241:
70:
A Coxeter group has a longest element if and only if it is finite; "only if" is because the size of the group is bounded by the number of words of length less than or equal to the maximum.
130:
445:
348:
162:
409:
375:
322:
572:
547:
20:
167:
564:
280:
81:
591:
87:
276:
39:
42:
with respect to the chosen generating set consisting of simple reflections. It is often denoted by
248:
In a reduced expression for the longest element, every simple reflection must occur at least once.
132:), by uniqueness of maximal length (the inverse of an element has the same length as the element).
556:
568:
543:
414:
291:
507:
327:
138:
54:
384:
287:
353:
300:
476:
465:
452:
35:
290:; topologically, it is the top dimensional cell of the decomposition, and represents the
73:
The longest element of a
Coxeter group is the unique maximal element with respect to the
471:
585:
259:
74:
534:
27:
245:
A reduced expression for the longest element is not in general unique.
451:
odd, when it is –1 multiplied by the order 2 automorphism of the
53:, Section 1.8: Simple transitivity and the longest element,
16:
Unique element of maximal length in a finite
Coxeter group
297:
The longest element is the central element –1 except for
417:
387:
356:
330:
303:
170:
141:
90:
236:{\displaystyle \ell (w_{0}w)=\ell (w_{0})-\ell (w).}
439:
403:
369:
342:
316:
251:If the Coxeter group is finite then the length of
235:
156:
124:
8:
536:The Geometry and Topology of Coxeter Groups
503:
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140:
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100:
95:
89:
50:
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520:
58:
7:
561:Reflection groups and Coxeter groups
468:, a different distinguished element
32:longest element of a Coxeter group
21:Coxeter element of a Coxeter group
14:
34:is the unique element of maximal
125:{\displaystyle w_{0}^{-1}=w_{0}}
434:
428:
227:
221:
212:
199:
190:
174:
1:
61:, Section 4.6, pp. 51–53).
608:
565:Cambridge University Press
533:Davis, Michael W. (2007),
281:semisimple algebraic group
80:The longest element is an
18:
523:, Remark 13.1.8, p. 259)
440:{\displaystyle I_{2}(p)}
19:Not to be confused with
343:{\displaystyle n\geq 2}
157:{\displaystyle w\in W,}
441:
405:
404:{\displaystyle E_{6},}
371:
344:
318:
237:
158:
126:
442:
406:
372:
370:{\displaystyle D_{n}}
345:
319:
317:{\displaystyle A_{n}}
258:is the number of the
238:
164:the length satisfies
159:
127:
415:
385:
354:
328:
301:
277:Bruhat decomposition
168:
139:
88:
40:finite Coxeter group
557:Humphreys, James E.
108:
437:
401:
367:
340:
314:
233:
154:
122:
91:
574:978-0-521-43613-7
549:978-0-691-13138-2
292:fundamental class
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577:
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541:
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373:
368:
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365:
349:
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341:
323:
321:
320:
315:
313:
312:
288:Zariski topology
242:
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239:
234:
211:
210:
186:
185:
163:
161:
160:
155:
131:
129:
128:
123:
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120:
107:
99:
607:
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602:
601:
600:
598:
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596:
582:
581:
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575:
555:
550:
539:
532:
528:
527:
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490:
485:
477:Length function
466:Coxeter element
462:
453:Coxeter diagram
418:
413:
412:
388:
383:
382:
357:
352:
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136:
112:
86:
85:
67:
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24:
17:
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11:
5:
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594:
592:Coxeter groups
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583:
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553:
548:
529:
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525:
512:
504:Humphreys 1992
487:
486:
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472:Coxeter number
469:
461:
458:
457:
456:
436:
433:
430:
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421:
400:
395:
391:
364:
360:
339:
336:
333:
311:
307:
295:
269:
265:The open cell
263:
260:positive roots
255:
249:
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232:
229:
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220:
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214:
209:
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201:
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184:
180:
176:
173:
153:
150:
147:
144:
133:
119:
115:
111:
106:
103:
98:
94:
84:(has order 2:
78:
71:
66:
63:
51:Humphreys 1992
46:
15:
13:
10:
9:
6:
4:
3:
2:
604:
593:
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551:
545:
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531:
530:
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337:
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331:
309:
305:
296:
293:
289:
285:
282:
278:
274:
268:
264:
261:
254:
250:
247:
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230:
224:
218:
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207:
203:
196:
193:
187:
182:
178:
171:
151:
148:
145:
142:
134:
117:
113:
109:
104:
101:
96:
92:
83:
79:
76:
72:
69:
68:
64:
62:
60:
56:
52:
45:
41:
37:
33:
29:
22:
560:
535:
515:
448:
378:
286:is dense in
283:
272:
266:
252:
75:Bruhat order
43:
31:
25:
28:mathematics
521:Davis 2007
483:References
82:involution
65:Properties
59:Davis 2007
335:≥
219:ℓ
216:−
197:ℓ
172:ℓ
146:∈
102:−
55:pp. 15–16
586:Category
559:(1992),
460:See also
135:For any
275:in the
57:) and (
49:. See (
571:
546:
36:length
30:, the
540:(PDF)
508:p. 16
381:odd,
279:of a
38:in a
569:ISBN
544:ISBN
447:for
411:and
377:for
350:),
26:In
588::
567:,
563:,
542:,
506:,
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455:.
267:Bw
519:(
510:)
502:(
449:p
435:)
432:p
429:(
424:2
420:I
399:,
394:6
390:E
379:n
363:n
359:D
338:2
332:n
324:(
310:n
306:A
294:.
284:G
273:B
270:0
262:.
256:0
253:w
231:.
228:)
225:w
222:(
213:)
208:0
204:w
200:(
194:=
191:)
188:w
183:0
179:w
175:(
152:,
149:W
143:w
118:0
114:w
110:=
105:1
97:0
93:w
77:.
47:0
44:w
23:.
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