In the classification of symplectic reflection groups by Cohen (1980) one portion is made up of the groups which are symplectically primitive and complex primitive. There are in total only 13, so they can be considered as the exceptional ones in the classification. Cohen lists in Table III some information about the groups and gives constructions (in terms of root systems) as well. Below, 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, especially Section 6 and Remark 6.2.
| Name | Dimension | Order | GAP | Magma |
|---|---|---|---|---|
| O₁ | 4 | 120 | ||
| O₂ | 4 | 720 | ||
| O₃ | 4 | 1440 | ||
| P₁ | 4 | 320 | ||
| P₂ | 4 | 1920 | ||
| P₃ | 4 | 3840 | ||
| Q | 6 | 2⁶ · 3³ · 7 = 12,096 | ||
| R | 6 | 2⁸ · 3³ · 5² · 7 = 1,209,600 | R.g | R.m |
| S₁ | 8 | 2⁸ · 3³ = 6,912 | ||
| S₂ | 8 | 2¹⁰ · 3⁴ = 82,944 | ||
| S₃ | 8 | 2¹³ · 3⁴ · 5 = 3,317,760 | ||
| T | 8 | 2⁸ · 3⁴ · 5³ = 2,592,000 | ||
| U | 10 | 2¹¹ · 3⁵ · 5 · 11 = 27,371,520 |
References
- Cohen, A. M. (1980). Finite quaternionic reflection groups. J. Algebra, 64, 293–324.
- Bellamy, G., Schmitt, J., & Thiel, U. (2021). Towards the classification of symplectic quotient singularities admitting a symplectic resolution. Math. Z.