[Davids-MBP-100:~/math/atlasofliegroupsMASTER/atlas-scripts] dav% ../atlas all This is 'atlas' (version 1.0.9, axis language version 1.0), the Atlas of Lie Groups and Representations interpreter, compiled on Dec 23 2020 at 16:01:25. http://www.liegroups.org/ atlas> set G=Sp(4,R) Variable G: RealForm atlas> show_sigma_L(G.character_table) Computing character table for simple root datum: C2 ...LOTS of output saying how the calculation is progressing... O i rd_int M H_M v L sigma_int sigma dim deg sgn deg_sgn char [0,0] 0 C2 2T1 [0,0] [0,0] C2 4 4 1 0 0 4 [1,1,1,1,1] [1,0] 0 2A1 A1+T1 [1,1] [0,0] 2A1 2 2 2 1 2 1 [2,0,0,-2,0] [1,0] 1 2A1 A1+T1 [1,1] [1,1]/2 2A1 2 2 2 1 2 1 [2,0,0,-2,0] [0,2] 0 C2 A1+T1 [1,0] [0,0] C2 2 2 2 1 2 1 [2,0,0,-2,0] [0,2] 1 C2 2A1 [-1,0] [-1,-1]/2 2A1 1 1 1 2 3 2 [1,1,-1,1,-1] [2,2] 0 C2 C2 [3,4] [0,0] C2 0 0 1 4 4 0 [1,-1,-1,1,1] [2,2] 1 C2 C2 [3,4] [1,2]/2 C2 0 0 1 4 4 0 [1,-1,-1,1,1] atlas> show_reps (G.character_table) Computing character table for simple root datum: C2 ...LITTLE output saying how the calculation is progressing... i dim degree 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> G.character_table.table Computing character table for simple root datum: C2 Updating degrees in character table Done updating degrees Done computing character table Value: | 1, -1, -1, 1, 1 | | 1, 1, -1, 1, -1 | | 2, 0, 0, -2, 0 | | 1, -1, 1, 1, -1 | | 1, 1, 1, 1, 1 | atlas> set G3=Sp(6,R) Variable G3: RealForm atlas> show_sigma_L(G3.character_table) Computing character table for simple root datum: C3 ...LOTS OF output saying how the calculation is progressing... O i rd_int M H_M v L sigma_int sigma dim deg sgn deg_sgn char [0,0,0] 0 C3 3T1 [0,0,0] [0,0,0] C3 9 9 1 0 0 9 [1,1,1,1,1,1,1,1,1,1] [1,0,0] 0 A1+C2 A1+2T1 [1,1,1] [0,0,0] A1+C2 8 6 3 1 3 4 [3,1,0,1,-1,-1,1,-3,-1,0] [1,0,0] 1 A1+C2 A1+2T1 [1,1,1] [1,1,1]/2 A1+C2 8 6 3 1 3 4 [3,1,0,1,-1,-1,1,-3,-1,0] [0,1,0] 0 A1+C2 A1+2T1 [1,0,0] [0,0,0] A1+C2 5 6 3 1 3 4 [3,1,0,1,-1,-1,1,-3,-1,0] [0,1,0] 1 A1+C2 2A1+T1 [-1,0,0] [-1,-1,-1]/2 3A1 3 4 3 2 5 3 [3,1,0,-1,1,-1,-1,3,-1,0] [0,0,2] 0 C3 2A1+T1 [1,0,1] [0,0,0] C3 4 4 3 2 5 3 [3,1,0,-1,1,-1,-1,3,-1,0] [0,0,2] 1 C3 2A1+T1 [1,0,1] [0,0,1]/2 A1+C2 4 4 3 2 5 3 [3,1,0,-1,1,-1,-1,3,-1,0] [0,2,0] 0 C3 A2+T1 [2,2,0] [0,0,0] C3 5 5 3 3 4 2 [3,-1,0,1,1,-1,-1,-3,1,0] [2,1,0] 0 A1+C2 C2+T1 [3,4,4] [0,0,0] A1+C2 1 3 3 4 6 1 [3,-1,0,-1,-1,-1,1,3,1,0] [2,1,0] 1 A1+C2 C2+T1 [3,4,4] [1,2,2]/2 A1+C2 1 3 3 4 6 1 [3,-1,0,-1,-1,-1,1,3,1,0] [2,0,2] 0 C3 A1+C2 [-1,2,3] [-2,-1,0]/2 A1+C2 1 1 2 5 8 2 [2,0,-1,-2,0,2,0,-2,0,1] [2,0,2] 1 C3 A1+C2 [-1,2,3] [-1,-1,-1]/2 A1+C2 1 1 2 5 8 2 [2,0,-1,-2,0,2,0,-2,0,1] [2,0,2] 2 C3 C3 [3,4,5] [0,0,0] C3 3 3 3 4 6 1 [3,-1,0,-1,-1,-1,1,3,1,0] [2,0,2] 3 C3 C3 [3,4,5] [1,2,3]/2 C3 3 3 3 4 6 1 [3,-1,0,-1,-1,-1,1,3,1,0] [2,2,2] 0 C3 C3 [5,8,9] [0,0,0] C3 0 0 1 9 9 0 [1,-1,1,-1,1,1,1,-1,-1,-1] [2,2,2] 1 C3 C3 [5,8,9] [1,2,3]/2 C3 0 0 1 9 9 0 [1,-1,1,-1,1,1,1,-1,-1,-1] atlas> show_reps (G3.character_table) Computing character table for simple root datum: C3 Updating degrees in character table Done updating degrees Done computing character table i dim degree name 0 1 9 [][1,1,1] 1 2 5 [][2,1] 2 1 3 [][3] 3 3 4 [1][1,1] 4 3 2 [1][2] 5 3 3 [1,1][1] 6 3 1 [2][1] 7 1 6 [1,1,1][] 8 2 2 [2,1][] 9 1 0 [3][]