atlas> show_lusztig_cells(Sp(6,R)) G=C3 {This computes the Lusztig map x in A(O) to sigma(x,trivial for the DUAL group G^v. In order to get the computation for Sp(6), I should have plugged in SO(7). That's done below.} #orbits: 6 orbit 0: B3 H=[ 6, 4, 2 ] diagram=[2,2,2] normalized diagram=[[2,2,2]] orbit 1: B3 H=[ 4, 2, 0 ] diagram=[2,2,0] normalized diagram=[[2,2,0]] orbit 2: B3 H=[ 2, 2, 0 ] diagram=[0,2,0] normalized diagram=[[0,2,0]] orbit 3: B3 H=[ 2, 1, 1 ] diagram=[1,0,1] normalized diagram=[[1,0,1]] orbit 4: B3 H=[ 2, 0, 0 ] diagram=[2,0,0] normalized diagram=[[2,0,0]] orbit 5: B3 H=[ 0, 0, 0 ] diagram=[0,0,0] normalized diagram=[[0,0,0]] O^v dim i rd_int M H_M v L sigma dim deg fdeg char [2,2,2] 18 0 C3 B3 [6,4,2] [0,0,0] C3 sigma=0 1 9 9 [1,-1,1,-1,1,1,1,-1,-1,-1] [2,2,0] 16 0 C3 B2+T1 [0,4,2] [0,0,0] C3 sigma=3 3 4 4 [3,-1,0,-1,-1,-1,1,3,1,0] [2,2,0] 16 1 C3 A3 [0,-2,-4] [-1,-1,-1]/2 C3 sigma=7 1 6 4 [1,-1,1,1,-1,1,-1,1,-1,1] [0,2,0] 14 0 C3 A2+T1 [2,0,-2] [0,0,0] C3 sigma=5 3 3 3 [3,-1,0,1,1,-1,-1,-3,1,0] [0,2,0] 14 1 C3 3A1 [0,-2,2] [-1,-1,0]/2 A1+C2 sigma=5 3 3 3 [3,-1,0,1,1,-1,-1,-3,1,0] [1,0,1] 12 0 3A1 2A1+T1 [1,-1,2] [0,0,0] C3 sigma=4 3 2 2 [3,1,0,-1,1,-1,-1,3,-1,0] [2,0,0] 10 0 C3 A1+2T1 [0,0,2] [0,0,0] C3 sigma=6 3 1 1 [3,1,0,1,-1,-1,1,-3,-1,0] [2,0,0] 10 1 C3 2A1+T1 [0,-2,0] [-1,-1,0]/2 A1+C2 sigma=8 2 2 1 [2,0,-1,2,0,2,0,2,0,-1] [0,0,0] 0 0 C3 3T1 [0,0,0] [0,0,0] C3 sigma=9 1 0 0 [1,1,1,1,1,1,1,1,1,1] atlas> show_lusztig_cells(SO(4,3)) G=B3 {NOW we're running over special orbits in Sp(6,C)!} #orbits: 6 orbit 0: C3 H=[ 5, 3, 1 ] diagram=[2,2,2] normalized diagram=[[2,2,2]] orbit 1: C3 H=[ 3, 1, 1 ] diagram=[2,0,2] normalized diagram=[[2,0,2]] orbit 2: C3 H=[ 2, 2, 0 ] diagram=[0,2,0] normalized diagram=[[0,2,0]] orbit 3: C3 H=[ 1, 1, 1 ] diagram=[0,0,2] normalized diagram=[[0,0,2]] orbit 4: C3 H=[ 1, 1, 0 ] diagram=[0,1,0] normalized diagram=[[0,1,0]] orbit 5: C3 H=[ 0, 0, 0 ] diagram=[0,0,0] normalized diagram=[[0,0,0]] O^v dim i rd_int M H_M v L sigma dim deg fdeg char [2,2,2] 18 0 B3 C3 [5,3,1] [0,0,0] B3 sigma=0 1 9 9 [1,-1,1,-1,1,1,1,-1,-1,-1] [2,2,2] 18 1 B3 C3 [5,3,1] [1,1,1]/2 A3 sigma=0 1 9 9 [1,-1,1,-1,1,1,1,-1,-1,-1] [2,0,2] 16 0 B3 A1+C2 [-1,3,1] [0,1,1]/2 3A1 sigma=1 2 5 4 [2,0,-1,-2,0,2,0,-2,0,1] [2,0,2] 16 1 B3 A1+C2 [-1,3,1] [-1,0,0]/2 B2+T1 sigma=1 2 5 4 [2,0,-1,-2,0,2,0,-2,0,1] [2,0,2] 16 2 B3 C3 [3,1,1] [0,0,0] B3 sigma=3 3 4 4 [3,-1,0,-1,-1,-1,1,3,1,0] [2,0,2] 16 3 B3 C3 [3,1,1] [1,1,1]/2 A3 sigma=3 3 4 4 [3,-1,0,-1,-1,-1,1,3,1,0] [0,2,0] 14 0 B3 A2+T1 [2,0,-2] [0,0,0] B3 sigma=5 3 3 3 [3,-1,0,1,1,-1,-1,-3,1,0] [0,0,2] 12 0 B3 2A1+T1 [1,-1,1] [0,0,0] B3 sigma=4 3 2 2 [3,1,0,-1,1,-1,-1,3,-1,0] [0,0,2] 12 1 B3 2A1+T1 [1,-1,1] [0,0,1]/2 B2+T1 sigma=4 3 2 2 [3,1,0,-1,1,-1,-1,3,-1,0] [0,1,0] 10 0 A1+B2 A1+2T1 [1,-1,0] [0,0,0] B3 sigma=6 3 1 1 [3,1,0,1,-1,-1,1,-3,-1,0] [0,1,0] 10 1 A1+B2 2A1+T1 [-1,0,1] [0,0,1]/2 B2+T1 sigma=2 1 3 1 [1,1,1,-1,-1,1,-1,-1,1,-1] [0,0,0] 0 0 B3 3T1 [0,0,0] [0,0,0] B3 sigma=9 1 0 0 [1,1,1,1,1,1,1,1,1,1] atlas> atlas> atlas> atlas> set G=PSp(4,R) Variable G: RealForm (overriding previous instance, which had type RealForm) atlas> set ct=G.character_table Variable ct: CharacterTable atlas> set cells=W_cells_of(G.trivial) Variable cells: [WCell] {These are the six cells in the block of the trivial rep of PSp(4,R)} atlas> set chars = for C in cells do cell_character(ct,C) od Variable chars: [[int]] atlas> for char in chars do prints(new_line); view(ct,char) od # mult dim deg fdeg 4 1 1 0 # mult dim deg fdeg 2 1 2 1 3 1 1 2 # mult dim deg fdeg 2 1 2 1 3 1 1 2 # mult dim deg fdeg 2 1 2 1 3 1 1 2 # mult dim deg fdeg 0 1 1 4 # mult dim deg fdeg 0 1 1 4 Value: [(),(),(),(),(),()] atlas> set P=all_parameters_x_gamma(x_open(G),G.rho) {all reps attached to open KGB orbit, infl char rho} Variable P: [Param] (overriding previous instance, which had type [Param]) atlas> #P Value: 4 atlas> for i:4 do #block_of(P[i]) od Value: [12,12,12,12] atlas> {to get different cells, need infl char not rho, still integral} atlas> set P31=all_parameters_x_gamma(x_open(G),[3,1]) {all reps attached to open KGB orbit,[3,1]}} ^ syntax error, unexpected invalid token, expecting '\n' or ',' atlas> set P31=all_parameters_x_gamma(x_open(G),[3,1]) {all reps attached to open KGB orbit,[3,1]} Variable P31: [Param] atlas> for i:4 do #block_of(P31[i]) od Value: [4,1,4,1] atlas> set cells31=W_cells_of(P31[0]) Variable cells31: [WCell] atlas> atlas> set cellchars31 = for C in cells31 do cell_character(ct,C) od Variable cellchars31: [[int]] atlas> for char in cellchars31 do prints(new_line); view(ct,char) od # mult dim deg fdeg 4 1 1 0 # mult dim deg fdeg 1 1 1 2 2 1 2 1 atlas> show_orbits_and_reps (G) type C2 symbol large symbol rep Sp. orbit [[0,1,2],[1,2]] [[0,2,4],[2,4]] [][1,1] * [1,1,1,1] [[0,1],[2]] [[0,2],[3]] [][2] [] [[0,2],[1]] [[0,3],[2]] [1][1] * [2,2] [[1,2],[0]] [[1,3],[1]] [1,1][] [2,1,1] [[2],[]] [[2],[]] [2][] * [4] atlas> {family = all W reps for which multiset of integers in symbol is same} atlas> show_lusztig_cells_classical (G) support: [0,1,1,2,2] symbol rep [[0,1,2],[1,2]] [][1,1] support: [0,1,2] symbol rep [[0,1],[2]] [][2] [[0,2],[1]] [1][1] [[1,2],[0]] [1,1][] support: [2] symbol rep [[2],[]] [2][]