokyo, Tome 17 (1941), pp. 57-59 | Article | MR 0004034 | Zbl 0025.01101

[11] Mühlherr, Bernhard; Petersson, Holger P.; Weiss, Richard M. Descent in buildings, Annals of Mathematics Studies, Tome 190, Princeton University Press, 2015, xvi+336 pages | Article | MR 3364836 | Zbl 1338.51002

[12] Mühlherr, Bernhard; Van Maldeghem, Hendrik Exceptional Moufang quadrangles of type F 4 , Can. J. Math., Tome 51 (1999) no. 2, pp. 347-371 | Article | MR 1697148 | Zbl 0942.51002

[13] Mühlherr, Bernhard; Van Maldeghem, Hendrik Moufang sets from groups of mixed type, J. Algebra, Tome 300 (2006) no. 2, pp. 820-833 | Article | MR 2228223 | Zbl 1101.51003

[14] Mühlherr, Bernhard; Weiss, Richard M. Galois involutions and exceptional buildings, Enseign. Math., Tome 62 (2016) no. 1-2, pp. 207-260 | Article | MR 3605817 | Zbl 1366.20018

[15] Mühlherr, Bernhard; Weiss, Richard M. Rhizospheres in spherical buildings, Math. Ann., Tome 369 (2017) no. 1-2, pp. 839-868 | Article

[16] Steinberg, Robert Lectures on Chevalley groups, Yale University, 1968, iii+277 pages (Notes prepared by John Faulkner and Robert Wilson) | MR 0466335 | Zbl 1196.22001

[17] Struyve, Koen Moufang sets related to polarities in exceptional Moufang quadrangles of type F 4 , Innov. Incidence Geom., Tome 10 (2009), pp. 121-132 | MR 2665197 | Zbl 1263.51002

[18] Tits, Jacques Classification of algebraic semisimple groups, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), American Mathematical Society, 1966, pp. 33-62 | MR 0224710 | Zbl 0238.20052

[19] Tits, Jacques Buildings of spherical type and finite BN-pairs, Lecture Notes in Mathematics, Tome 386, Springer, 1974, x+299 pages | MR 0470099 | Zbl 0295.20047

[20] Tits, Jacques Moufang octagons and the Ree groups of type 2 F 4 , Am. J. Math., Tome 105 (1983) no. 2, pp. 539-594 | Article | MR 701569 | Zbl 0521.20016

[21] Tits, Jacques Twin buildings and groups of Kac-Moody type, Groups, combinatorics & geometry (Durham, 1990) (Lond. Math. Soc. Lect. Note Ser.) Tome 165, Cambridge University Press, 1992, pp. 249-286 | Article | MR 1200265 | Zbl 0851.22023

[22] Tits, Jacques Les groupes simples de Suzuki et de Ree, Séminaire Bourbaki, Vol. 6, Société Mathématique de France, 1995, pp. Exp. No. 210, 65-82 | MR 1611778 | Zbl 0267.20041

[23] Tits, Jacques; Weiss, Richard M. Moufang polygons, Springer Monographs in Mathematics, Springer, 2002, x+535 pages | Article | MR 1938841 | Zbl 1010.20017

[24] Weiss, Richard M. The structure of spherical buildings, Princeton University Press, 2003, xiv+135 pages | MR 2034361 | Zbl 1061.51011

[25] Weiss, Richard M. Quadrangular algebras, Mathematical Notes, Tome 46, Princeton University Press, 2006, x+131 pages | MR 2177056 | Zbl 129.17001

[26] Weiss, Richard M. The structure of affine buildings, Annals of Mathematics Studies, Tome 168, Princeton University Press, 2009, xii+368 pages | MR 2468338 | Zbl 1166.51001

Cité par Sources :