G=real points of E6adjoint quaternionic (quasisplit compact inner class) G^v = E6 simply connected split G,G^v both connected Cells and representations are labelled as in output G=E6_ad_c_21 G^v=E6_sc_s_12 (not dualblock for E6_ad_c) quaternionic split special special orbit cells dualorbit cells diagram #O_R (see below) E6 0 (LDS) 0 (C) 31 000000 1 E6(a1) 3 A1 30 010000 D5 1,4 2A1 28,29 100001 E6(a3) 5,11 A2 21,26 020000 2 D5(a1) 6,7 A2+A1 24,25 110001 A4+A1 12,13 A2+2A1 19,20 001010 A4 8,14 A3 18,23 120001 1 D4 2,9 2A2 22,27 200002 1 D4(a1) 10,15,16 D4(a1) 13,14,17 000200 2 A3 19,21 A4 12,16 220002 1 A2+2A1 17,18,22 A4+A1 9,10,11 121011 2A2 23 D4 8 020200 1 A2+A1 24,25 D5(a1) 6,7 121011 A2 20,26,27 E6(a3) 4,5,15 200202 2 2A1 28,29 D5 2,3 220202 1 A1 30 E6(a1) 1 222022 1 0 (C) 31 E6 0 (LFS) 222222 1 LFS = large fundamental (not discrete) series #O_R is the number of real orbits of real points of E6_sc_split. Since this is connected, can be read off from Collingwood-McGovern. | abcdef means a-c-d-e-f the diagram gives the simple roots at which lambda is singular example: 220202 means lambda is singular at roots 3,5 Duality of Cells E6_ad_quaternionic E6_sc_split 31 0 30 1 29 2 28 3 27 4 26 5 24 (or 25?A) 6 25 (or 24?A) 7 23 8 22 9 17 (or 18?A) 10 18 (or 17?A) 11 21 12 16 13 15 14 20 15 19 16 10 17 14 18 13 (or 12?B) 19 12 (or 13?B) 20 11 21 9 22 8 23 6 (or 7?B) 24 7 (or 6?B) 25 5 26 2 27 4 28 1 29 3 30 0 31 A/B: matching of cells is determined by the "squash" graphs, except the four cells labelled A can all be switched (simultaneously), independently the 4 cells labelled B. A(lambda) cells computed using dualblocku and dualwcells cell aq 0: 0 1: 2,3 2: 7,10 3: 4: 1,4,6,8,13,14,16 5: 5,9,15 6: 7: 8: 50 9: 37 10: 11: 12: 51 13: 18,49,75 14: 48 15: 16: 17: 18: 19: 20: 21: 213 22: 231 23: 24: 25: 26: 432 27: 28: 29: 30: 31: 1863 ------------------------------------------------------------------- special special orbit cells dualorbit cells diagram #O_R (see below) E6 0 (LDS) 0 (C) 31 000000 1 cell stable sum 0 18 (LDS) %stable -d -c 31 -S 1,2,3,4,5,6 lambda is singular at simple roots: 1,2,3,4,5,6 cells:31 Parameters (living at lambda): 18 18( 18,981): 0 0 [i1,i1,i1,i1,i1,i1] 19 20 21 22 23 24 ( 86, *) ( 80, *) ( 72, *) ( 57, *) ( 55, *) ( 45, *) Dual parameters (to those living at lambda): 1863 1863(981, 18): 20 4 [r2,r2,r2,r2,r2,r2] 1864 1865 1866 1867 1868 1869 (1785, *) (1799, *) (1811, *) (1816, *) (1834, *) (1844, *) 1,2,3,1,4,2,3,1,4,3,5,4,2,3,1,4,3,5,4,2,6,5,4,2,3,1,4,3,5,4,2,6,5,4,3,1 Dimension of space of stable characters: 1 Everything is stable ------------------------------------------------------------------- special special orbit cells dualorbit cells diagram #O_R (see below) E6(a3) 5,11 A2 21,26 020000 2 two cells each containing A(lambda), two real forms => two stable sums, one from each cell cells stable sums 5 705+1441 11 317+1667 one extra: 317+705 (for example) %stable -d -c 21 -S 1,3,4,5,6 lambda is singular at simple roots: 1,3,4,5,6 cells:21 Parameters (living at lambda): 317,1667 317( 317,914): 4 2 [C+,C-,C+,i1,C+,C+] 490 188 463 316 463 451 ( *, *) ( *, *) ( *, *) ( 405, *) ( *, *) ( *, *) 2,4,3,5,4,2 1667(1622,193): 14 4 [rn,C-,C+,rn,C+,rn] 1667 1569 1738 1667 1746 1667 ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) 1,2,4,2,3,1,5,4,2,3,4,5,6,5,4,2,3,1,4,3,5,4,2,6 Dual parameters (to those living at lambda): 1563,213 1563(914, 317): 16 2 [C-,C+,C-,r2,C-,C-] 1392 1692 1419 1562 1419 1431 ( *, *) ( *, *) ( *, *) (1475, *) ( *, *) ( *, *) 1,3,1,4,2,3,1,5,4,2,3,1,4,3,6,5,4,2,3,1,4,3,5,4,2,6,5,4,3,1 213(193,1622): 6 0 [ic,C+,C-,ic,C-,ic] 213 309 142 213 132 213 ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) 3,1,4,3,5,4,3,1,6,5,4,3 Dimension of space of stable characters: 1 Basis of stable characters expressed as sums of irreducibles 317,1667: 1 1 %stable -d -c 26 -S 1,3,4,5,6 lambda is singular at simple roots: 1,3,4,5,6 cells:26 Parameters (living at lambda): 705,1441 705( 705,773): 7 3 [C+,C-,C+,C+,C+,C+] 892 543 873 869 892 873 ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) 2,4,3,1,5,4,3,6,5,4,2 1441(1421,359): 11 1 [i1,C-,i1,i1,i1,i1] 1449 1307 1446 1448 1445 1447 (1560, *) ( *, *) (1543, *) (1540, *) (1536, *) (1535, *) 2,4,3,1,5,4,2,3,4,5,6,5,4,2,3,1,4,3,5,4,2 Dual parameters (to those living at lambda): 1175,432 1175(773, 705): 13 1 [C-,C+,C-,C-,C-,C-] 988 1337 1008 1011 988 1008 ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) 1,3,1,4,3,1,5,4,2,3,1,4,5,6,5,4,2,3,1,4,3,5,4,6,5 432(359,1421): 9 3 [r2,C+,r2,r2,r2,r2] 440 566 437 439 436 438 ( 318, *) ( *, *) ( 337, *) ( 340, *) ( 342, *) ( 347, *) 1,3,1,4,3,1,5,4,3,1,6,5,4,3,1 Dimension of space of stable characters: 1 Basis of stable characters expressed as sums of irreducibles 705,1441: 1 1 %stable -d -c 21,26 -S 1,3,4,5,6 lambda is singular at simple roots: 1,3,4,5,6 cells:21,26 Parameters (living at lambda): 317,705,1441,1667 317( 317,914): 4 2 [C+,C-,C+,i1,C+,C+] 490 188 463 316 463 451 ( *, *) ( *, *) ( *, *) ( 405, *) ( *, *) ( *, *) 2,4,3,5,4,2 705( 705,773): 7 3 [C+,C-,C+,C+,C+,C+] 892 543 873 869 892 873 ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) 2,4,3,1,5,4,3,6,5,4,2 1441(1421,359): 11 1 [i1,C-,i1,i1,i1,i1] 1449 1307 1446 1448 1445 1447 (1560, *) ( *, *) (1543, *) (1540, *) (1536, *) (1535, *) 2,4,3,1,5,4,2,3,4,5,6,5,4,2,3,1,4,3,5,4,2 1667(1622,193): 14 4 [rn,C-,C+,rn,C+,rn] 1667 1569 1738 1667 1746 1667 ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) 1,2,4,2,3,1,5,4,2,3,4,5,6,5,4,2,3,1,4,3,5,4,2,6 Dual parameters (to those living at lambda): 1563,1175,432,213 1563(914, 317): 16 2 [C-,C+,C-,r2,C-,C-] 1392 1692 1419 1562 1419 1431 ( *, *) ( *, *) ( *, *) (1475, *) ( *, *) ( *, *) 1,3,1,4,2,3,1,5,4,2,3,1,4,3,6,5,4,2,3,1,4,3,5,4,2,6,5,4,3,1 1175(773, 705): 13 1 [C-,C+,C-,C-,C-,C-] 988 1337 1008 1011 988 1008 ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) 1,3,1,4,3,1,5,4,2,3,1,4,5,6,5,4,2,3,1,4,3,5,4,6,5 432(359,1421): 9 3 [r2,C+,r2,r2,r2,r2] 440 566 437 439 436 438 ( 318, *) ( *, *) ( 337, *) ( 340, *) ( 342, *) ( 347, *) 1,3,1,4,3,1,5,4,3,1,6,5,4,3,1 213(193,1622): 6 0 [ic,C+,C-,ic,C-,ic] 213 309 142 213 132 213 ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) 3,1,4,3,5,4,3,1,6,5,4,3 Dimension of space of stable characters: 3 Basis of stable characters expressed as sums of irreducibles 317,705,1441,1667: 1 0 0 1 -1 0 1 0 1 1 0 0 ------------------------------------------------------------------- special special orbit cells dualorbit cells diagram #O_R (see below) D4 2,9 2A2 22,27 200002 1 cells stable sums 2,9 523+1649 %stable -d -c 22,27 -S 2,3,4,5 lambda is singular at simple roots: 2,3,4,5 cells:22,27 Parameters (living at lambda): 523,1649 523( 523,851): 5 1 [C-,C+,i1,i1,i1,C-] 351 685 526 527 520 379 ( *, *) ( *, *) ( 635, *) ( 632, *) ( 627, *) ( *, *) 1,3,4,5,6,5,4,3,1 1649(1609,204): 13 2 [C-,i1,C+,i1,C+,C-] 1526 1650 1728 1648 1728 1552 ( *, *) (1708, *) ( *, *) (1698, *) ( *, *) ( *, *) 1,3,4,2,5,4,3,1,6,5,4,2,3,1,4,3,5,4,2,6,5,4,3,1 Dual parameters (to those living at lambda): 1354,231 1354(851, 523): 15 3 [C+,C-,r2,r2,r2,C+] 1524 1194 1357 1358 1351 1492 ( *, *) ( *, *) (1245, *) (1248, *) (1255, *) ( *, *) 2,3,4,2,3,1,4,5,4,2,3,1,4,3,5,4,6,5,4,2,3,1,4,3,5,4,2 231(204,1609): 7 2 [C+,r2,C-,r2,C-,C+] 356 232 150 230 150 328 ( *, *) ( 172, *) ( *, *) ( 182, *) ( *, *) ( *, *) 2,3,4,2,3,4,5,4,2,3,4,5 Dimension of space of stable characters: 1 Basis of stable characters expressed as sums of irreducibles 523,1649: 1 1 ------------------------------------------------------------------- special special orbit cells dualorbit cells diagram #O_R (see below) D4(a1) 10,15,16 D4(a1) 13,14,17 000200 2 Hard Case (too many stable sums) stability information seems not to determine packets and associated varieties cell dual 16 13 15 14 10 17 cells 13,14 have A(lambda), cell 17 doesn't AV(cell 13) = real form #1 of D4(a1) AV(cell 14) = real form #2 of D4(a1) AV(cell 17) = all cases allowed Three possibilities: 1) AV(cell 17) = AV(cell 13) packets: real form #1 = AV(cell 13) -> stable sum on cells 10,16 real form #2 = AV(cell 14) -> stable sum on cells 15 extra stable sum on cells 16 extra stable sum on cells 10,15 2) AV(cell 17) = AV(cell 14) packets: real form #1 = AV(cell 13) -> stable sum on cells 16 real form #2 = AV(cell 14) -> stable sum on cells 10,15 extra stable sum on cells 15 extra stable sum on cells 10,15 3) AV(cell 17) = both real forms real form #1 = AV(cell 13) -> stable sum on cells 10,16 real form #2 = AV(cell 14) -> stable sum on cells 10,15 extra stable sum on cell 15 extra stable sum on cell 16 %stable -d -c 13,14,17 -S 1,2,3,5,6 -a Cells Dimension of space of stable sums 16 1 15 1 10 0 15,16 2 10,16 1 10,15 2 10,15,16 4 %stable -d -c 13,14,17 -S 1,2,3,5,6 lambda is singular at simple roots: 1,2,3,5,6 cells:13,14,17 Parameters (living at lambda): 509,1080,1183,1396,1583,1805,1830 509( 509,853): 5 1 [C+,i1,i1,C-,i1,C+] 680 505 502 373 501 667 ( *, *) ( 600, *) ( 584, *) ( *, *) ( 572, *) ( *, *) 4,2,3,4,5,4,2,3,4 1080(1074,567): 9 2 [C+,i1,C+,C-,C+,C+] 1279 1082 1244 945 1240 1279 ( *, *) (1196, *) ( *, *) ( *, *) ( *, *) ( *, *) 4,2,3,1,4,3,5,4,3,1,6,5,4,2,3,4 1183(1171,527): 10 3 [C+,C+,C+,C-,C+,C+] 1357 1340 1332 1045 1329 1330 ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) 4,2,3,1,4,5,4,2,3,1,4,6,5,4,2,3,4 1396(1376,374): 11 2 [i1,C+,C+,C-,C+,i1] 1395 1540 1527 1266 1521 1395 (1492, *) ( *, *) ( *, *) ( *, *) ( *, *) (1467, *) 4,2,3,1,5,4,2,3,4,5,6,5,4,2,3,1,4,3,5,4 1583(1543,257): 13 3 [C+,C+,C+,C-,C+,C+] 1696 1685 1679 1484 1679 1676 ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) 4,2,3,1,4,5,4,2,3,1,4,3,5,6,5,4,2,3,1,4,3,5,4 1805(1735, 73): 16 3 [C+,i2,C+,C-,C+,C+] 1846 1805 1842 1760 1846 1842 ( *, *) (1831,1832) ( *, *) ( *, *) ( *, *) ( *, *) 4,2,3,1,4,3,5,4,2,3,1,4,3,5,4,6,5,4,2,3,1,4,3,5,4,2,6,5,4 1830(1760, 48): 17 4 [C+,rn,C+,C-,C+,C+] 1858 1830 1856 1793 1858 1856 ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) 2,4,2,3,1,4,3,5,4,2,3,1,4,3,5,4,6,5,4,2,3,1,4,3,5,4,2,6,5,4 Dual parameters (to those living at lambda): 1380,798,697,484,297,75,48 1380(853, 509): 15 3 [C-,r2,r2,C+,r2,C-] 1209 1376 1373 1506 1372 1216 ( *, *) (1282, *) (1296, *) ( *, *) (1308, *) ( *, *) 1,2,3,1,4,2,5,4,2,3,1,6,5,4,2,3,1,4,3,5,4,2,6,5,4,3,1 798(567,1074): 11 2 [C-,r2,C-,C+,C-,C-] 599 800 636 935 638 599 ( *, *) ( 684, *) ( *, *) ( *, *) ( *, *) ( *, *) 1,2,3,1,5,4,2,3,4,5,6,5,4,2,3,1,4,3,5,6 697(527,1171): 10 1 [C-,C-,C-,C+,C-,C-] 523 540 548 835 551 550 ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) 1,2,3,1,4,5,4,2,3,6,5,4,2,3,1,4,3,5,6 484(374,1376): 9 2 [r2,C-,C-,C+,C-,r2] 483 340 351 616 357 483 ( 388, *) ( *, *) ( *, *) ( *, *) ( *, *) ( 413, *) 1,2,3,1,4,3,5,4,3,1,6,5,4,2,3,1 297(257,1543): 7 1 [C-,C-,C-,C+,C-,C-] 184 195 201 396 201 204 ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) 1,2,3,1,4,5,4,6,5,4,2,3,1 75( 73,1735): 4 1 [C-,r1,C-,C+,C-,C-] 34 75 38 120 34 38 ( *, *) ( 49, 50) ( *, *) ( *, *) ( *, *) ( *, *) 1,2,3,1,5,6,5 48( 48,1760): 3 0 [C-,ic,C-,C+,C-,C-] 20 48 24 85 20 24 ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) 1,3,1,5,6,5 Dimension of space of stable characters: 4 Basis of stable characters expressed as sums of irreducibles 509,1080,1183,1396,1583,1805,1830: 1 0 0 -1 0 0 1 -1 -1 0 0 0 1 0 0 1 0 1 1 0 0 1 1 1 0 0 0 0 ------------------------------------------------------------------- special special orbit cells dualorbit cells diagram #O_R (see below) A3 19,21 A4 12,16 220002 1 cells stable sum 19,21 17)8+1827 %stable -d -c 12,16 -S 3,4,5 lambda is singular at simple roots: 3,4,5 cells:12,16 Parameters (living at lambda): 1708,1827 1708(1658,152): 14 3 [C-,r1,C+,C+,C+,C-] 1594 1708 1776 1774 1776 1620 ( *, *) (1649,1650) ( *, *) ( *, *) ( *, *) ( *, *) 1,2,3,4,2,5,4,3,1,6,5,4,2,3,1,4,3,5,4,2,6,5,4,3,1 1827(1757, 51): 16 2 [C-,C-,i1,i1,i1,C-] 1783 1784 1829 1828 1829 1789 ( *, *) ( *, *) (1853, *) (1851, *) (1850, *) ( *, *) 1,2,3,4,2,3,1,5,4,2,3,1,4,5,6,5,4,2,3,1,4,3,5,4,2,6,5,4,3,1 Dual parameters (to those living at lambda): 172,51 172(152,1658): 6 1 [C+,i2,C-,C-,C-,C+] 286 172 104 106 104 260 ( *, *) ( 231, 232) ( *, *) ( *, *) ( *, *) ( *, *) 3,4,2,3,4,5,4,2,3,4,5 51( 51,1757): 4 2 [C+,C+,r2,r2,r2,C+] 99 94 53 52 53 93 ( *, *) ( *, *) ( 27, *) ( 29, *) ( 30, *) ( *, *) 3,4,3,5,4,3 Dimension of space of stable characters: 1 Basis of stable characters expressed as sums of irreducibles 1708,1827: 1 1 ------------------------------------------------------------------- special special orbit cells dualorbit cells diagram #O_R (see below) 2A2 23 D4 8 020200 1 cell stable sum 23 1832 %stable -d -c 8 -S 1,3,5,6 lambda is singular at simple roots: 1,3,5,6 cells:8 Parameters (living at lambda): 1832 1832(1760, 50): 17 4 [C+,r2,C+,C-,C+,C+] 1860 1831 1857 1795 1860 1857 ( *, *) (1805, *) ( *, *) ( *, *) ( *, *) ( *, *) 2,4,2,3,1,4,3,5,4,2,3,1,4,3,5,4,6,5,4,2,3,1,4,3,5,4,2,6,5,4 Dual parameters (to those living at lambda): 50 50( 50,1760): 3 0 [C-,i1,C-,C+,C-,C-] 22 49 25 87 22 25 ( *, *) ( 75, *) ( *, *) ( *, *) ( *, *) ( *, *) 1,3,1,5,6,5 Dimension of space of stable characters: 1 Basis of stable characters expressed as sums of irreducibles 1832: 1 ------------------------------------------------------------------- special special orbit cells dualorbit cells diagram #O_R (see below) A2 20,26,27 E6(a3) 4,5,15 200202 2 cells 4,5 (duals 26,27) have A(lambda), 15 (dual 20) does not 4 dual 27 5 dual 26 15 dual 20 AV(cell 26) = real form #1 of E6(a3) AV(cell 27) = real form #2 of E6(a3) AV(cell 20) = AV(cell 26) = real form #1 OR AV(cell 26)U AV(cell 27) = both real forms Summary: All combinations of cells with non-zero stable sums Cells Dimension of space of stable sums 4 1 5 0 15 0 4,15 1 5,15 1 4,5 1 4,5,15 3 AV(cell 20) = AV(cell 27) disallowed: then dual to cell 26 (i.e. 5) would support a stable sum, but it doesn't Two possibilities: 1) AV(cell 20) = AV(cell 26) = real form #1 packets: real form #1 = AV(cell 26) -> stable sum on cells 5,15 real form #2 = AV(cell 27) -> stable sum on cell 4 third stable sum on cells 4,5,15 is extra 2) AV(cell 20) = AV(cell 26) U AV(cell 27) = both real forms real form #1 = AV(cell 26) -> stable sum on cells 5,15 real form #2 = AV(cell 27) -> stable sum on cell 4,15 stable sum on cell 4 alone is extra %stable -d -c 4,5 -S 2,3,5 lambda is singular at simple roots: 2,3,5 cells:4,5 Parameters (living at lambda): 1786,1816,1866,1873 1786(1726, 90): 15 2 [C-,C+,i1,C-,i1,C-] 1716 1829 1784 1730 1784 1726 ( *, *) ( *, *) (1818, *) ( *, *) (1815, *) ( *, *) 1,3,4,2,3,1,5,4,2,3,1,4,6,5,4,2,3,1,4,3,5,4,2,6,5,4,3,1 1816(1746, 62): 16 3 [C-,C+,C+,C-,C+,C-] 1751 1851 1848 1776 1847 1766 ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) 1,3,4,2,3,1,4,5,4,2,3,1,4,6,5,4,2,3,1,4,3,5,4,2,6,5,4,3,1 1866(1782, 14): 18 3 [C-,i2,C+,C-,C+,C-] 1842 1866 1876 1852 1876 1846 ( *, *) (1874,1875) ( *, *) ( *, *) ( *, *) ( *, *) 1,3,4,2,3,1,4,3,5,4,2,3,1,4,3,5,4,6,5,4,2,3,1,4,3,5,4,2,6,5,4,3,1 1873(1787, 5): 19 4 [C-,rn,C+,C-,C+,C-] 1856 1873 1880 1863 1880 1858 ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) 1,2,3,4,2,3,1,4,3,5,4,2,3,1,4,3,5,4,6,5,4,2,3,1,4,3,5,4,2,6,5,4,3,1 Dual parameters (to those living at lambda): 96,64,14,5 96( 90,1726): 5 2 [C+,C-,r2,C+,r2,C+] 162 53 94 152 94 154 ( *, *) ( *, *) ( 62, *) ( *, *) ( 65, *) ( *, *) 2,3,4,3,5,4,2,3 64( 62,1746): 4 1 [C+,C-,C-,C+,C-,C+] 129 29 32 104 33 114 ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) 2,3,4,5,4,2,3 14( 14,1782): 2 1 [C+,r1,C-,C+,C-,C+] 38 14 4 28 4 34 ( *, *) ( 6, 7) ( *, *) ( *, *) ( *, *) ( *, *) 2,3,5 5( 5,1787): 1 0 [C+,ic,C-,C+,C-,C+] 24 5 2 19 2 20 ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) 3,5 Dimension of space of stable characters: 1 Basis of stable characters expressed as sums of irreducibles 1786,1816,1866,1873: 0 1 1 0 %stable -d -c 4,5,15 -S 2,3,5 lambda is singular at simple roots: 2,3,5 cells:4,5,15 Parameters (living at lambda): 845,1397,1786,1816,1866,1873 845( 845,715): 7 1 [C-,i1,C+,C-,C+,C-] 662 843 999 689 1014 675 ( *, *) ( 977, *) ( *, *) ( *, *) ( *, *) ( *, *) 1,4,2,3,4,5,6,5,4,2,3,1,4 1397(1377,374): 11 2 [ic,C+,C+,C-,C+,ic] 1397 1539 1529 1265 1522 1397 ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) 4,2,3,1,5,4,2,3,4,5,6,5,4,2,3,1,4,3,5,4 1786(1726, 90): 15 2 [C-,C+,i1,C-,i1,C-] 1716 1829 1784 1730 1784 1726 ( *, *) ( *, *) (1818, *) ( *, *) (1815, *) ( *, *) 1,3,4,2,3,1,5,4,2,3,1,4,6,5,4,2,3,1,4,3,5,4,2,6,5,4,3,1 1816(1746, 62): 16 3 [C-,C+,C+,C-,C+,C-] 1751 1851 1848 1776 1847 1766 ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) 1,3,4,2,3,1,4,5,4,2,3,1,4,6,5,4,2,3,1,4,3,5,4,2,6,5,4,3,1 1866(1782, 14): 18 3 [C-,i2,C+,C-,C+,C-] 1842 1866 1876 1852 1876 1846 ( *, *) (1874,1875) ( *, *) ( *, *) ( *, *) ( *, *) 1,3,4,2,3,1,4,3,5,4,2,3,1,4,3,5,4,6,5,4,2,3,1,4,3,5,4,2,6,5,4,3,1 1873(1787, 5): 19 4 [C-,rn,C+,C-,C+,C-] 1856 1873 1880 1863 1880 1858 ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) 1,2,3,4,2,3,1,4,3,5,4,2,3,1,4,3,5,4,6,5,4,2,3,1,4,3,5,4,2,6,5,4,3,1 Dual parameters (to those living at lambda): 1034,485,96,64,14,5 1034(715, 845): 13 3 [C+,r2,C-,C+,C-,C+] 1211 1032 880 1198 875 1204 ( *, *) ( 901, *) ( *, *) ( *, *) ( *, *) ( *, *) 2,3,1,4,2,5,4,2,3,1,4,3,5,4,6,5,4,2,3,1,4,3,5 485(374,1377): 9 2 [rn,C-,C-,C+,C-,rn] 485 339 353 615 358 485 ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) 1,2,3,1,4,3,5,4,3,1,6,5,4,2,3,1 96( 90,1726): 5 2 [C+,C-,r2,C+,r2,C+] 162 53 94 152 94 154 ( *, *) ( *, *) ( 62, *) ( *, *) ( 65, *) ( *, *) 2,3,4,3,5,4,2,3 64( 62,1746): 4 1 [C+,C-,C-,C+,C-,C+] 129 29 32 104 33 114 ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) 2,3,4,5,4,2,3 14( 14,1782): 2 1 [C+,r1,C-,C+,C-,C+] 38 14 4 28 4 34 ( *, *) ( 6, 7) ( *, *) ( *, *) ( *, *) ( *, *) 2,3,5 5( 5,1787): 1 0 [C+,ic,C-,C+,C-,C+] 24 5 2 19 2 20 ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) 3,5 Dimension of space of stable characters: 3 Basis of stable characters expressed as sums of irreducibles 845,1397,1786,1816,1866,1873: 1 1 1 0 0 1 -1 0 -1 0 1 0 1 0 1 1 0 0 ------------------------------------------------------------------- special special orbit cells dualorbit cells diagram #O_R (see below) 2A1 28,29 D5 2,3 220202 1 cells stable sum 28,29 1851+1875 %stable -d -c 2,3 -S 3,5 lambda is singular at simple roots: 3,5 cells:2,3 Parameters (living at lambda): 1851,1875 1851(1773, 29): 17 3 [C-,C-,C+,r1,C+,C-] 1810 1816 1869 1851 1868 1823 ( *, *) ( *, *) ( *, *) (1827,1828) ( *, *) ( *, *) 1,2,3,4,2,3,1,4,5,4,2,3,1,4,5,6,5,4,2,3,1,4,3,5,4,2,6,5,4,3,1 1875(1787, 7): 19 4 [C-,r2,C+,C-,C+,C-] 1857 1874 1878 1861 1878 1860 ( *, *) (1866, *) ( *, *) ( *, *) ( *, *) ( *, *) 1,2,3,4,2,3,1,4,3,5,4,2,3,1,4,3,5,4,6,5,4,2,3,1,4,3,5,4,2,6,5,4,3,1 Dual parameters (to those living at lambda): 29,7 29( 29,1773): 3 1 [C+,C+,C-,i2,C-,C+] 70 64 11 29 12 57 ( *, *) ( *, *) ( *, *) ( 51, 52) ( *, *) ( *, *) 3,4,5,4,3 7( 7,1787): 1 0 [C+,i1,C-,C+,C-,C+] 25 6 0 17 0 22 ( *, *) ( 14, *) ( *, *) ( *, *) ( *, *) ( *, *) 3,5 Dimension of space of stable characters: 1 Basis of stable characters expressed as sums of irreducibles 1851,1875: 1 1 ------------------------------------------------------------------- special special orbit cells dualorbit cells diagram #O_R (see below) A1 30 E6(a1) 1 222022 1 cells stable sums 1 1877 %stable -d -c 1 -S 4 lambda is singular at simple roots: 4 cells:1 Parameters (living at lambda): 1877 1877(1789, 3): 19 3 [C-,C-,C-,i2,C-,C-] 1865 1867 1868 1877 1869 1865 ( *, *) ( *, *) ( *, *) (1878,1880) ( *, *) ( *, *) 1,2,3,1,4,2,3,1,4,5,4,2,3,1,4,3,5,4,2,6,5,4,2,3,1,4,3,5,4,2,6,5,4,3,1 Dual parameters (to those living at lambda): 3 3( 3,1789): 1 1 [C+,C+,C+,r1,C+,C+] 15 13 12 3 11 15 ( *, *) ( *, *) ( *, *) ( 0, 2) ( *, *) ( *, *) 4 Dimension of space of stable characters: 1 Basis of stable characters expressed as sums of irreducibles 1877: 1 ------------------------------------------------------------------- special special orbit cells dualorbit cells diagram #O_R (see below) 0 (C) 31 E6 0 (LFS) 222222 1 cell stable sums 31 1878 (trivial representation)