In the classification of symplectic reflection groups by Cohen [1] one portion is made up of the groups which are symplectically primitive and, at the same time, complex primitive. There are in total only 13, so they can be considered as the exceptional ones in the classification. Cohen lists in Table II and III some information about the groups and gives constructions (in terms of root systems) as well. In this repository, you can find explicit matrix models for GAP and Magma obtained from the paper.
All the work was done by Johannes Schmitt.
For questions concerning symplectic resolutions, see our paper [2], especially Section 6 and Remark 6.2.
| Name | Dimension | Order |
|---|---|---|
| O₁ | 4 | 120 |
| O₂ | 4 | 720 |
| O₃ | 4 | 1440 |
| P₁ | 4 | 320 |
| P₂ | 4 | 1920 |
| P₃ | 4 | 3840 |
| Q | 6 | 12096 |
| R | 6 | 1209600 |
| S₁ | 8 | 6912 |
| S₂ | 8 | 82944 |
| S₃ | 8 | 3317760 |
| T | 8 | 2592000 |
| U | 10 | 27371520 |
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Cohen, A. (1980). Finite quaternionic reflection groups. J. Algebra 64 (1980), no. 2, 293–324.
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Bellamy, G., Schmitt, J. & Thiel, U. (2023). On parabolic subgroups of symplectic reflection groups. Glasg. Math. J. 65 (2023), no. 2, 401–413. [DOI] · [arXiv] · [MR]
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