In mathematics, the Steinberg triality groups of type D4 form a family of Steinberg or twisted Chevalley groups. They are quasi-split forms of D4, depending on a cubic Galois extension of fields K ⊂ L, and using the triality automorphism of the Dynkin diagram D4. Unfortunately the notation for the group is not standardized, as some authors write it as D4(K) (thinking of D4 as an algebraic group taking values in K) and some as D4(L) (thinking of the group as a subgroup of D4(L) fixed by an outer automorphism of order 3). The group D4 is very similar to an orthogonal or spin group in dimension 8.
Over finite fields these groups form one of the 18 infinite families of finite simple groups, and were introduced by Steinberg (1959). They were independently discovered by Jacques Tits in Tits (1958) and Tits (1959).
Construction
The simply connected split algebraic group of type D4 has a triality automorphism σ of order 3 coming from an order 3 automorphism of its Dynkin diagram. If L is a field with an automorphism τ of order 3, then this induced an order 3 automorphism τ of the group D4(L). The group D4(L) is the subgroup of D4(L) of points fixed by στ. It has three 8-dimensional representations over the field L, permuted by the outer automorphism τ of order 3.
Over finite fields
The group D4(q) has order q (q + q + 1) (q − 1) (q − 1). For comparison, the split spin group D4(q) in dimension 8 has order q (q − 2q + 1) (q − 1) (q − 1) and the quasisplit spin group D4(q) in dimension 8 has order q (q − 1) (q − 1) (q − 1).
The group D4(q) is always simple. The Schur multiplier is always trivial. The outer automorphism group is cyclic of order f where q = p and p is prime.
This group is also sometimes called D4(q), D4(q), or a twisted Chevalley group.
D4(2)
The smallest member of this family of groups has several exceptional properties not shared by other members of the family. It has order 211341312 = 2⋅3⋅7⋅13 and outer automorphism group of order 3.
The automorphism group of D4(2) is a maximal subgroup of the Thompson sporadic group, and is also a subgroup of the compact Lie group of type F4 of dimension 52. In particular it acts on the 26-dimensional representation of F4. In this representation it fixes a 26-dimensional lattice that is the unique 26-dimensional even lattice of determinant 3 with no norm 2 vectors, studied by Elkies & Gross (1996). The dual of this lattice has 819 pairs of vectors of norm 8/3, on which D4(2) acts as a rank 4 permutation group.
The group D4(2) has 9 classes of maximal subgroups, of structure
- 2:L2(8) fixing a point of the rank 4 permutation representation on 819 points.
- :(7 × S3)
- U3(3):2
- S3 × L2(8)
- (7 × L2(7)):2
- 3.2S4
- 7:2A4
- 3:2A4
- 13:4
See also
References
- Carter, Roger W. (1989) , Simple groups of Lie type, Wiley Classics Library, New York: John Wiley & Sons, ISBN 978-0-471-50683-6, MR 0407163
- Elkies, Noam D.; Gross, Benedict H. (1996), "The exceptional cone and the Leech lattice", International Mathematics Research Notices, 1996 (14): 665–698, doi:10.1155/S1073792896000426, ISSN 1073-7928, MR 1411589
- Steinberg, Robert (1959), "Variations on a theme of Chevalley", Pacific Journal of Mathematics, 9 (3): 875–891, doi:10.2140/pjm.1959.9.875, ISSN 0030-8730, MR 0109191
- Steinberg, Robert (1968), Lectures on Chevalley groups, Yale University, New Haven, Conn., MR 0466335, archived from the original on 2012-09-10
- Tits, Jacques (1958), Les "formes réelles" des groupes de type E6, Séminaire Bourbaki; 10e année: 1957/1958. Textes des conférences; Exposés 152 à 168; 2e èd. corrigée, Exposé 162, vol. 15, Paris: Secrétariat math'ematique, MR 0106247
- Tits, Jacques (1959), "Sur la trialité et certains groupes qui s'en déduisent", Inst. Hautes Études Sci. Publ. Math., 2: 13–60, doi:10.1007/BF02684706, S2CID 120426125