3D4

In mathematics, the Steinberg triality groups of type ³D₄ 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 ³D₄(K) (thinking of ³D₄ as an algebraic group taking values in K) and some as ³D₄(L) (thinking of the group as a subgroup of D₄(L) fixed by an outer automorphism of order 3). The group ³D₄ 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).

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 ³D₄(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.

The group ³D₄(q³) has order q¹² (q⁸ + q⁴ + 1) (q⁶ − 1) (q² − 1). For comparison, the split spin group D₄(q) in dimension 8 has order q¹² (q⁸ − 2q⁴ + 1) (q⁶ − 1) (q² − 1) and the quasisplit spin group ²D₄(q²) in dimension 8 has order q¹² (q⁸ − 1) (q⁶ − 1) (q² − 1).

The group ³D₄(q³) is always simple. The Schur multiplier is always trivial. The outer automorphism group is cyclic of order f where q³ = p^f and p is prime.

This group is also sometimes called ³D₄(q), D₄²(q³), or a twisted Chevalley group.

The smallest member of this family of groups has several exceptional properties not shared by other members of the family. It has order 211341312 = 2¹²⋅3⁴⋅7²⋅13 and outer automorphism group of order 3.

The automorphism group of ³D₄(2³) 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 ³D₄(2³) acts as a rank 4 permutation group.

The group ³D₄(2³) has 9 classes of maximal subgroups, of structure

2¹⁺⁸:L₂(8) fixing a point of the rank 4 permutation representation on 819 points.

[2¹¹]:(7 × S₃)

U₃(3):2

S₃ × L₂(8)

(7 × L₂(7)):2

3¹⁺².2S₄

7²:2A₄

3²:2A₄

13:4

• List of finite simple groups
• ²E₆

• Carter, Roger W. (1989) [1972], 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

• ³D₄(2³) at the atlas of finite groups
• ³D₄(3³) at the atlas of finite groups