atlas> show_lusztig_cells(Sp(4,R)) {first few calculations I did before the seminar} G=C2 #orbits: 3 orbit 0: B2 H=[ 4, 2 ] diagram=[2,2] normalized diagram=[[2,2]] orbit 1: B2 H=[ 2, 0 ] diagram=[2,0] normalized diagram=[[2,0]] orbit 2: B2 H=[ 0, 0 ] diagram=[0,0] normalized diagram=[[0,0]] O^v dim i rd_int M H_M v L sigma dim deg fdeg char [2,2] 8 0 C2 B2 [4,2] [0,0] C2 sigma=0 1 4 4 [1,-1,-1,1,1] [2,0] 6 0 C2 A1+T1 [0,2] [0,0] C2 sigma=2 2 1 1 [2,0,0,-2,0] [2,0] 6 1 C2 2A1 [0,-2] [-1,-1]/2 C2 sigma=3 1 2 1 [1,-1,1,1,-1] [0,0] 0 0 C2 2T1 [0,0] [0,0] C2 sigma=4 1 0 0 [1,1,1,1,1] atlas> atlas> atlas> show_lusztig_cells(SO(4,4)) G=D4 #orbits: 11 {these are the 11 SPECIAL orbits} orbit 0: D4 H=[ 6, 4, 2, 0 ] diagram=[2,2,2,2] normalized diagram=[[2,2,2,2]] orbit 1: D4 H=[ 4, 2, 2, 0 ] diagram=[2,0,2,2] normalized diagram=[[2,0,2,2]] orbit 2: D4 H=[ 3, 3, 1, 1 ] diagram=[0,2,0,2] normalized diagram=[[0,2,0,2]] orbit 3: D4 H=[ 3, 3, 1, -1 ] diagram=[0,2,2,0] normalized diagram=[[0,2,2,0]] orbit 4: D4 H=[ 4, 2, 0, 0 ] diagram=[2,2,0,0] normalized diagram=[[2,2,0,0]] orbit 5: D4 H=[ 2, 2, 0, 0 ] diagram=[0,2,0,0] normalized diagram=[[0,2,0,0]] orbit 6: D4 H=[ 1, 1, 1, 1 ] diagram=[0,0,0,2] normalized diagram=[[0,0,0,2]] orbit 7: D4 H=[ 1, 1, 1, -1 ] diagram=[0,0,2,0] normalized diagram=[[0,0,2,0]] orbit 8: D4 H=[ 2, 0, 0, 0 ] diagram=[2,0,0,0] normalized diagram=[[2,0,0,0]] orbit 9: D4 H=[ 1, 1, 0, 0 ] diagram=[0,1,0,0] normalized diagram=[[0,1,0,0]] orbit 10: D4 H=[ 0, 0, 0, 0 ] diagram=[0,0,0,0] normalized diagram=[[0,0,0,0]] O^v dim i rd_int M H_M v L sigma dim deg fdeg char [2,2,2,2] 24 0 D4 D4 [6,4,2,0] [0,0,0,0] D4 sigma=4 1 12 12 [1,1,1,-1,1,-1,1,-1,1,1,1,-1,-1] [2,2,2,2] 24 1 D4 D4 [6,4,2,0] [1,1,1,1]/2 D4 sigma=4 1 12 12 [1,1,1,-1,1,-1,1,-1,1,1,1,-1,-1] [2,0,2,2] 22 0 D4 D4 [4,2,2,0] [0,0,0,0] D4 sigma=1 4 7 7 [-4,0,-1,0,0,2,4,-2,0,0,1,0,0] [2,0,2,2] 22 1 D4 D4 [4,2,2,0] [1,1,1,1]/2 D4 sigma=1 4 7 7 [-4,0,-1,0,0,2,4,-2,0,0,1,0,0] [0,2,0,2] 20 0 D4 A3+T1 [3,1,-1,-3] [0,0,0,0] D4 sigma=10 3 6 6 [3,-1,0,1,-1,-1,3,-1,3,-1,0,-1,1] [0,2,2,0] 20 0 D4 A3+T1 [3,1,-1,3] [0,0,0,0] D4 sigma=9 3 6 6 [3,-1,0,1,-1,-1,3,-1,-1,3,0,1,-1] [2,2,0,0] 20 0 D4 A3+T1 [0,4,2,0] [0,0,0,0] D4 sigma=5 3 6 6 [3,-1,0,-1,3,-1,3,-1,-1,-1,0,1,1] [2,2,0,0] 20 1 D4 A3+T1 [0,4,2,0] [0,1,1,1]/2 A3+T1 sigma=5 3 6 6 [3,-1,0,-1,3,-1,3,-1,-1,-1,0,1,1] [0,2,0,0] 18 0 D4 A2+2T1 [2,0,-2,0] [0,0,0,0] D4 sigma=2 8 3 3 [-8,0,1,0,0,0,8,0,0,0,-1,0,0] [0,2,0,0] 18 1 D4 4A1 [0,-2,2,0] [0,0,1,1]/2 4A1 sigma=6 2 4 3 [2,2,-1,0,2,0,2,0,2,2,-1,0,0] [0,0,0,2] 12 0 D4 2A1+2T1 [1,-1,1,-1] [0,0,0,0] D4 sigma=12 3 2 2 [3,-1,0,-1,-1,1,3,1,3,-1,0,1,-1] [0,0,2,0] 12 0 D4 2A1+2T1 [1,-1,1,1] [0,0,0,0] D4 sigma=11 3 2 2 [3,-1,0,-1,-1,1,3,1,-1,3,0,-1,1] [2,0,0,0] 12 0 D4 2A1+2T1 [0,0,2,0] [0,0,0,0] D4 sigma=7 3 2 2 [3,-1,0,1,3,1,3,1,-1,-1,0,-1,-1] [2,0,0,0] 12 1 D4 2A1+2T1 [0,0,2,0] [0,0,1,1]/2 4A1 sigma=7 3 2 2 [3,-1,0,1,3,1,3,1,-1,-1,0,-1,-1] [0,1,0,0] 10 0 4A1 A1+3T1 [1,-1,0,0] [0,0,0,0] D4 sigma=3 4 1 1 [-4,0,-1,0,0,-2,4,2,0,0,1,0,0] [0,0,0,0] 0 0 D4 4T1 [0,0,0,0] [0,0,0,0] D4 sigma=8 1 0 0 [1,1,1,1,1,1,1,1,1,1,1,1,1] atlas> show_nilpotent_orbits(Sp(4,R)) 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 T1 [1,2] 3 [3,1] [2,2] 8 C2 e [1,2] atlas> show_nilpotent_orbits_long(Sp(4,R)) complex nilpotent orbits for connected split real group with Lie algebra 'sp(4,R)' i: orbit number H: semisimple element BC Levi: Bala-Carter Levi Cent: identity component of Cent(SL(2)) Z(Cent^0): order of center of derived group of id. comp. of Centralizer A(O): orders of conj. classes in component group of centralizer #RF(O): number of real forms of O C_2: conjugacy classes in Cent(SL(2))_0 with square 1 i H diagram dim BC Levi Cent Z C_2 A(O) #RF(O) 0 [0,0] [0,0] 0 2T1 B2 2 3 [1] 1 1 [1,0] [1,0] 4 A1+T1 A1 2 2 [1,2] 2 2 [1,1] [0,2] 6 A1+T1 T1 1 2 [1,2] 3 3 [3,1] [2,2] 8 C2 e 1 1 [1,2] 2 atlas> show_lusztig_cells(Sp(4,R)) G=C2 #orbits: 3 orbit 0: B2 H=[ 4, 2 ] diagram=[2,2] normalized diagram=[[2,2]] orbit 1: B2 H=[ 2, 0 ] diagram=[2,0] normalized diagram=[[2,0]] orbit 2: B2 H=[ 0, 0 ] diagram=[0,0] normalized diagram=[[0,0]] O^v dim i rd_int M H_M v L sigma dim deg fdeg char [2,2] 8 0 C2 B2 [4,2] [0,0] C2 sigma=0 1 4 4 [1,-1,-1,1,1] [2,0] 6 0 C2 A1+T1 [0,2] [0,0] C2 sigma=2 2 1 1 [2,0,0,-2,0] [2,0] 6 1 C2 2A1 [0,-2] [-1,-1]/2 C2 sigma=3 1 2 1 [1,-1,1,1,-1] [0,0] 0 0 C2 2T1 [0,0] [0,0] C2 sigma=4 1 0 0 [1,1,1,1,1] atlas> show(character_table(Sp(4,R))) Classes: i order class_size |cent| sgn(w) chi_ref(w)) name 0 1 1 8 1 2 [1+,1+] 1 2 2 4 -1 0 [2+] 2 2 2 4 -1 0 [1-,1+] 3 2 1 8 1 -2 [1-,1-] 4 4 2 4 1 0 [2-] Representations: i dim deg fdeg name 0 1 4 [][1,1] 1 1 2 [][2] 2 2 1 [1][1] 3 1 2 [1,1][] 4 1 0 [2][] atlas> show_lusztig_cells(Sp(4,R)) G=C2 #orbits: 3 {THIS IS JUST SPECIAL ORBITS} orbit 0: B2 H=[ 4, 2 ] diagram=[2,2] normalized diagram=[[2,2]] orbit 1: B2 H=[ 2, 0 ] diagram=[2,0] normalized diagram=[[2,0]] orbit 2: B2 H=[ 0, 0 ] diagram=[0,0] normalized diagram=[[0,0]] O^v dim i rd_int M H_M v L sigma dim deg fdeg char [2,2] 8 0 C2 B2 [4,2] [0,0] C2 sigma=0 1 4 4 [1,-1,-1,1,1] [2,0] 6 0 C2 A1+T1 [0,2] [0,0] C2 sigma=2 2 1 1 [2,0,0,-2,0] [2,0] 6 1 C2 2A1 [0,-2] [-1,-1]/2 C2 sigma=3 1 2 1 [1,-1,1,1,-1] [0,0] 0 0 C2 2T1 [0,0] [0,0] C2 sigma=4 1 0 0 [1,1,1,1,1] atlas> show_lusztig_cells(F4_s) G=F4 #orbits: 11 {THIS IS JUST SPECIAL ORBITS} orbit 0: F4 H=[ 2, 2, 2, 2 ] diagram=[2,2,2,2] normalized diagram=[[2,2,2,2]] orbit 1: F4 H=[ 2, 0, 2, 2 ] diagram=[2,0,2,2] normalized diagram=[[2,2,0,2]] orbit 2: F4 H=[ 2, 0, 2, 0 ] diagram=[2,0,2,0] normalized diagram=[[0,2,0,2]] orbit 3: F4 H=[ 0, 0, 2, 2 ] diagram=[0,0,2,2] normalized diagram=[[2,2,0,0]] orbit 4: F4 H=[ 2, 1, 0, 1 ] diagram=[2,1,0,1] normalized diagram=[[1,0,1,2]] orbit 5: F4 H=[ 0, 0, 2, 0 ] diagram=[0,0,2,0] normalized diagram=[[0,2,0,0]] orbit 6: F4 H=[ 0, 0, 0, 2 ] diagram=[0,0,0,2] normalized diagram=[[2,0,0,0]] orbit 7: F4 H=[ 2, 0, 0, 0 ] diagram=[2,0,0,0] normalized diagram=[[0,0,0,2]] orbit 8: F4 H=[ 0, 0, 1, 0 ] diagram=[0,0,1,0] normalized diagram=[[0,1,0,0]] orbit 9: F4 H=[ 1, 0, 0, 0 ] diagram=[1,0,0,0] normalized diagram=[[0,0,0,1]] orbit 10: F4 H=[ 0, 0, 0, 0 ] diagram=[0,0,0,0] normalized diagram=[[0,0,0,0]] O^v dim i rd_int M H_M v L sigma dim deg fdeg char [2,2,2,2] 48 0 F4 F4 [2,2,2,2] [0,0,0,0] F4 sigma=3 1 24 24 [1,1,1,1,1,1,1,1,1,1,1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1] [2,0,2,2] 46 0 F4 B4 [-10,2,2,2] [-1,0,0,0]/2 A1+C3 sigma=7 2 16 13 [2,2,2,-1,-1,2,2,2,-1,-1,-1,0,0,0,0,0,-2,-2,1,1,-2,0,0,0,0] [2,0,2,2] 46 1 F4 F4 [2,0,2,2] [0,0,0,0] F4 sigma=19 4 13 13 [4,-4,0,1,-1,0,1,-1,-2,2,0,-2,2,1,-1,0,-2,2,1,-1,0,0,2,-2,0] [2,0,2,0] 44 0 F4 A1+C3 [2,2,2,-10] [0,0,0,-1]/2 B4 sigma=12 9 10 10 [9,9,1,0,0,-3,0,0,0,0,0,-3,-3,0,0,1,-3,-3,0,0,1,1,1,1,-1] [2,0,2,0] 44 1 F4 F4 [2,0,2,0] [0,0,0,0] F4 sigma=12 9 10 10 [9,9,1,0,0,-3,0,0,0,0,0,-3,-3,0,0,1,-3,-3,0,0,1,1,1,1,-1] [0,0,2,2] 42 0 F4 B3+T1 [-6,2,2,2] [0,0,0,0] F4 sigma=23 8 9 9 [8,-8,0,-1,1,0,2,-2,2,-2,0,0,0,0,0,0,-4,4,-1,1,0,0,0,0,0] [2,1,0,1] 42 0 A1+B3 C3+T1 [2,2,2,-9] [0,0,0,0] F4 sigma=21 8 9 9 [8,-8,0,2,-2,0,-1,1,2,-2,0,-4,4,-1,1,0,0,0,0,0,0,0,0,0,0] [0,0,2,0] 40 0 F4 2A2 [2,2,-6,2] [0,0,-1,0]/3 B3+T1 sigma=13 6 6 4 [6,6,-2,0,0,2,0,0,3,3,-1,0,0,0,0,0,0,0,0,0,0,2,-2,-2,0] [0,0,2,0] 40 1 F4 A1+A3 [2,2,-6,2] [3,0,-2,0]/4 A1+C2+T1 sigma=17 4 7 4 [4,-4,0,1,-1,0,1,-1,-2,2,0,2,-2,-1,1,0,-2,2,1,-1,0,0,-2,2,0] [0,0,2,0] 40 2 F4 A1+C3 [2,0,2,-6] [0,0,0,-1]/2 B4 sigma=24 16 5 4 [16,-16,0,-2,2,0,-2,2,-2,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0] [0,0,2,0] 40 3 F4 B4 [-4,0,2,0] [-1,0,0,0]/2 A1+C3 sigma=10 9 6 4 [9,9,1,0,0,-3,0,0,0,0,0,3,3,0,0,-1,-3,-3,0,0,1,-1,-1,-1,1] [0,0,2,0] 40 4 F4 F4 [0,0,2,0] [0,0,0,0] F4 sigma=15 12 4 4 [12,12,-4,0,0,4,0,0,-3,-3,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0] [0,0,0,2] 30 0 F4 A2+2T1 [0,-2,2,2] [0,0,0,0] F4 sigma=20 8 3 3 [8,-8,0,2,-2,0,-1,1,2,-2,0,4,-4,1,-1,0,0,0,0,0,0,0,0,0,0] [0,0,0,2] 30 1 F4 3A1+T1 [2,-2,2,-2] [0,0,0,-1]/2 B4 sigma=20 8 3 3 [8,-8,0,2,-2,0,-1,1,2,-2,0,4,-4,1,-1,0,0,0,0,0,0,0,0,0,0] [2,0,0,0] 30 0 F4 A2+2T1 [2,2,-4,0] [0,0,0,0] F4 sigma=22 8 3 3 [8,-8,0,-1,1,0,2,-2,2,-2,0,0,0,0,0,0,4,-4,1,-1,0,0,0,0,0] [0,0,1,0] 28 0 A1+B3 2A1+2T1 [2,-2,2,-1] [0,0,0,0] F4 sigma=9 9 2 2 [9,9,1,0,0,-3,0,0,0,0,0,3,3,0,0,-1,3,3,0,0,-1,1,1,1,-1] [1,0,0,0] 22 0 C4 A1+3T1 [2,-1,0,0] [0,0,0,0] F4 sigma=16 4 1 1 [4,-4,0,1,-1,0,1,-1,-2,2,0,2,-2,-1,1,0,2,-2,-1,1,0,0,2,-2,0] [1,0,0,0] 22 1 C4 2A1+2T1 [0,-1,2,-2] [0,0,0,-1]/2 B4 sigma=4 2 4 1 [2,2,2,2,2,2,-1,-1,-1,-1,-1,2,2,-1,-1,2,0,0,0,0,0,0,0,0,0] [0,0,0,0] 0 0 F4 4T1 [0,0,0,0] [0,0,0,0] F4 sigma=0 1 0 0 [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] atlas> set F4cells=W_cells_of(F4_s.trivial) Variable F4cells: [WCell] atlas> #F4cells Value: 25 atlas> {Ones corr to S_4 family are the ones containing W rep #15= special} atlas> set F4ct=F4_s.character_table Variable F4ct: CharacterTable atlas> set F4cellchars = for C in F4cells do cell_character(F4ct,C) od Variable F4cellchars: [[int]] atlas> view(F4ct,F4cellchars[13]) # mult dim deg fdeg 1 1 1 12 10 2 9 6 14 1 6 6 15 1 12 4 17 1 4 7 24 1 16 5 atlas> {for cell 13, get W reps for classes 1, g2, g2' (twice), g4 (once)} atlas> {has to be dihedral group of order 8, 2-Sylow in S_4} atlas> view(F4ct,F4cellchars[9]) # mult dim deg fdeg 8 1 4 8 10 1 9 6 11 1 9 6 14 1 6 6 15 1 12 4 24 2 16 5 atlas> {get Lusztig reps #15 = sigma(1,1), #10 = sigma(g2',1), } atlas> {#24 = sigma(g_2,1)} atlas> {CONCLUDE: S should be subgroup of S_4 having elts conj to} atlas> {1, (12) (two of these), (12)(34) (one of these)} atlas> { S = S_2 x S_2 } atlas>