²E₆ (mathematics)

In mathematics, ²E₆ is a family of Steinberg or twisted Chevalley groups. It is a quasi-split form of E₆, depending on a quadratic extension of fields K⊂L. Unfortunately the notation for the group is not standardized, as some authors write it as ²E₆(K) (thinking of ²E₆ as an algebraic group taking values in K) and some as ²E₆(L) (thinking of the group as a subgroup of E₆(L) fixed by an outer involution).

Over finite fields these groups form one of the 18 infinite families of finite simple groups, and were introduced independently by Tits (1958) and Steinberg (1959).

Over finite fields

The group ²E₆(q²) has order q³⁶ (q¹² − 1) (q⁹ + 1) (q⁸ − 1) (q⁶ − 1) (q⁵ + 1) (q² − 1) /(3,q + 1).^[1] This is similar to the order q³⁶ (q¹² − 1) (q⁹ − 1) (q⁸ − 1) (q⁶ − 1) (q⁵ − 1) (q² − 1) /(3,q − 1) of E₆(q).

Its Schur multiplier has order (3, q + 1) except for q=2, i. e. ²E₆(2²), when it has order 12 and is a product of cyclic groups of orders 2,2,3. One of the exceptional double covers of ²E₆(2²) is a subgroup of the baby monster group, and the exceptional central extension by the elementary abelian group of order 4 is a subgroup of the monster group.

The outer automorphism group has order (3, q + 1) · f where q² = p^f.

Over the real numbers

Over the real numbers, ²E₆ is the quasisplit form of E₆, and is one of the five real forms of E₆ classified by Élie Cartan. Its maximal compact subgroup is of type F₄.

Remarks

^ Reading example: If q²=2² in ²E₆(q²) then q=2 in the order formula q³⁶ (q¹² − 1) (q⁹ + 1) (q⁸ − 1) (q⁶ − 1) (q⁵ + 1) (q² − 1) /(3,q + 1). However, the group ²E₆(2²) is sometimes also written ²E₆(2) (e. g. in Wilson's Atlas).

References

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
Steinberg, Robert (1959), "Variations on a theme of Chevalley", Pacific Journal of Mathematics, 9: 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
Robert Wilson: Atlas of Finite Group Representations: Sporadic groups

[1] Reading example: If q²=2² in ²E₆(q²) then q=2 in the order formula q³⁶ (q¹² − 1) (q⁹ + 1) (q⁸ − 1) (q⁶ − 1) (q⁵ + 1) (q² − 1) /(3,q + 1). However, the group ²E₆(2²) is sometimes also written ²E₆(2) (e. g. in Wilson's Atlas).

[1]