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Symplectic representation

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is the field of complex numbers, then by introducing a compatible unitary structure (which exists by an averaging argument), one can show that any complex symplectic representation is a
32: 604: 527: 485:. Quaternionic representations of finite or compact groups are often called symplectic representations, and may be identified using the 90: 62: 328: 69: 623: 499: 223: 161: 47: 486: 76: 511: 628: 482: 58: 597: 503: 132: 136: 633: 590: 116: 83: 124: 206: 128: 541: 523: 574: 515: 439: 39: 537: 533: 617: 470: 474: 562: 312: 112: 21: 519: 545: 570: 514:, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. 392:{\displaystyle \omega (\xi \cdot v,w)+\omega (v,\xi \cdot w)=0} 15: 430:
is equivalently a group or Lie algebra homomorphism from
578: 281:{\displaystyle \omega (g\cdot v,g\cdot w)=\omega (v,w)} 191:{\displaystyle \omega \colon V\times V\to \mathbb {F} } 43: 331: 226: 164: 391: 280: 190: 155:is a nondegenerate skew symmetric bilinear form 598: 8: 48:introducing citations to additional sources 605: 591: 330: 225: 184: 183: 163: 209:of scalars. A representation of a group 38:Relevant discussion may be found on the 147:) which preserves the symplectic form 508:Representation theory. A first course 7: 559: 557: 577:. You can help Knowledge (XXG) by 14: 561: 311:, whereas a representation of a 31:relies largely or entirely on a 20: 380: 362: 353: 335: 275: 263: 254: 230: 180: 1: 512:Graduate Texts in Mathematics 483:quaternionic representation 422:. Thus a representation of 59:"Symplectic representation" 650: 556: 520:10.1007/978-1-4612-0979-9 487:Frobenius–Schur indicator 121:symplectic representation 137:symplectic vector space 573:-related article is a 393: 282: 192: 624:Representation theory 450:) or its Lie algebra 394: 283: 193: 117:representation theory 329: 224: 162: 44:improve this article 629:Symplectic geometry 389: 278: 188: 586: 585: 529:978-0-387-97495-8 109: 108: 94: 641: 607: 600: 593: 565: 558: 549: 473:(for example, a 440:symplectic group 398: 396: 395: 390: 287: 285: 284: 279: 197: 195: 194: 189: 187: 104: 101: 95: 93: 52: 24: 16: 649: 648: 644: 643: 642: 640: 639: 638: 614: 613: 612: 611: 554: 530: 500:Fulton, William 498: 495: 327: 326: 222: 221: 160: 159: 105: 99: 96: 53: 51: 37: 25: 12: 11: 5: 647: 645: 637: 636: 631: 626: 616: 615: 610: 609: 602: 595: 587: 584: 583: 566: 552: 551: 528: 494: 491: 400: 399: 388: 385: 382: 379: 376: 373: 370: 367: 364: 361: 358: 355: 352: 349: 346: 343: 340: 337: 334: 289: 288: 277: 274: 271: 268: 265: 262: 259: 256: 253: 250: 247: 244: 241: 238: 235: 232: 229: 199: 198: 186: 182: 179: 176: 173: 170: 167: 125:representation 107: 106: 42:. Please help 28: 26: 19: 13: 10: 9: 6: 4: 3: 2: 646: 635: 634:Algebra stubs 632: 630: 627: 625: 622: 621: 619: 608: 603: 601: 596: 594: 589: 588: 582: 580: 576: 572: 567: 564: 560: 555: 547: 543: 539: 535: 531: 525: 521: 517: 513: 509: 505: 501: 497: 496: 492: 490: 488: 484: 480: 476: 472: 471:compact group 468: 463: 461: 457: 453: 449: 445: 441: 437: 433: 429: 425: 421: 417: 413: 409: 405: 386: 383: 377: 374: 371: 368: 365: 359: 356: 350: 347: 344: 341: 338: 332: 325: 324: 323: 321: 317: 314: 310: 306: 302: 298: 294: 272: 269: 266: 260: 257: 251: 248: 245: 242: 239: 236: 233: 227: 220: 219: 218: 216: 212: 208: 204: 177: 174: 171: 168: 165: 158: 157: 156: 154: 150: 146: 142: 138: 134: 130: 126: 122: 118: 114: 103: 92: 89: 85: 82: 78: 75: 71: 68: 64: 61: â€“  60: 56: 55:Find sources: 49: 45: 41: 35: 34: 33:single source 29:This article 27: 23: 18: 17: 579:expanding it 568: 553: 507: 478: 475:finite group 466: 464: 459: 455: 451: 447: 443: 435: 431: 427: 423: 419: 415: 411: 407: 403: 401: 319: 315: 308: 304: 300: 296: 292: 290: 214: 210: 202: 200: 152: 148: 144: 140: 120: 113:mathematical 110: 97: 87: 80: 73: 66: 54: 30: 504:Harris, Joe 313:Lie algebra 133:Lie algebra 618:Categories 493:References 318:preserves 213:preserves 70:newspapers 546:246650103 375:⋅ 372:ξ 360:ω 342:⋅ 339:ξ 333:ω 261:ω 249:⋅ 237:⋅ 228:ω 181:→ 175:× 169:: 166:ω 115:field of 40:talk page 506:(1991). 402:for all 291:for all 100:May 2024 571:algebra 538:1153249 477:), and 438:to the 205:is the 151:. Here 84:scholar 544:  536:  526:  201:where 86:  79:  72:  65:  57:  569:This 469:is a 207:field 135:on a 131:or a 129:group 127:of a 123:is a 91:JSTOR 77:books 575:stub 542:OCLC 524:ISBN 410:and 299:and 119:, a 63:news 516:doi 465:If 442:Sp( 434:or 426:or 418:in 406:in 322:if 307:in 295:in 217:if 111:In 46:by 620:: 540:. 534:MR 532:. 522:. 510:. 502:; 489:. 462:) 452:sp 414:, 303:, 143:, 606:e 599:t 592:v 581:. 550:. 548:. 518:: 479:F 467:G 460:ω 458:, 456:V 454:( 448:ω 446:, 444:V 436:g 432:G 428:g 424:G 420:V 416:w 412:v 408:g 404:Îľ 387:0 384:= 381:) 378:w 369:, 366:v 363:( 357:+ 354:) 351:w 348:, 345:v 336:( 320:ω 316:g 309:V 305:w 301:v 297:G 293:g 276:) 273:w 270:, 267:v 264:( 258:= 255:) 252:w 246:g 243:, 240:v 234:g 231:( 215:ω 211:G 203:F 185:F 178:V 172:V 153:ω 149:ω 145:ω 141:V 139:( 102:) 98:( 88:· 81:· 74:· 67:· 50:. 36:.

Index


single source
talk page
improve this article
introducing citations to additional sources
"Symplectic representation"
news
newspapers
books
scholar
JSTOR
mathematical
representation theory
representation
group
Lie algebra
symplectic vector space
field
Lie algebra
symplectic group
compact group
finite group
quaternionic representation
Frobenius–Schur indicator
Fulton, William
Harris, Joe
Graduate Texts in Mathematics
doi
10.1007/978-1-4612-0979-9
ISBN

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