atlas> atlas> atlas> set G=PSp(4) Variable G: RootDatum (overriding previous instance, which had type RealForm) atlas> show_nilpotent_orbits(G) i H diagram dim BC Levi Cent A(O) 0 [0,0] [0,0] 0 2T1 B2 [1] 1 [1,0] [1,0] 4 A1+T1 A1 [1] 2 [0,2] [0,2] 6 A1+T1 e [1,2] 3 [2,2] [2,2] 8 C2 e [1] {This is the component group information, for G adjoint, directly provided by Sommers' paper.} atlas> set G=Sp(4) Variable G: RootDatum (overriding previous instance, which had type RootDatum) atlas> show_nilpotent_orbits(G) i H diagram dim BC Levi Cent A(O) 0 [0,0] [0,0] 0 2T1 B2 [1] 1 [1,0] [1,0] 4 A1+T1 A1 [1,2] 2 [1,1] [0,2] 6 A1+T1 e [1,2] 3 [3,1] [2,2] 8 C2 e [1,2] {These (larger) component groups in the simply connected cover can be computed by a variant of Sommers' ideas.} atlas> set H= simply_connected(G2) Variable H: RootDatum (overriding previous instance, which had type string (constant)) atlas> show_nilpotent_orbits(H) i H diagram dim BC Levi Cent A(O) 0 [0,0] [0,0] 0 2T1 G2 [1] 1 [1,2] [0,1] 6 A1+T1 A1 [1] 2 [2,3] [1,0] 8 A1+T1 A1 [1] 3 [2,4] [0,2] 10 G2 e [1,2,3] 4 [6,10] [2,2] 12 G2 e [1] atlas> {fact: if G simple adjoint, A(O) is _either_ (Z/2Z)^m _or_ S_3, S_4, or S_5. } atlas> set rd=adjoint(B4) Variable rd: RootDatum (overriding previous instance, which had type RootDatum) atlas> show_nilpotent_orbits(rd) i H diagram dim BC Levi Cent A(O) 0 [0,0,0,0] [0,0,0,0] 0 4T1 B4 [1] 1 [0,1,0,0] [0,1,0,0] 12 A1+3T1 A1+C2 [1] 2 [2,0,0,0] [2,0,0,0] 14 A1+3T1 A3 [1,2] 3 [0,0,0,1] [0,0,0,1] 16 2A1+2T1 B2 [1] 4 [1,0,1,0] [1,0,1,0] 20 2A1+2T1 A1+T1 [1,2] 5 [0,2,0,0] [0,2,0,0] 22 A2+2T1 A1+T1 [1,2] 6 [2,2,0,0] [2,2,0,0] 24 B2+2T1 2A1 [1,2] 7 [0,0,2,0] [0,0,2,0] 24 A1+A2+T1 A1 [1] 8 [0,2,0,1] [0,2,0,1] 26 A3+T1 A1 [1] 9 [2,1,0,1] [2,1,0,1] 26 A1+B2+T1 A1 [1] 10 [2,0,2,0] [2,0,2,0] 28 B4 e [1,2,2,2] 11 [2,2,2,0] [2,2,2,0] 30 B3+T1 e [1,2] 12 [2,2,2,2] [2,2,2,2] 32 B4 e [1] atlas> {O = #10: A(O) = Z/2 x Z/2} atlas> {orbits for so(N) <---> partitions of N, even parts have even multiplicity} atlas> {N=9 = 5+3+1 (orbit #10)} atlas> {A(O) has ONE Z/2Z factor for each odd size in partition, less one (det 1)} atlas> {Component group in O(N) has _exactly_ one Z/2Z factor for each odd size in partition} atlas> set rd=simply_connected(A5) {SL(6): nilps <--> partitions of 6} Variable rd: RootDatum (overriding previous instance, which had type RootDatum) atlas> show_nilpotent_orbits(rd) i H diagram dim BC Levi Cent A(O) 0 [0,0,0,0,0] [0,0,0,0,0] 0 5T1 A5 [1] 1 [1,1,1,1,1] [1,0,0,0,1] 10 A1+4T1 A3+T1 [1] 2 [1,2,2,2,1] [0,1,0,1,0] 16 2A1+3T1 2A1+T1 [1] 3 [2,2,2,2,2] [2,0,0,0,2] 18 A2+3T1 A2+T1 [1] 4 [1,2,3,2,1] [0,0,2,0,0] 18 3A1+2T1 A2 [1,2] 5 [2,3,3,3,2] [1,1,0,1,1] 22 A1+A2+2T1 e [1] 6 [3,4,4,4,3] [2,1,0,1,2] 24 A3+2T1 A1+T1 [1] 7 [2,4,4,4,2] [0,2,0,2,0] 24 2A2+T1 A1 [1,3,3] 8 [3,4,5,4,3] [2,0,2,0,2] 26 A1+A3+T1 e [1,2] 9 [4,6,6,6,4] [2,2,0,2,2] 28 A4+T1 e [1] 10 [5,8,9,8,5] [2,2,2,2,2] 30 A5 e [1,2,3,3,6,6] atlas> {A(principal): Z/6Z (better: 6th roots of 1)} atlas> {orbit 7: one class order 1, 2 order 3} atlas> {# conj classes in Z/3Z)^m is 3^m. : A(orbit 7)= (sixth roots of 1)/(\pm 1)} atlas> {SL(n): A(O) _cyclic_, order = gcd(row sizes)} atlas> {orbit 10: partition 6} atlas> {orbit 7: partition 3+3} atlas> {4+2, 2+2+2}