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Symmetric power

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568: 22: 117: 487: 217: 398: 353: 251: 410: 32: 527: 609: 90: 47: 62: 628: 69: 548: 177: 633: 513: 307: 76: 365: 312: 135: 602: 58: 225: 139: 131: 595: 544: 359: 292: 482:{\displaystyle \operatorname {Spec} ((A\otimes _{k}\dots \otimes _{k}A)^{{\mathfrak {S}}_{n}})} 362:, a symmetric power is defined in a way similar to that in algebraic topology. For example, if 523: 300: 273: 83: 517: 220: 579: 509: 401: 285: 261: 622: 405: 575: 21: 567: 256:
More precisely, the notion exists at least in the following three areas:
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3264 and All That: A Second Course in Algebraic Geometry
583: 43: 413: 368: 315: 228: 180: 481: 392: 347: 245: 211: 126:may lack focus or may be about more than one topic 212:{\displaystyle X^{n}:=X\times \cdots \times X} 130:Please help improve this article, possibly by 603: 8: 48:introducing citations to additional sources 610: 596: 393:{\displaystyle X=\operatorname {Spec} (A)} 468: 462: 461: 459: 446: 433: 412: 367: 348:{\displaystyle X^{n}/{\mathfrak {S}}_{n}} 339: 333: 332: 326: 320: 314: 237: 231: 230: 227: 185: 179: 38:Relevant discussion may be found on the 355:, as in the beginning of this article. 268:-th symmetric power of a vector space 7: 564: 562: 134:the article and/or by introducing a 463: 334: 246:{\displaystyle {\mathfrak {S}}_{n}} 232: 14: 219:by the permutation action of the 566: 549:"Symmetric powers of the sphere" 284:elements (here the product is a 115: 31:relies largely or entirely on a 20: 138:, or discuss this issue on the 522:, Cambridge University Press, 476: 456: 423: 420: 387: 381: 272:is the vector subspace of the 1: 582:. You can help Knowledge by 650: 561: 299:-th symmetric power of a 493:-th symmetric power of 170:is the quotient of the 578:-related article is a 483: 394: 349: 247: 213: 484: 395: 350: 280:consisting of degree- 248: 214: 411: 366: 313: 226: 178: 159:In mathematics, the 44:improve this article 629:Symmetric relations 545:Hopkins, Michael J. 164:-th symmetric power 136:disambiguation page 479: 390: 360:algebraic geometry 345: 293:algebraic topology 243: 209: 634:Mathematics stubs 591: 590: 529:978-1-107-01708-5 301:topological space 274:symmetric algebra 157: 156: 109: 108: 94: 59:"Symmetric power" 641: 612: 605: 598: 570: 563: 555: 553: 532: 488: 486: 485: 480: 475: 474: 473: 472: 467: 466: 451: 450: 438: 437: 399: 397: 396: 391: 354: 352: 351: 346: 344: 343: 338: 337: 330: 325: 324: 252: 250: 249: 244: 242: 241: 236: 235: 218: 216: 215: 210: 190: 189: 152: 149: 143: 119: 118: 111: 104: 101: 95: 93: 52: 24: 16: 649: 648: 644: 643: 642: 640: 639: 638: 619: 618: 617: 616: 559: 551: 543: 540: 535: 530: 510:Eisenbud, David 508: 504: 460: 455: 442: 429: 409: 408: 364: 363: 331: 316: 311: 310: 229: 224: 223: 221:symmetric group 181: 176: 175: 153: 147: 144: 129: 120: 116: 105: 99: 96: 53: 51: 37: 25: 12: 11: 5: 647: 645: 637: 636: 631: 621: 620: 615: 614: 607: 600: 592: 589: 588: 571: 557: 556: 547:(March 2018). 539: 538:External links 536: 534: 533: 528: 505: 503: 500: 499: 498: 478: 471: 465: 458: 454: 449: 445: 441: 436: 432: 428: 425: 422: 419: 416: 402:affine variety 389: 386: 383: 380: 377: 374: 371: 356: 342: 336: 329: 323: 319: 308:quotient space 289: 286:tensor product 262:linear algebra 240: 234: 208: 205: 202: 199: 196: 193: 188: 184: 174:-fold product 155: 154: 123: 121: 114: 107: 106: 42:. Please help 28: 26: 19: 13: 10: 9: 6: 4: 3: 2: 646: 635: 632: 630: 627: 626: 624: 613: 608: 606: 601: 599: 594: 593: 587: 585: 581: 577: 572: 569: 565: 560: 550: 546: 542: 541: 537: 531: 525: 521: 520: 515: 511: 507: 506: 501: 496: 492: 469: 452: 447: 443: 439: 434: 430: 426: 417: 414: 407: 403: 384: 378: 375: 372: 369: 361: 357: 340: 327: 321: 317: 309: 305: 302: 298: 294: 290: 287: 283: 279: 275: 271: 267: 263: 259: 258: 257: 254: 238: 222: 206: 203: 200: 197: 194: 191: 186: 182: 173: 169: 166:of an object 165: 163: 151: 141: 137: 133: 127: 124:This article 122: 113: 112: 103: 92: 89: 85: 82: 78: 75: 71: 68: 64: 61: –  60: 56: 55:Find sources: 49: 45: 41: 35: 34: 33:single source 29:This article 27: 23: 18: 17: 584:expanding it 573: 558: 518: 494: 490: 406:GIT quotient 303: 296: 281: 277: 269: 265: 255: 171: 167: 161: 160: 158: 145: 125: 97: 87: 80: 73: 66: 54: 30: 576:mathematics 514:Harris, Joe 404:, then the 623:Categories 502:References 100:April 2024 70:newspapers 444:⊗ 440:⋯ 431:⊗ 418:⁡ 379:⁡ 204:× 201:⋯ 198:× 140:talk page 132:splitting 40:talk page 148:May 2024 489:is the 306:is the 84:scholar 526:  400:is an 295:, the 264:, the 86:  79:  72:  65:  57:  574:This 552:(PDF) 91:JSTOR 77:books 580:stub 524:ISBN 415:Spec 376:Spec 63:news 358:In 291:In 276:of 260:In 46:by 625:: 516:, 512:; 288:). 253:. 192::= 611:e 604:t 597:v 586:. 554:. 497:. 495:X 491:n 477:) 470:n 464:S 457:) 453:A 448:k 435:k 427:A 424:( 421:( 388:) 385:A 382:( 373:= 370:X 341:n 335:S 328:/ 322:n 318:X 304:X 297:n 282:n 278:V 270:V 266:n 239:n 233:S 207:X 195:X 187:n 183:X 172:n 168:X 162:n 150:) 146:( 142:. 128:. 102:) 98:( 88:· 81:· 74:· 67:· 50:. 36:.

Index


single source
talk page
improve this article
introducing citations to additional sources
"Symmetric power"
news
newspapers
books
scholar
JSTOR
splitting
disambiguation page
talk page
symmetric group
linear algebra
symmetric algebra
tensor product
algebraic topology
topological space
quotient space
algebraic geometry
affine variety
GIT quotient
Eisenbud, David
Harris, Joe
3264 and All That: A Second Course in Algebraic Geometry
ISBN
978-1-107-01708-5
Hopkins, Michael J.

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