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:
39:
110:
15:
519:
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:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.