Abstract
We develop a theory of Ennola duality for subgroups of finite groups of Lie type, relating subgroups of twisted and untwisted groups of the same type. Roughly speaking, one finds that subgroups H of GUd(q) correspond to subgroups of GLd(−q), where −q is interpreted modulo |H|. Analogous results for types other than A are established, including for those exceptional types where the maximal subgroups are known, although the result for type D is still conjectural. Let M denote the Gram matrix of a non-zero orthogonal form for a real, irreducible representation of a finite group, and consider α = √det(M). If the representation has twice odd dimension, we conjecture that α lies in some cyclotomic field. This does not hold for representations of dimension a multiple of 4, with a specific example of the Janko group J1 in dimension 56 given. (This tallies with Ennola duality for representations, where type D2n has no Ennola duality with 2D2n.)
Original language | English |
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Pages (from-to) | 785–799 |
Number of pages | 15 |
Journal | Monatshefte fur Mathematik |
Volume | 199 |
Issue number | 4 |
Early online date | 5 Feb 2022 |
DOIs | |
Publication status | Published - Dec 2022 |
Keywords
- Maximal subgroups
- Ennola duality
- Subgroup structure of groups
- Representations of finite groups