The following table is an extended version about data of the Calogero–Moser hyperplane arrangements for exceptional complex reflection groups as discussed by G. Bellamy, T. Schedler, and myself in Hyperplane arrangements associated to symplectic quotient singularities. The hyperplanes have been computed by myself with methods described in CHAMP: A Cherednik Algebra Magma Package, and by C. Bonnafé and myself in Calogero-Moser families and cellular characters: computational aspects (in preparation). The 2-reflection groups are missing in the table since here the situation is clear (see the paper). The other missing ones are so far too complicated to compute.
If you click on a group name you get a Sage file describing the corresponding hyperplane arrangement ℰ inside a vector space 𝔠. This file can be loaded into Sage using the load function as follows:
load("G4.sage") #loads file with arrangement for G4
A.is_central() #check if A is central
A.is_essential() #check if A is essential
P=A.intersection_poset() #the intersection poset of A
L=LatticePoset(P) #the intersection lattice of A
L.is_supersolvable() #check if A is supersolvable
The column “ss” describes whether the arrangement is supersolvable or not. If it is, then it is already free, and it is also K(π,1) by Falk-Randell and Terao.
| Group | #ℰ | dim 𝔠 | π(t) | exp | ss | free | K(π,1) |
|---|---|---|---|---|---|---|---|
| G4 | 6 | 2 | (5t + 1)(t + 1) | 1,5 | yes | yes | yes |
| G5 | 33 | 4 | (116t2 +21t +1)(11t +1)(t +1) | – | no | no | ? |
| G6 | 16 | 3 | (8t +1)(7t +1)(t +1) | 1,7,8 | no | yes | ? |
| G7 | 61 | 5 | (98644t4 +18462t3 +1489t2 +60t +1)(t +1) | – | no | no | ? |
| G8 | 25 | 3 | (13t +1)(11t +1)(t +1) | 1,11,13 | no | yes | ? |
| G9 | 54 | 4 | (6499t3 +983t2 +53t +1)(t +1) | – | no | no | ? |
| G10 | 111 | 5 | (1001586t4 + 107662t3 + 4913t2 + 110t + 1)(t + 1) | – | no | no | ? |
| G11 | 196 | 6 | (383999826t5 + 25688824t4 + 857259t3 + 17047t2 + 195t + 1)(t + 1) | – | no | no | ? |
| G13 | 6 | 2 | (5t +1)(t +1) | 1,5 | yes | yes | yes |
| G14 | 22 | 3 | (116t2 +21t +1)(t +1) | – | no | no | ? |
| G15 | 65 | 4 | (13982t3 +1529t2 +32t +1)(1+t) | – | no | no | ? |
| G20 | 12 | 2 | (11t + 1)(t + 1) | 1,11 | yes | yes | yes |
| G25 | 12 | 2 | (11t + 1)(t + 1) | 1,11 | yes | yes | yes |
| G26 | 37 | 3 | (335t2 +36t +1)(t +1) | – | no | no | ? |
| F4=G28 | 8 | 2 | (7t +1)(t +1) | 1,7 | yes | yes | yes |