Dimensions of the irreducible representations of the exceptional simple Lie algebras

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Numbers

$\mathfrak g$

$\lambda$

$\dim V_{\mathfrak g}(\lambda)$

$G_2$

(1, 0):

$G_2$

(0, 1):

$G_2$

(2, 0):

$G_2$

(1, 1):

$G_2$

(0, 2):

$G_2$

(3, 0):

$G_2$

(4, 0):

182

$G_2$

(2, 1):

189

$G_2$

(0, 3):

273

$G_2$

(1, 2):

286

$G_2$

(5, 0):

378

$G_2$

(3, 1):

448

$G_2$

(6, 0):

714

$G_2$

(2, 2):

729

$G_2$

(0, 4):

748

$G_2$

(1, 3):

896

$G_2$

(4, 1):

924

$G_2$

(7, 0):

1254

$G_2$

(3, 2):

1547

$G_2$

(5, 1):

1728

$G_2$

(0, 5):

1729

$G_2$

(2, 3):

2079

$G_2$

(8, 0):

2079

$G_2$

(1, 4):

2261

$G_2$

(4, 2):

2926

$G_2$

(6, 1):

3003

$G_2$

(9, 0):

3289

$G_2$

(0, 6):

3542

$G_2$

(3, 3):

4096

$G_2$

(2, 4):

4914

$G_2$

(1, 5):

4928

$G_2$

(7, 1):

4928

$G_2$

(10, 0):

5005

$G_2$

(5, 2):

5103

$G_2$

(0, 7):

6630

$G_2$

(4, 3):

7293

$G_2$

(11, 0):

7371

$G_2$

(8, 1):

7722

$G_2$

(6, 2):

8372

$G_2$

(3, 4):

9177

$G_2$

(1, 6):

9660

$G_2$

(2, 5):

10206

$G_2$

(12, 0):

10556

$G_2$

(0, 8):

11571

$G_2$

(9, 1):

11648

$G_2$

(5, 3):

12096

$G_2$

(7, 2):

13090

$G_2$

(13, 0):

14756

$G_2$

(4, 4):

15625

$G_2$

(10, 1):

17017

$G_2$

(1, 7):

17472

$G_2$

(3, 5):

18304

$G_2$

(6, 3):

19019

$G_2$

(0, 9):

19096

$G_2$

(2, 6):

19278

$G_2$

(8, 2):

19683

$G_2$

(14, 0):

20196

$G_2$

(11, 1):

24192

$G_2$

(5, 4):

24948

$G_2$

(15, 0):

27132

$G_2$

(9, 2):

28652

$G_2$

(7, 3):

28672

$G_2$

(1, 8):

29667

$G_2$

(4, 5):

30107

$G_2$

(0, 10):

30107

$G_2$

(3, 6):

33495

$G_2$

(12, 1):

33592

$G_2$

(2, 7):

33858

$G_2$

(16, 0):

35853

$G_2$

(6, 4):

37961

$G_2$

(10, 2):

40579

$G_2$

(8, 3):

41769

$G_2$

(0, 11):

45695

$G_2$

(13, 1):

45696

$G_2$

(5, 5):

46656

$G_2$

(17, 0):

46683

$G_2$

(1, 9):

47872

$G_2$

(4, 6):

53599

$G_2$

(7, 4):

55614

$G_2$

(2, 8):

56133

$G_2$

(11, 2):

56133

$G_2$

(3, 7):

57344

$G_2$

(9, 3):

59136

$G_2$

(18, 0):

59983

$G_2$

(14, 1):

61047

$G_2$

(0, 12):

67158

$G_2$

(6, 5):

69160

$G_2$

(1, 10):

74074

$G_2$

(12, 2):

76076

$G_2$

(19, 0):

76153

$G_2$

(8, 4):

79002

$G_2$

(15, 1):

80256

$G_2$

(5, 6):

81081

$G_2$

(10, 3):

81719

$G_2$

(2, 9):

88803

$G_2$

(4, 7):

89726

$G_2$

(3, 8):

93093

$G_2$

(20, 0):

95634

$G_2$

(0, 13):

96019

$G_2$

(7, 5):

99008

$G_2$

(13, 2):

101269

$G_2$

(16, 1):

104006

$G_2$

(9, 4):

109375

$G_2$

(11, 3):

110592

$G_2$

(1, 11):

110656

$G_2$

(6, 6):

117649

$G_2$

(21, 0):

118910

$G_2$

(14, 2):

132678

$G_2$

(5, 7):

133056

$G_2$

(17, 1):

133056

$G_2$

(0, 14):

134044

$G_2$

(2, 10):

135135

$G_2$

(8, 5):

137781

$G_2$

(4, 8):

142968

$G_2$

(3, 9):

144704

$G_2$

(22, 0):

146510

$G_2$

(12, 3):

146965

$G_2$

(10, 4):

148148

$G_2$

(1, 12):

160433

$G_2$

(7, 6):

165242

$G_2$

(18, 1):

168245

$G_2$

(15, 2):

171380

$G_2$

(23, 0):

179010

$G_2$

(0, 15):

183260

$G_2$

(9, 5):

187264

$G_2$

(6, 7):

189658

$G_2$

(13, 3):

192192

$G_2$

(11, 4):

196911

$G_2$

(2, 11):

199017

$G_2$

(5, 8):

208494

$G_2$

(19, 1):

210496

$G_2$

(3, 10):

216931

$G_2$

(24, 0):

217035

$G_2$

(16, 2):

218569

$G_2$

(4, 9):

218750

$G_2$

(8, 6):

226044

$G_2$

(1, 13):

226688

$G_2$

(0, 16):

245973

$G_2$

(14, 3):

247779

$G_2$

(10, 5):

249458

$G_2$

(12, 4):

257439

$G_2$

(20, 1):

260820

$G_2$

(25, 0):

261261

$G_2$

(7, 7):

262144

$G_2$

(17, 2):

275562

$G_2$

(2, 12):

285012

$G_2$

(6, 8):

292740

$G_2$

(9, 6):

302498

$G_2$

(26, 0):

312417

$G_2$

(1, 14):

313208

$G_2$

(5, 9):

314496

$G_2$

(3, 11):

315392

$G_2$

(15, 3):

315392

$G_2$

(21, 1):

320320

$G_2$

(4, 10):

323532

$G_2$

(0, 17):

324786

$G_2$

(11, 5):

326592

$G_2$

(13, 4):

331702

$G_2$

(18, 2):

343805

$G_2$

(8, 7):

353430

$G_2$

(27, 0):

371287

$G_2$

(22, 1):

390195

$G_2$

(16, 3):

396865

$G_2$

(10, 6):

397320

$G_2$

(2, 13):

398412

$G_2$

(7, 8):

399126

$G_2$

(12, 5):

421135

$G_2$

(14, 4):

421875

$G_2$

(0, 18):

422617

$G_2$

(1, 15):

424320

$G_2$

(19, 2):

424879

$G_2$

(6, 9):

435897

$G_2$

(28, 0):

438712

$G_2$

(3, 12):

446641

$G_2$

(5, 10):

459459

$G_2$

(4, 11):

464899

$G_2$

(9, 7):

466752

$G_2$

(23, 1):

471744

$G_2$

(17, 3):

494208

$G_2$

(11, 6):

513513

$G_2$

(29, 0):

515592

$G_2$

(20, 2):

520506

$G_2$

(15, 4):

530348

$G_2$

(8, 8):

531441

$G_2$

(13, 5):

535808

$G_2$

(0, 19):

542717

$G_2$

(2, 14):

545292

$G_2$

(1, 16):

564927

$G_2$

(24, 1):

566370

$G_2$

(7, 9):

587328

$G_2$

(30, 0):

602888

$G_2$

(10, 7):

605682

$G_2$

(18, 3):

609615

$G_2$

(3, 13):

618240

$G_2$

(6, 10):

629706

$G_2$

(21, 2):

632555

$G_2$

(4, 12):

651651

$G_2$

(5, 11):

653184

$G_2$

(12, 6):

654381

$G_2$

(16, 4):

659736

$G_2$

(14, 5):

673596

$G_2$

(25, 1):

675584

$G_2$

(0, 20):

688688

$G_2$

(9, 8):

693861

$G_2$

(31, 0):

701624

$G_2$

(2, 15):

732564

$G_2$

(1, 17):

740544

$G_2$

(19, 3):

745472

$G_2$

(22, 2):

763048

$G_2$

(8, 9):

773604

$G_2$

(11, 7):

774144

$G_2$

(26, 1):

801009

$G_2$

(17, 4):

812889

$G_2$

(32, 0):

812889

$G_2$

(13, 6):

823543

$G_2$

(15, 5):

837760

$G_2$

(3, 14):

838831

$G_2$

(7, 10):

839762

$G_2$

(0, 21):

864501

$G_2$

(6, 11):

886445

$G_2$

(10, 8):

890967

$G_2$

(4, 13):

893893

$G_2$

(20, 3):

904365

$G_2$

(5, 12):

906984

$G_2$

(23, 2):

914166

$G_2$

(33, 0):

937839

$G_2$

(27, 1):

944384

$G_2$

(1, 18):

957334

$G_2$

(2, 16):

968031

$G_2$

(12, 7):

976430

$G_2$

(18, 4):

992902

$G_2$

(9, 9):

1000000

$F_4$

(0, 0, 0, 1):

$F_4$

(1, 0, 0, 0):

$F_4$

(0, 0, 1, 0):

273

$F_4$

(0, 0, 0, 2):

324

$F_4$

(1, 0, 0, 1):

1053

$F_4$

(2, 0, 0, 0):

1053

$F_4$

(0, 1, 0, 0):

1274

$F_4$

(0, 0, 0, 3):

2652

$F_4$

(0, 0, 1, 1):

4096

$F_4$

(1, 0, 1, 0):

8424

$F_4$

(1, 0, 0, 2):

10829

$F_4$

(3, 0, 0, 0):

12376

$F_4$

(0, 0, 0, 4):

16302

$F_4$

(2, 0, 0, 1):

17901

$F_4$

(0, 1, 0, 1):

19278

$F_4$

(0, 0, 2, 0):

19448

$F_4$

(1, 1, 0, 0):

29172

$F_4$

(0, 0, 1, 2):

34749

$F_4$

(1, 0, 0, 3):

76076

$F_4$

(0, 0, 0, 5):

81081

$F_4$

(4, 0, 0, 0):

100776

$F_4$

(1, 0, 1, 1):

106496

$F_4$

(0, 1, 1, 0):

107406

$F_4$

(2, 0, 1, 0):

119119

$F_4$

(0, 1, 0, 2):

160056

$F_4$

(2, 0, 0, 2):

160056

$F_4$

(3, 0, 0, 1):

184756

$F_4$

(0, 0, 2, 1):

205751

$F_4$

(0, 0, 1, 3):

212992

$F_4$

(0, 2, 0, 0):

226746

$F_4$

(2, 1, 0, 0):

340119

$F_4$

(0, 0, 0, 6):

342056

$F_4$

(1, 1, 0, 1):

379848

$F_4$

(1, 0, 0, 4):

412776

$F_4$

(1, 0, 2, 0):

420147

$F_4$

(5, 0, 0, 0):

627912

$F_4$

(0, 0, 3, 0):

629356

$F_4$

(1, 0, 1, 2):

787644

$F_4$

(0, 1, 0, 3):

952952

$F_4$

(2, 0, 0, 3):

1002456

$F_4$

(0, 0, 1, 4):

1042899

$F_4$

(3, 0, 1, 0):

1074944

$F_4$

(0, 1, 1, 1):

1118208

$F_4$

(0, 0, 0, 7):

1264120

$F_4$

(2, 0, 1, 1):

1327104

$F_4$

(0, 0, 2, 2):

1341522

$F_4$

(4, 0, 0, 1):

1360476

$F_4$

(3, 0, 0, 2):

1484406

$F_4$

(1, 1, 1, 0):

1801371

$F_4$

(1, 0, 0, 5):

1850212

$F_4$

(0, 2, 0, 1):

2488563

$F_4$

(3, 1, 0, 0):

2674763

$F_4$

(1, 1, 0, 2):

2792556

$F_4$

(6, 0, 0, 0):

3187041

$F_4$

(1, 2, 0, 0):

3195192

$F_4$

(0, 1, 2, 0):

3508596

$F_4$

(1, 0, 2, 1):

3921372

$F_4$

(2, 1, 0, 1):

3955952

$F_4$

(0, 0, 0, 8):

4188834

$F_4$

(1, 0, 1, 3):

4313088

$F_4$

(0, 0, 1, 5):

4313088

$F_4$

(0, 1, 0, 4):

4528953

$F_4$

(2, 0, 2, 0):

4582656

$F_4$

(2, 0, 0, 4):

4940676

$F_4$

(0, 0, 3, 1):

5218304

$F_4$

(0, 0, 2, 3):

6680856

$F_4$

(0, 1, 1, 2):

7113106

$F_4$

(4, 0, 1, 0):

7142499

$F_4$

(1, 0, 0, 6):

7147140

$F_4$

(5, 0, 0, 1):

7822737

$F_4$

(3, 0, 0, 3):

8498776

$F_4$

(2, 0, 1, 2):

8843094

$F_4$

(0, 2, 1, 0):

9683388

$E_6$

(0, 0, 0, 0, 0, 1):

$E_6$

(1, 0, 0, 0, 0, 0):

$E_6$

(0, 1, 0, 0, 0, 0):

$E_6$

(0, 0, 0, 0, 1, 0):

351

$E_6$

(0, 0, 1, 0, 0, 0):

351

$E_6$

(0, 0, 0, 0, 0, 2):

351

$E_6$

(2, 0, 0, 0, 0, 0):

351

$E_6$

(1, 0, 0, 0, 0, 1):

650

$E_6$

(0, 1, 0, 0, 0, 1):

1728

$E_6$

(1, 1, 0, 0, 0, 0):

1728

$E_6$

(0, 2, 0, 0, 0, 0):

2430

$E_6$

(0, 0, 0, 1, 0, 0):

2925

$E_6$

(0, 0, 0, 0, 0, 3):

3003

$E_6$

(3, 0, 0, 0, 0, 0):

3003

$E_6$

(0, 0, 0, 0, 1, 1):

5824

$E_6$

(1, 0, 1, 0, 0, 0):

5824

$E_6$

(0, 0, 1, 0, 0, 1):

7371

$E_6$

(1, 0, 0, 0, 1, 0):

7371

$E_6$

(1, 0, 0, 0, 0, 2):

7722

$E_6$

(2, 0, 0, 0, 0, 1):

7722

$E_6$

(0, 1, 0, 0, 1, 0):

17550

$E_6$

(0, 1, 1, 0, 0, 0):

17550

$E_6$

(0, 1, 0, 0, 0, 2):

19305

$E_6$

(2, 1, 0, 0, 0, 0):

19305

$E_6$

(0, 0, 0, 0, 0, 4):

19305

$E_6$

(4, 0, 0, 0, 0, 0):

19305

$E_6$

(0, 0, 0, 0, 2, 0):

34398

$E_6$

(0, 0, 2, 0, 0, 0):

34398

$E_6$

(1, 1, 0, 0, 0, 1):

34749

$E_6$

(0, 3, 0, 0, 0, 0):

43758

$E_6$

(0, 2, 0, 0, 0, 1):

46332

$E_6$

(1, 2, 0, 0, 0, 0):

46332

$E_6$

(0, 0, 0, 1, 0, 1):

51975

$E_6$

(1, 0, 0, 1, 0, 0):

51975

$E_6$

(0, 0, 0, 0, 1, 2):

54054

$E_6$

(2, 0, 1, 0, 0, 0):

54054

$E_6$

(1, 0, 0, 0, 0, 3):

61425

$E_6$

(3, 0, 0, 0, 0, 1):

61425

$E_6$

(0, 0, 1, 0, 1, 0):

70070

$E_6$

(0, 0, 1, 0, 0, 2):

78975

$E_6$

(2, 0, 0, 0, 1, 0):

78975

$E_6$

(2, 0, 0, 0, 0, 2):

85293

$E_6$

(0, 0, 0, 0, 0, 5):

100386

$E_6$

(5, 0, 0, 0, 0, 0):

100386

$E_6$

(0, 1, 0, 1, 0, 0):

105600

$E_6$

(1, 0, 0, 0, 1, 1):

112320

$E_6$

(1, 0, 1, 0, 0, 1):

112320

$E_6$

(0, 1, 0, 0, 0, 3):

146432

$E_6$

(3, 1, 0, 0, 0, 0):

146432

$E_6$

(0, 1, 0, 0, 1, 1):

252252

$E_6$

(1, 1, 1, 0, 0, 0):

252252

$E_6$

(0, 1, 1, 0, 0, 1):

314496

$E_6$

(1, 1, 0, 0, 1, 0):

314496

$E_6$

(0, 0, 0, 0, 1, 3):

359424

$E_6$

(1, 1, 0, 0, 0, 2):

359424

$E_6$

(2, 1, 0, 0, 0, 1):

359424

$E_6$

(3, 0, 1, 0, 0, 0):

359424

$E_6$

(1, 0, 0, 0, 0, 4):

371800

$E_6$

(4, 0, 0, 0, 0, 1):

371800

$E_6$

(0, 0, 0, 1, 1, 0):

386100

$E_6$

(0, 0, 1, 1, 0, 0):

386100

$E_6$

(0, 2, 0, 0, 1, 0):

393822

$E_6$

(0, 2, 1, 0, 0, 0):

393822

$E_6$

(0, 0, 0, 0, 2, 1):

412776

$E_6$

(1, 0, 2, 0, 0, 0):

412776

$E_6$

(0, 0, 0, 0, 0, 6):

442442

$E_6$

(6, 0, 0, 0, 0, 0):

442442

$E_6$

(0, 2, 0, 0, 0, 2):

459459

$E_6$

(2, 2, 0, 0, 0, 0):

459459

$E_6$

(0, 0, 0, 1, 0, 2):

494208

$E_6$

(2, 0, 0, 1, 0, 0):

494208

$E_6$

(0, 4, 0, 0, 0, 0):

537966

$E_6$

(0, 0, 1, 0, 0, 3):

579150

$E_6$

(3, 0, 0, 0, 1, 0):

579150

$E_6$

(0, 0, 2, 0, 0, 1):

600600

$E_6$

(1, 0, 0, 0, 2, 0):

600600

$E_6$

(2, 0, 0, 0, 0, 3):

638820

$E_6$

(3, 0, 0, 0, 0, 2):

638820

$E_6$

(0, 3, 0, 0, 0, 1):

741312

$E_6$

(1, 3, 0, 0, 0, 0):

741312

$E_6$

(1, 2, 0, 0, 0, 1):

812175

$E_6$

(1, 0, 0, 1, 0, 1):

852930

$E_6$

(0, 1, 0, 0, 0, 4):

853281

$E_6$

(4, 1, 0, 0, 0, 0):

853281

$E_6$

(0, 0, 1, 0, 1, 1):

967680

$E_6$

(1, 0, 1, 0, 1, 0):

967680

$E_6$

(1, 0, 0, 0, 1, 2):

972972

$E_6$

(2, 0, 1, 0, 0, 1):

972972

$E_6$

(1, 0, 1, 0, 0, 2):

1123200

$E_6$

(2, 0, 0, 0, 1, 1):

1123200

$E_6$

(0, 1, 0, 0, 2, 0):

1253070

$E_6$

(0, 1, 2, 0, 0, 0):

1253070

$E_6$

(0, 0, 0, 2, 0, 0):

1337050

$E_6$

(0, 0, 0, 0, 3, 0):

1559376

$E_6$

(0, 0, 3, 0, 0, 0):

1559376

$E_6$

(0, 1, 0, 1, 0, 1):

1640925

$E_6$

(1, 1, 0, 1, 0, 0):

1640925

$E_6$

(0, 0, 0, 0, 0, 7):

1706562

$E_6$

(7, 0, 0, 0, 0, 0):

1706562

$E_6$

(1, 0, 0, 0, 0, 5):

1837836

$E_6$

(5, 0, 0, 0, 0, 1):

1837836

$E_6$

(0, 0, 0, 0, 1, 4):

1896180

$E_6$

(4, 0, 1, 0, 0, 0):

1896180

$E_6$

(0, 2, 0, 1, 0, 0):

1911195

$E_6$

(0, 1, 0, 0, 1, 2):

2088450

$E_6$

(2, 1, 1, 0, 0, 0):

2088450

$E_6$

(0, 1, 1, 0, 1, 0):

2453814

$E_6$

(1, 1, 0, 0, 0, 3):

2559843

$E_6$

(3, 1, 0, 0, 0, 1):

2559843

$E_6$

(0, 1, 1, 0, 0, 2):

2977975

$E_6$

(2, 1, 0, 0, 1, 0):

2977975

$E_6$

(0, 0, 0, 0, 2, 2):

3007368

$E_6$

(2, 0, 2, 0, 0, 0):

3007368

$E_6$

(0, 2, 0, 0, 0, 3):

3162159

$E_6$

(3, 2, 0, 0, 0, 0):

3162159

$E_6$

(0, 0, 1, 0, 0, 4):

3281850

$E_6$

(4, 0, 0, 0, 1, 0):

3281850

$E_6$

(0, 0, 0, 1, 0, 3):

3309696

$E_6$

(3, 0, 0, 1, 0, 0):

3309696

$E_6$

(2, 1, 0, 0, 0, 2):

3490695

$E_6$

(2, 0, 0, 0, 0, 4):

3675672

$E_6$

(4, 0, 0, 0, 0, 2):

3675672

$E_6$

(0, 1, 0, 0, 0, 5):

4088448

$E_6$

(5, 1, 0, 0, 0, 0):

4088448

$E_6$

(1, 1, 0, 0, 1, 1):

4200768

$E_6$

(1, 1, 1, 0, 0, 1):

4200768

$E_6$

(3, 0, 0, 0, 0, 3):

4548180

$E_6$

(0, 0, 1, 0, 2, 0):

4582656

$E_6$

(0, 0, 2, 0, 1, 0):

4582656

$E_6$

(0, 0, 0, 1, 1, 1):

4752384

$E_6$

(1, 0, 1, 1, 0, 0):

4752384

$E_6$

(0, 5, 0, 0, 0, 0):

4969107

$E_6$

(0, 2, 0, 0, 1, 1):

5054400

$E_6$

(1, 2, 1, 0, 0, 0):

5054400

$E_6$

(0, 3, 0, 0, 1, 0):

5501925

$E_6$

(0, 3, 1, 0, 0, 0):

5501925

$E_6$

(0, 0, 2, 0, 0, 2):

5553900

$E_6$

(2, 0, 0, 0, 2, 0):

5553900

$E_6$

(0, 0, 1, 1, 0, 1):

5776056

$E_6$

(1, 0, 0, 1, 1, 0):

5776056

$E_6$

(0, 0, 0, 0, 0, 8):

5895396

$E_6$

(8, 0, 0, 0, 0, 0):

5895396

$E_6$

(1, 0, 0, 0, 1, 3):

6110208

$E_6$

(3, 0, 1, 0, 0, 1):

6110208

$E_6$

(0, 2, 1, 0, 0, 1):

6243237

$E_6$

(1, 2, 0, 0, 1, 0):

6243237

$E_6$

(0, 3, 0, 0, 0, 2):

6675669

$E_6$

(2, 3, 0, 0, 0, 0):

6675669

$E_6$

(1, 0, 0, 0, 2, 1):

6747300

$E_6$

(1, 0, 2, 0, 0, 1):

6747300

$E_6$

(1, 2, 0, 0, 0, 2):

7528950

$E_6$

(2, 2, 0, 0, 0, 1):

7528950

$E_6$

(1, 0, 0, 1, 0, 2):

7601958

$E_6$

(2, 0, 0, 1, 0, 1):

7601958

$E_6$

(0, 0, 1, 0, 1, 2):

7757100

$E_6$

(2, 0, 1, 0, 1, 0):

7757100

$E_6$

(1, 0, 0, 0, 0, 6):

7757100

$E_6$

(6, 0, 0, 0, 0, 1):

7757100

$E_6$

(1, 0, 1, 0, 0, 3):

7779200

$E_6$

(3, 0, 0, 0, 1, 1):

7779200

$E_6$

(0, 4, 0, 0, 0, 1):

8281845

$E_6$

(1, 4, 0, 0, 0, 0):

8281845

$E_6$

(0, 0, 0, 0, 1, 5):

8401536

$E_6$

(5, 0, 1, 0, 0, 0):

8401536

$E_6$

(2, 0, 0, 0, 1, 2):

9189180

$E_6$

(2, 0, 1, 0, 0, 2):

9189180

$E_7$

(0, 0, 0, 0, 0, 0, 1):

$E_7$

(1, 0, 0, 0, 0, 0, 0):

133

$E_7$

(0, 1, 0, 0, 0, 0, 0):

912

$E_7$

(0, 0, 0, 0, 0, 0, 2):

1463

$E_7$

(0, 0, 0, 0, 0, 1, 0):

1539

$E_7$

(1, 0, 0, 0, 0, 0, 1):

6480

$E_7$

(2, 0, 0, 0, 0, 0, 0):

7371

$E_7$

(0, 0, 1, 0, 0, 0, 0):

8645

$E_7$

(0, 0, 0, 0, 0, 0, 3):

24320

$E_7$

(0, 0, 0, 0, 1, 0, 0):

27664

$E_7$

(0, 1, 0, 0, 0, 0, 1):

40755

$E_7$

(0, 0, 0, 0, 0, 1, 1):

51072

$E_7$

(1, 1, 0, 0, 0, 0, 0):

86184

$E_7$

(1, 0, 0, 0, 0, 0, 2):

150822

$E_7$

(1, 0, 0, 0, 0, 1, 0):

152152

$E_7$

(3, 0, 0, 0, 0, 0, 0):

238602

$E_7$

(0, 2, 0, 0, 0, 0, 0):

253935

$E_7$

(0, 0, 0, 0, 0, 0, 4):

293930

$E_7$

(2, 0, 0, 0, 0, 0, 1):

320112

$E_7$

(0, 0, 1, 0, 0, 0, 1):

362880

$E_7$

(0, 0, 0, 1, 0, 0, 0):

365750

$E_7$

(1, 0, 1, 0, 0, 0, 0):

573440

$E_7$

(0, 0, 0, 0, 0, 2, 0):

617253

$E_7$

(0, 1, 0, 0, 0, 1, 0):

861840

$E_7$

(0, 1, 0, 0, 0, 0, 2):

885248

$E_7$

(0, 0, 0, 0, 0, 1, 2):

915705

$E_7$

(0, 0, 0, 0, 1, 0, 1):

980343

$E_7$

(1, 0, 0, 0, 0, 0, 3):

2273920

$E_7$

(1, 0, 0, 0, 1, 0, 0):

2282280

$E_7$

(0, 0, 0, 0, 0, 0, 5):

2785552

$E_7$

(1, 1, 0, 0, 0, 0, 1):

3424256

$E_7$

(2, 1, 0, 0, 0, 0, 0):

3635840

$E_7$

(0, 1, 1, 0, 0, 0, 0):

3792096

$E_7$

(1, 0, 0, 0, 0, 1, 1):

4522000

$E_7$

(4, 0, 0, 0, 0, 0, 0):

5248750

$E_7$

(2, 0, 0, 0, 0, 1, 0):

6619239

$E_7$

(2, 0, 0, 0, 0, 0, 2):

6760390

$E_7$

(0, 0, 1, 0, 0, 1, 0):

7142499

$E_7$

(0, 0, 1, 0, 0, 0, 2):

7482618

$E_7$

(3, 0, 0, 0, 0, 0, 1):

9405760

$E_7$

(0, 2, 0, 0, 0, 0, 1):

9480240

$E_7$

(0, 1, 0, 0, 1, 0, 0):

11316305

$E_7$

(0, 0, 0, 0, 0, 1, 3):

11376288

$E_7$

(0, 1, 0, 0, 0, 0, 3):

12609597

$E_7$

(0, 0, 0, 1, 0, 0, 1):

13069056

$E_7$

(0, 0, 2, 0, 0, 0, 0):

13728792

$E_7$

(0, 0, 0, 0, 0, 2, 1):

14220360

$E_7$

(0, 0, 0, 0, 1, 1, 0):

14910896

$E_7$

(0, 0, 0, 0, 1, 0, 2):

17926272

$E_7$

(1, 2, 0, 0, 0, 0, 0):

18372354

$E_7$

(2, 0, 1, 0, 0, 0, 0):

19046664

$E_7$

(1, 0, 1, 0, 0, 0, 1):

21633248

$E_7$

(0, 0, 0, 0, 0, 0, 6):

21737254

$E_7$

(0, 1, 0, 0, 0, 1, 1):

23969792

$E_7$

(1, 0, 0, 1, 0, 0, 0):

24386670

$E_7$

(1, 0, 0, 0, 0, 0, 4):

25273458

$E_7$

(0, 3, 0, 0, 0, 0, 0):

32995248

$E_7$

(1, 0, 0, 0, 0, 2, 0):

48316905

$E_7$

(1, 1, 0, 0, 0, 1, 0):

63431424

$E_7$

(1, 1, 0, 0, 0, 0, 2):

67395888

$E_7$

(1, 0, 0, 0, 1, 0, 1):

72847026

$E_7$

(1, 0, 0, 0, 0, 1, 2):

73853000

$E_7$

(5, 0, 0, 0, 0, 0, 0):

85709988

$E_7$

(0, 0, 1, 0, 1, 0, 0):

86141440

$E_7$

(2, 0, 0, 0, 1, 0, 0):

86184000

$E_7$

(2, 0, 0, 0, 0, 0, 3):

93768192

$E_7$

(3, 1, 0, 0, 0, 0, 0):

94057600

$E_8$

(0, 0, 0, 0, 0, 0, 0, 1):

248

$E_8$

(1, 0, 0, 0, 0, 0, 0, 0):

3875

$E_8$

(0, 0, 0, 0, 0, 0, 0, 2):

27000

$E_8$

(0, 0, 0, 0, 0, 0, 1, 0):

30380

$E_8$

(0, 1, 0, 0, 0, 0, 0, 0):

147250

$E_8$

(1, 0, 0, 0, 0, 0, 0, 1):

779247

$E_8$

(0, 0, 0, 0, 0, 0, 0, 3):

1763125

$E_8$

(0, 0, 0, 0, 0, 1, 0, 0):

2450240

$E_8$

(0, 0, 0, 0, 0, 0, 1, 1):

4096000

$E_8$

(2, 0, 0, 0, 0, 0, 0, 0):

4881384

$E_8$

(0, 0, 1, 0, 0, 0, 0, 0):

6696000

$E_8$

(0, 1, 0, 0, 0, 0, 0, 1):

26411008

$E_8$

(1, 0, 0, 0, 0, 0, 0, 2):

70680000

$E_8$

(1, 0, 0, 0, 0, 0, 1, 0):

76271625

$E_8$

(0, 0, 0, 0, 0, 0, 0, 4):

79143000

$E_8$

(0, 0, 0, 0, 1, 0, 0, 0):

146325270

$E_8$

(0, 0, 0, 0, 0, 0, 2, 0):

203205000

$E_8$

(0, 0, 0, 0, 0, 0, 1, 2):

281545875

$E_8$

(1, 1, 0, 0, 0, 0, 0, 0):

301694976

$E_8$

(0, 0, 0, 0, 0, 1, 0, 1):

344452500

$E_8$

(2, 0, 0, 0, 0, 0, 0, 1):

820260000

$E_8$

(0, 0, 1, 0, 0, 0, 0, 1):

1094951000

$E_8$

(0, 1, 0, 0, 0, 0, 0, 2):

2172667860

$E_8$

(0, 1, 0, 0, 0, 0, 1, 0):

2275896000

$E_8$

(0, 0, 0, 0, 0, 0, 0, 5):

2642777280

$E_8$

(3, 0, 0, 0, 0, 0, 0, 0):

2903770000

$E_8$

(1, 0, 0, 0, 0, 0, 0, 3):

3929713760

$E_8$

(0, 2, 0, 0, 0, 0, 0, 0):

4076399250

$E_8$

(1, 0, 0, 0, 0, 1, 0, 0):

4825673125

$E_8$

(0, 0, 0, 1, 0, 0, 0, 0):

6899079264

$E_8$

(1, 0, 1, 0, 0, 0, 0, 0):

8634368000

$E_8$

(1, 0, 0, 0, 0, 0, 1, 1):

8634368000

$E_8$

(0, 0, 0, 0, 0, 0, 1, 3):

12692520960

$E_8$

(0, 0, 0, 0, 0, 0, 2, 1):

17535336000

$E_8$

(0, 0, 0, 0, 1, 0, 0, 1):

20288765952

$E_8$

(0, 0, 0, 0, 0, 1, 1, 0):

21039669000

$E_8$

(0, 0, 0, 0, 0, 1, 0, 2):

23592339045

$E_8$

(1, 1, 0, 0, 0, 0, 0, 1):

45329752170

$E_8$

(2, 0, 0, 0, 0, 0, 0, 2):

63513702720

$E_8$

(2, 0, 0, 0, 0, 0, 1, 0):

66393847000

$E_8$

(0, 0, 0, 0, 0, 0, 0, 6):

69176971200

$E_8$

(0, 0, 1, 0, 0, 0, 0, 2):

83080364250

$E_8$

(0, 0, 1, 0, 0, 0, 1, 0):

85424220000

$E_8$

(0, 1, 0, 0, 0, 0, 0, 3):

110977024000

$E_8$

(0, 1, 0, 0, 0, 1, 0, 0):

124436480000

$E_8$

(1, 0, 0, 0, 0, 0, 0, 4):

152883490500

$E_8$

(1, 0, 0, 0, 1, 0, 0, 0):

220778105625

$E_8$

(2, 1, 0, 0, 0, 0, 0, 0):

223850628000

$E_8$

(0, 1, 0, 0, 0, 0, 1, 1):

234550030000

$E_8$

(0, 1, 1, 0, 0, 0, 0, 0):

267413986840

$E_8$

(1, 0, 0, 0, 0, 0, 2, 0):

355647996000

$E_8$

(3, 0, 0, 0, 0, 0, 0, 1):

417532087000

$E_8$

(0, 0, 0, 0, 0, 0, 1, 4):

417933862500

$E_8$

(0, 0, 0, 0, 0, 0, 3, 0):

492957660000

$E_8$

(1, 0, 0, 0, 0, 0, 1, 2):

508731738750

$E_8$

(0, 2, 0, 0, 0, 0, 0, 1):

560213725500

$E_8$

(1, 0, 0, 0, 0, 1, 0, 1):

574197082368

$E_8$

(0, 0, 0, 0, 0, 2, 0, 0):

627099023250

$E_8$

(0, 0, 0, 0, 0, 0, 2, 2):

841900509450

$E_8$

(0, 0, 0, 1, 0, 0, 0, 1):

919045960000

$E_8$

(4, 0, 0, 0, 0, 0, 0, 0):

976217435000

Definition

For an exceptional complex simple Lie algebra $\mathfrak g$ [1], $V_{\mathfrak g}(\lambda)$ is the irreducible representation of highest weight $\lambda$. This table stores $\dim V_{\mathfrak g}(\lambda)$ for nonzero dominant integral weights $\lambda$.

Parameters

$\mathfrak g$

— exceptional simple Lie algebra (one of $G_2$, $F_4$, $E_6$, $E_7$ or $E_8$)

$\lambda$

— highest weight (the Dynkin labels of $\lambda=\sum_{i=1}^r a_i\varpi_i$ in Bourbaki node numbering for the fundamental weights $\varpi_i$: a nonzero tuple $(a_1,\ldots,a_r)$ of nonnegative integers, where $r$ is the rank of $\mathfrak g$)

Formulas

(1)

Weyl's dimension formula is $\dim V_{\mathfrak g}(\lambda)=\prod_{\alpha\in\Phi^+} \frac{\langle\lambda+\rho,\alpha^\vee\rangle} {\langle\rho,\alpha^\vee\rangle}$, where $\Phi^+$ is the set of positive roots and $\rho$ is the half-sum of the positive roots [2].

Comments

(2)

Weights are written as Dynkin labels in Bourbaki node numbering, the numbering used by Sage's root-system data [3]. For example, in this convention the $E_8$ row for $(1,0,0,0,0,0,0,0)$ has dimension $3875$, while $(0,0,0,0,0,0,0,1)$ has dimension $248$.

(3)

The zero highest weight, whose representation has dimension $1$, is omitted.

(4)

Weights with the same dimension remain separate rows. This includes dual pairs such as the two $27$-dimensional representations of $E_6$ and repeated dimensions such as the two $77$-dimensional representations of $G_2$.

(5)

The classical simple Lie algebras are not included.

Programs

(P1)

Sage

from sage.combinat.root_system.root_system import RootSystem
from sage.rings.rational_field import QQ

ambient = RootSystem("E8").ambient_space()
weights = ambient.fundamental_weights()
rho = ambient.rho()
labels = (0, 0, 0, 0, 0, 0, 0, 1)
lam = ambient.zero()
for a, i in zip(labels, ambient.index_set()):
    lam += QQ(a) * weights[i]
dimension = QQ(1)
for alpha in ambient.positive_roots():
    coroot = alpha.associated_coroot()
    dimension *= QQ((lam + rho).scalar(coroot)) / QQ(rho.scalar(coroot))
dimension                                  # 248

Links

[1]

Wikipedia: Exceptional Lie algebra

[2]

Wikipedia: Weyl character formula

[3]

SageMath: root systems

[4]

OEIS A104599: dimensions of irreducible representations of $G_2$

[5]

OEIS A121738: dimensions of irreducible representations of $F_4$

[6]

OEIS A121737: dimensions of irreducible representations of $E_6$

[7]

OEIS A121736: dimensions of irreducible representations of $E_7$

[8]

OEIS A121732: dimensions of irreducible representations of $E_8$

Similar tables

Orders of finite simple groups of Lie type — uses the same exceptional Dynkin types for the corresponding finite groups of Lie type

Schur polynomials — gives the type $A$ characters whose specialisations include dimensions of polynomial representations of general linear groups

Data properties

Entries are of type: integer

How well the digits are known: exact

Table is complete: no (it holds every nontrivial irreducible representation with $\dim V_{\mathfrak g}(\lambda)\leq10^6$ for $G_2$, $\dim V_{\mathfrak g}(\lambda)\leq10^7$ for $F_4$ and $E_6$, $\dim V_{\mathfrak g}(\lambda)\leq10^8$ for $E_7$, and $\dim V_{\mathfrak g}(\lambda)\leq10^{12}$ for $E_8$)

How they were obtained:

All entries are exact integers computed from Weyl's dimension formula (1). The generator reads the positive roots and fundamental weights from Sage's root-system data [3], evaluates the product over positive coroots with exact rational arithmetic, and checks that each result is integral.

The finite range is checked by enumerating the dominant weights a second way, by total label sum. The fundamental representation dimensions are checked, in Bourbaki node order, against the standard values $7,14$ for $G_2$, $52,1274,273,26$ for $F_4$, $27,78,351,2925,351,27$ for $E_6$, $133,912,8645,365750,27664,1539,56$ for $E_7$, and $3875,147250,6696000,6899079264,146325270,2450240,30380,248$ for $E_8$. For each algebra, the set of distinct stored dimensions, together with the omitted trivial dimension $1$, is checked against the corresponding OEIS b-file through the stated bound: [4], [5], [6], [7] and [8].