Unipotent/Stability computation at Thu Dec 4 12:52:01 2008 Unipotent representations for G=F4 split G6v=G^v=F4 split Atlas version 0.3./Build date: Nov 19 2007 at 06:09:46. Special Special Orbit Cells Diagram #R A Dual Orbit Cells Diagram #R A F4 0 2222 1 1 0 24 0000 1 1 Parameters: 7* 7( 7,228): 0 0 [i1,i1,i1,i1] 8 9 11 10 ( 23, *) ( 21, *) ( 16, *) ( 12, *) e Dual Parameters: 331* 331(228, 7): 14 7 [r2,r2,r2,r2] 332 333 335 334 (321, *) (319, *) (314, *) (310, *) 4,3,4,2,3,4,2,3,2,1,2,3,4,2,3,2,1,2,3,4,2,3,2,1 Everything is stable Dimension of space of stable characters: 1 ---------------------------------------------------------------------------- Special Special Orbit Cells Diagram #R A Dual Orbit Cells Diagram #R A F4(a3) 9,13,14 0200 3 S4 F4(a3) 9,13,14 0020 3 S4 Parameters: 35,81,98,146,161,191,192,193,194,225,244,285*,295*,328* 35( 35,218): 2 1 [C+,i1,C-,C+] 58 34 18 53 ( *, *) ( 46, *) ( *, *) ( *, *) 3,2,3 81( 75,195): 4 2 [C+,i1,C-,i2] 120 79 55 81 ( *, *) (100, *) ( *, *) ( 94, 95) 3,2,3,4,3,2,3 98( 90,186): 5 4 [C+,C+,C-,C+] 137 130 72 129 ( *, *) ( *, *) ( *, *) ( *, *) 3,2,1,3,4,3,2,3 146(125,153): 6 1 [i1,C+,C-,i1] 148 179 115 148 (169, *) ( *, *) ( *, *) (160, *) 3,2,1,4,3,2,1,3,2,4,3 161(135,140): 7 4 [C+,C+,C-,C+] 203 196 132 190 ( *, *) ( *, *) ( *, *) ( *, *) 3,2,1,3,4,3,2,1,3,2,4,3 191(156,116): 8 3 [i1,C+,C-,C+] 193 229 164 221 (218, *) ( *, *) ( *, *) ( *, *) 3,2,1,3,2,3,4,3,2,1,3,2,4,3 192(156,117): 8 3 [i1,C+,C-,C+] 194 228 163 222 (219, *) ( *, *) ( *, *) ( *, *) 3,2,1,3,2,3,4,3,2,1,3,2,4,3 193(157,116): 8 3 [i1,C+,C-,C+] 191 231 166 223 (218, *) ( *, *) ( *, *) ( *, *) 3,2,1,3,2,3,4,3,2,1,3,2,4,3 194(157,117): 8 3 [i1,C+,C-,C+] 192 230 165 224 (219, *) ( *, *) ( *, *) ( *, *) 3,2,1,3,2,3,4,3,2,1,3,2,4,3 225(179, 95): 9 4 [C+,i2,C-,C+] 263 225 198 257 ( *, *) (245,246) ( *, *) ( *, *) 3,2,1,3,2,4,3,2,1,3,2,3,4,3,2,3 244(193, 81): 10 6 [C+,rn,C-,C+] 279 244 216 272 ( *, *) ( *, *) ( *, *) ( *, *) 2,3,2,1,3,2,4,3,2,1,3,2,3,4,3,2,3 285(211, 47): 11 4 [C+,C+,C-,i2] 309 308 262 285 ( *, *) ( *, *) ( *, *) (293,294) 3,2,1,3,2,3,4,3,2,1,3,2,3,4,3,2,1,3,2,3 295(217, 31): 12 5 [C+,C+,C-,rn] 314 311 275 295 ( *, *) ( *, *) ( *, *) ( *, *) 3,2,1,3,2,3,4,3,2,1,3,2,3,4,3,2,1,3,2,3,4 328(228, 4): 14 7 [rn,rn,r2,rn] 328 328 335 328 ( *, *) ( *, *) (319, *) ( *, *) 1,2,1,3,2,1,3,2,3,4,3,2,1,3,2,3,4,3,2,1,3,2,3,4 Dual Parameters: 299,252,234,183,168,133,135,134,136,103,87,49*,31*,4* 299(218, 35): 12 5 [C-,r2,C+,C-] 278 298 316 273 ( *, *) (289, *) ( *, *) ( *, *) 4,3,2,3,4,2,1,2,3,4,2,3,2,1,2,3,4,2,3,2,1 252(195, 75): 10 6 [C-,r2,C+,r1] 219 250 275 252 ( *, *) (233, *) ( *, *) (221,223) 4,2,1,2,3,4,2,3,2,1,2,3,4,2,3,2,1 234(186, 90): 9 4 [C-,C-,C+,C-] 203 205 262 190 ( *, *) ( *, *) ( *, *) ( *, *) 4,2,3,2,1,2,3,4,2,3,2,1,2,3,4,2 183(153,125): 8 5 [r2,C-,C+,r2] 185 156 214 185 (169, *) ( *, *) ( *, *) (160, *) 4,2,3,4,1,2,3,4,2,3,2,1,2 168(140,135): 7 4 [C-,C-,C+,C-] 137 138 198 129 ( *, *) ( *, *) ( *, *) ( *, *) 4,2,3,1,2,3,4,2,3,2,1,2 133(116,156): 6 3 [r2,C-,C+,C-] 134 106 165 94 (119, *) ( *, *) ( *, *) ( *, *) 4,2,3,2,1,2,3,4,2,1 135(117,156): 6 3 [r2,C-,C+,C-] 136 104 163 96 (120, *) ( *, *) ( *, *) ( *, *) 4,2,3,2,1,2,3,4,2,1 134(116,157): 6 3 [r2,C-,C+,C-] 133 107 166 95 (119, *) ( *, *) ( *, *) ( *, *) 4,2,3,2,1,2,3,4,2,1 136(117,157): 6 3 [r2,C-,C+,C-] 135 105 164 97 (120, *) ( *, *) ( *, *) ( *, *) 4,2,3,2,1,2,3,4,2,1 103( 95,179): 5 4 [C-,r1,C+,C-] 73 103 132 65 ( *, *) ( 88, 89) ( *, *) ( *, *) 4,2,1,2,3,4,2,1 87( 81,193): 4 2 [C-,ic,C+,C-] 59 87 117 52 ( *, *) ( *, *) ( *, *) ( *, *) 4,1,2,3,4,2,1 49( 47,211): 3 4 [C-,C-,C+,r1] 28 27 72 49 ( *, *) ( *, *) ( *, *) ( 29, 30) 4,2,1,2 31( 31,217): 2 1 [C-,C-,C+,ic] 16 13 55 31 ( *, *) ( *, *) ( *, *) ( *, *) 2,1,2 4( 4,228): 0 0 [ic,ic,i1,ic] 4 4 11 4 ( *, *) ( *, *) ( 21, *) ( *, *) e Dimension of space of stable characters: 9 Basis of stable characters as sums of irreducibles: 35-191+328* 35-146-191-192+295* 35-191-192+285* 146+191+244 146+191+192+225 81+192+194 -35-81+146+191+193 81-146+161 -35-81+98 Basis of stable characters as matrix of coefficients: 1,0,0,0,0,-1,0,0,0,0,0,0,0,1 1,0,0,-1,0,-1,-1,0,0,0,0,0,1,0 1,0,0,0,0,-1,-1,0,0,0,0,1,0,0 0,0,0,1,0,1,0,0,0,0,1,0,0,0 0,0,0,1,0,1,1,0,0,1,0,0,0,0 0,1,0,0,0,0,1,0,1,0,0,0,0,0 -1,-1,0,1,0,1,0,1,0,0,0,0,0,0 0,1,0,-1,1,0,0,0,0,0,0,0,0,0 -1,-1,1,0,0,0,0,0,0,0,0,0,0,0 All combinations of cells with non-zero stable sums Cells Dimension of space of stable sums 14 1 13 2 9 2 14,13 4 14,9 4 13,9 5 14,13,9 9 ---------------------------------------------------------------------------- Special Special Orbit Cells Diagram #R A Dual Orbit Cells Diagram #R A C3 12 1012 1 1 A2~ 18 2000 1 1 Parameters: 212* 212(173,111): 8 1 [C-,i1,i1,i1] 181 210 211 210 ( *, *) (238, *) (236, *) (235, *) 1,2,3,2,1,4,3,2,1,3,2,4,3,2,1 Dual Parameters: 127* 127(111,173): 6 5 [C+,r2,r2,r2] 158 125 126 125 ( *, *) (111, *) (109, *) (102, *) 4,3,4,2,3,4,2,3,2 Dimension of space of stable characters: 1 Basis of stable characters as sums of irreducibles: 212* Basis of stable characters as matrix of coefficients: 1 All combinations of cells with non-zero stable sums Cells Dimension of space of stable sums 18 1 ---------------------------------------------------------------------------- Special Special Orbit Cells Diagram #R A Dual Orbit Cells Diagram #R A B3 2,6,7,8 2200 2 1 A2 10,15,16,21 0002 3 2/1 Parameters: 67,207*,251*,324* 67( 65,198): 4 3 [C+,rn,C+,C-] 105 67 95 45 ( *, *) ( *, *) ( *, *) ( *, *) 2,4,3,2,3,4 207(168,110): 8 2 [i1,i1,i2,C-] 209 208 207 178 (233, *) (226, *) (221,222) ( *, *) 4,3,2,1,3,2,3,4,3,2,1,3,2,3,4 251(195, 74): 10 5 [rn,C+,rn,C-] 251 276 251 220 ( *, *) ( *, *) ( *, *) ( *, *) 1,3,4,3,2,1,3,2,3,4,3,2,1,3,2,3,4 324(228, 0): 14 7 [rn,rn,rn,r2] 324 324 324 325 ( *, *) ( *, *) ( *, *) (320, *) 1,2,1,3,2,1,3,2,3,4,3,2,1,3,2,3,4,3,2,1,3,2,3,4 Dual Parameters: 258,122*,80*,0* 258(198, 65): 10 3 [C-,ic,C-,C+] 230 258 223 290 ( *, *) ( *, *) ( *, *) ( *, *) 3,2,3,1,2,3,4,2,3,2,1,2,3,4,2,3,2,1 122(110,168): 6 6 [r2,r2,r1,C+] 124 123 122 155 (100, *) ( 99, *) ( 94, 96) ( *, *) 3,2,3,2,1,2,3,2,1 80( 74,195): 4 1 [ic,C-,ic,C+] 80 56 80 121 ( *, *) ( *, *) ( *, *) ( *, *) 2,3,2,1,2,3,2 0( 0,228): 0 0 [ic,ic,ic,i1] 0 0 0 1 ( *, *) ( *, *) ( *, *) ( 22, *) e Dimension of space of stable characters: 3 Basis of stable characters as sums of irreducibles: 67+324* 67+251* 67+207* Basis of stable characters as matrix of coefficients: 1,0,0,1 1,0,1,0 1,1,0,0 All combinations of cells with non-zero stable sums Cells Dimension of space of stable sums 15,10 1 16,10 1 21,16 1 16,15 1 21,15 1 21,10 1 21,16,10 2 16,15,10 2 21,16,15 2 21,15,10 2 21,16,15,10 3 ---------------------------------------------------------------------------- Special Special Orbit Cells Diagram #R A Dual Orbit Cells Diagram #R A A2 10,15,16,21 2000 3 2/1 B3 2,6,7,8 0022 2 1 Parameters: 149,257,293*,325* 149(128,153): 6 1 [i1,C+,C-,ic] 147 182 118 149 (171, *) ( *, *) ( *, *) ( *, *) 3,2,1,4,3,2,1,3,2,4,3 257(197, 63): 10 4 [C+,i2,C-,C-] 287 257 226 225 ( *, *) (273,274) ( *, *) ( *, *) 3,2,3,4,3,2,1,3,2,3,4,3,2,1,3,2,3,4 293(217, 29): 12 5 [C+,C+,C-,r2] 313 310 277 294 ( *, *) ( *, *) ( *, *) (285, *) 3,2,1,3,2,3,4,3,2,1,3,2,3,4,3,2,1,3,2,3,4 325(228, 1): 14 7 [rn,rn,r2,r2] 325 325 326 324 ( *, *) ( *, *) (316, *) (320, *) 1,2,1,3,2,1,3,2,3,4,3,2,1,3,2,3,4,3,2,1,3,2,3,4 Dual Parameters: 186,65,29*,1* 186(153,128): 8 5 [r2,C-,C+,rn] 184 159 217 186 (170, *) ( *, *) ( *, *) ( *, *) 4,2,3,4,1,2,3,4,2,3,2,1,2 65( 63,197): 4 4 [C-,r1,C+,C+] 43 65 99 103 ( *, *) ( 53, 54) ( *, *) ( *, *) 2,1,2,3,2,1 29( 29,217): 2 1 [C-,C-,C+,i1] 15 12 57 30 ( *, *) ( *, *) ( *, *) ( 49, *) 2,1,2 1( 1,228): 0 0 [ic,ic,i1,i1] 1 1 2 0 ( *, *) ( *, *) ( 18, *) ( 22, *) e Dimension of space of stable characters: 2 Basis of stable characters as sums of irreducibles: 325* 149+257+293* Basis of stable characters as matrix of coefficients: 0,0,0,1 1,1,1,0 All combinations of cells with non-zero stable sums Cells Dimension of space of stable sums 7,2 1 8,7,2 1 8,7,6 1 2 1 6,2 1 8,6,2 1 8,2 1 7,6,2 1 8,7,6,2 2 ---------------------------------------------------------------------------- Special Special Orbit Cells Diagram #R A Dual Orbit Cells Diagram #R A A1+A1~ 17,19,20 0100 2 1 F4(a2) 4,5,11 2020 2 2/1 Parameters: 267,309*,334* 267(204, 70): 10 3 [C-,i1,C-,C+] 231 265 240 290 ( *, *) (283, *) ( *, *) ( *, *) 1,2,3,2,1,3,2,3,4,3,2,1,3,2,4,3,2,1 309(223, 28): 12 4 [C-,i2,C-,i2] 285 309 292 309 ( *, *) (321,323) ( *, *) (313,315) 1,2,3,2,1,3,2,3,4,3,2,1,3,2,3,4,3,2,1,3,2,3 334(228, 10): 14 7 [r2,rn,r2,rn] 330 334 331 334 (311, *) ( *, *) (317, *) ( *, *) 1,2,1,3,2,1,3,2,3,4,3,2,1,3,2,3,4,3,2,1,3,2,3,4 Dual Parameters: 75,28*,10* 75( 70,204): 4 3 [C+,r2,C+,C-] 107 74 112 45 ( *, *) ( 63, *) ( *, *) ( *, *) 4,3,2,3,4,2 28( 28,223): 2 4 [C+,r1,C+,r1] 49 28 48 28 ( *, *) ( 23, 25) ( *, *) ( 15, 17) 4,2 10( 10,228): 0 0 [i1,ic,i1,ic] 6 10 7 10 ( 13, *) ( *, *) ( 19, *) ( *, *) e Dimension of space of stable characters: 2 Basis of stable characters as sums of irreducibles: 267+334* 267+309* Basis of stable characters as matrix of coefficients: 1,0,1 1,1,0 All combinations of cells with non-zero stable sums Cells Dimension of space of stable sums 11,4 1 5,4 1 11,5 1 11,5,4 2 ---------------------------------------------------------------------------- Special Special Orbit Cells Diagram #R A Dual Orbit Cells Diagram #R A A1~ 22,23 0001 2 2 F4(a1) 1,3 2022 2 2 Parameters: 290,299,313*,332* 290(215, 44): 11 3 [C-,i1,ic,C-] 261 288 290 267 ( *, *) (300, *) ( *, *) ( *, *) 1,2,3,2,1,4,3,2,1,3,2,3,4,3,2,1,3,2,3,4 299(218, 35): 12 6 [C-,C+,r2,C-] 273 318 296 278 ( *, *) ( *, *) (289, *) ( *, *) 1,2,3,2,1,3,4,3,2,1,3,2,3,4,3,2,1,3,2,3,4 313(225, 15): 13 5 [C-,i2,C-,r2] 293 313 300 315 ( *, *) (331,335) ( *, *) (309, *) 1,2,3,2,1,3,2,3,4,3,2,1,3,2,3,4,3,2,1,3,2,3,4 332(228, 8): 14 7 [r2,rn,r2,r2] 331 332 330 329 (310, *) ( *, *) (318, *) (322, *) 1,2,1,3,2,1,3,2,3,4,3,2,1,3,2,3,4,3,2,1,3,2,3,4 Dual Parameters: 45,35,15*,8* 45( 44,215): 3 3 [C+,r2,rn,C+] 70 44 45 75 ( *, *) ( 36, *) ( *, *) ( *, *) 3,2,3,2 35( 35,218): 2 2 [C+,C-,i1,C+] 53 20 32 58 ( *, *) ( *, *) ( 46, *) ( *, *) 2,3,2 15( 15,225): 1 1 [C+,r1,C+,i1] 29 15 36 17 ( *, *) ( 7, 11) ( *, *) ( 28, *) 2 8( 8,228): 0 0 [i1,ic,i1,i1] 7 8 6 5 ( 12, *) ( *, *) ( 20, *) ( 24, *) e Dimension of space of stable characters: 2 Basis of stable characters as sums of irreducibles: 299+332* 290+313* Basis of stable characters as matrix of coefficients: 0,1,0,1 1,0,1,0 All combinations of cells with non-zero stable sums Cells Dimension of space of stable sums 1 1 3 1 3,1 2 ---------------------------------------------------------------------------- Special Special Orbit Cells Diagram #R A Dual Orbit Cells Diagram #R A 0 24(trivial) 0000 1 1 F4 0 2222 1 1 Parameters: 331* 331(228, 7): 14 7 [r2,r2,r2,r2] 332 335 334 333 (310, *) (313, *) (317, *) (321, *) 1,2,1,3,2,1,3,2,3,4,3,2,1,3,2,3,4,3,2,1,3,2,3,4 Dual Parameters: 7* 7( 7,228): 0 0 [i1,i1,i1,i1] 8 11 10 9 ( 12, *) ( 15, *) ( 19, *) ( 23, *) e Dimension of space of stable characters: 1 Basis of stable characters as sums of irreducibles: 331* Basis of stable characters as matrix of coefficients: 1 ----------------------------------------------------------------------------