290:). One may view the graph as an edge-labeled directed graph with edge labels coming from the set of reflections. (One could also define the Bruhat graph using multiplication on the right; as graphs, the resulting objects are isomorphic, but the edge labelings are different.)
372:
254:
The Bruhat graph is a directed graph related to the (strong) Bruhat order. The vertex set is the set of elements of the
Coxeter group and the edge set consists of directed edges (
558:
489:
457:
419:
296:
474:
Chevalley, C. (1958), "Sur les décompositions cellulaires des espaces G/B", in
Haboush, William J.; Parshall, Brian J. (eds.),
383:
481:
411:
243:
80:
374:, and thus this poset is Eulerian, meaning its Möbius function is produced by the rank function on the poset.
605:
556:
Verma, Daya-Nand (1968), "Structure of certain induced representations of complex semisimple Lie algebras",
610:
512:
96:
545:
52:
100:
577:
529:
485:
453:
415:
567:
537:
521:
445:
397:
64:
589:
499:
467:
429:
293:
The strong Bruhat order on the symmetric group (permutations) has Möbius function given by
585:
541:
495:
463:
441:
425:
477:
Algebraic groups and their generalizations: classical methods (University Park, PA, 1991)
127:
599:
507:
401:
68:
48:
44:
572:
72:
475:
405:
92:
581:
533:
449:
186:. (Note that here a substring is not necessarily a consecutive substring.)
95:, and introduced the name "Bruhat order" because of the relation to the
549:
106:
The left and right weak Bruhat orderings were studied by Björner (
525:
367:{\displaystyle \mu (\pi ,\sigma )=(-1)^{\ell (\sigma )-\ell (\pi )}}
440:, Graduate Texts in Mathematics, vol. 231, Berlin, New York:
480:, Proc. Sympos. Pure Math., vol. 56, Providence, R.I.:
91:
started the combinatorial study of the Bruhat order on the
134:, then the Bruhat order is a partial order on the group
178:
if some substring of some (or every) reduced word for
299:
230:
if some initial substring of some reduced word for
410:, Contemp. Math., vol. 34, Providence, R.I.:
366:
205:if some final substring of some reduced word for
508:"Sur la Topologie de Certains Espaces Homogènes"
407:Combinatorics and algebra (Boulder, Colo., 1983)
559:Bulletin of the American Mathematical Society
242:For more on the weak orders, see the article
216:The weak right (Bruhat) order is defined by
51:, that corresponds to the inclusion order on
8:
191:The weak left (Bruhat) order is defined by
138:. Recall that a reduced word for an element
436:Björner, Anders; Brenti, Francesco (2005),
571:
334:
298:
84:
76:
170:The (strong) Bruhat order is defined by
107:
520:(2), Annals of Mathematics: 396–443,
88:
7:
79:, and the analogue for more general
146:is a minimal length expression of
14:
166:is the length of a reduced word.
16:Partial order on a Coxeter group
573:10.1090/S0002-9904-1968-11921-4
438:Combinatorics of Coxeter groups
359:
353:
344:
338:
331:
321:
315:
303:
1:
516:, Second Series (in French),
482:American Mathematical Society
412:American Mathematical Society
398:"Orderings of Coxeter groups"
150:as a product of elements of
506:Ehresmann, Charles (1934),
234:is a reduced word for
209:is a reduced word for
182:is a reduced word for
81:semisimple algebraic groups
627:
384:Kazhdan–Lusztig polynomial
244:weak order of permutations
396:Björner, Anders (1984),
63:The Bruhat order on the
368:
41:Chevalley–Bruhat order
37:Bruhat–Chevalley order
513:Annals of Mathematics
450:10.1007/3-540-27596-7
369:
75:was first studied by
47:on the elements of a
414:, pp. 175–195,
297:
270:for some reflection
97:Bruhat decomposition
19:In mathematics, the
29:strong Bruhat order
364:
65:Schubert varieties
53:Schubert varieties
491:978-0-8218-1540-3
484:, pp. 1–23,
459:978-3-540-44238-7
421:978-0-8218-5029-9
282:) <
154:, and the length
23:(also called the
618:
592:
575:
552:
502:
470:
432:
373:
371:
370:
365:
363:
362:
130:with generators
85:Chevalley (1958)
77:Ehresmann (1934)
626:
625:
621:
620:
619:
617:
616:
615:
596:
595:
555:
526:10.2307/1968440
505:
492:
473:
460:
442:Springer-Verlag
435:
422:
395:
392:
380:
330:
295:
294:
252:
225:
200:
116:
101:François Bruhat
83:was studied by
61:
33:Chevalley order
17:
12:
11:
5:
624:
622:
614:
613:
608:
606:Coxeter groups
598:
597:
594:
593:
553:
503:
490:
471:
458:
433:
420:
402:Greene, Curtis
391:
388:
387:
386:
379:
376:
361:
358:
355:
352:
349:
346:
343:
340:
337:
333:
329:
326:
323:
320:
317:
314:
311:
308:
305:
302:
251:
248:
240:
239:
221:
214:
196:
188:
187:
128:Coxeter system
115:
112:
99:introduced by
60:
57:
15:
13:
10:
9:
6:
4:
3:
2:
623:
612:
609:
607:
604:
603:
601:
591:
587:
583:
579:
574:
569:
565:
561:
560:
554:
551:
547:
543:
539:
535:
531:
527:
523:
519:
515:
514:
509:
504:
501:
497:
493:
487:
483:
479:
478:
472:
469:
465:
461:
455:
451:
447:
443:
439:
434:
431:
427:
423:
417:
413:
409:
408:
403:
399:
394:
393:
389:
385:
382:
381:
377:
375:
356:
350:
347:
341:
335:
327:
324:
318:
312:
309:
306:
300:
291:
289:
285:
281:
277:
273:
269:
266: =
265:
261:
257:
249:
247:
245:
237:
233:
229:
224:
219:
215:
212:
208:
204:
199:
194:
190:
189:
185:
181:
177:
174: ≤
173:
169:
168:
167:
165:
161:
157:
153:
149:
145:
141:
137:
133:
129:
125:
121:
113:
111:
109:
104:
102:
98:
94:
90:
86:
82:
78:
74:
70:
69:flag manifold
66:
58:
56:
54:
50:
49:Coxeter group
46:
45:partial order
42:
38:
34:
30:
26:
22:
611:Order theory
563:
557:
517:
511:
476:
437:
406:
292:
287:
283:
279:
275:
271:
267:
263:
259:
255:
253:
250:Bruhat graph
241:
235:
231:
227:
222:
217:
210:
206:
202:
197:
192:
183:
179:
175:
171:
163:
159:
155:
151:
147:
143:
139:
135:
131:
123:
119:
117:
105:
89:Verma (1968)
73:Grassmannian
62:
40:
36:
32:
28:
25:strong order
24:
21:Bruhat order
20:
18:
566:: 160–166,
262:) whenever
600:Categories
542:60.1223.05
390:References
114:Definition
93:Weyl group
582:0002-9904
534:0003-486X
357:π
351:ℓ
348:−
342:σ
336:ℓ
325:−
313:σ
307:π
301:μ
378:See also
590:0218417
550:1968440
500:1278698
468:2133266
430:0777701
404:(ed.),
258:,
220: ≤
195: ≤
126:) is a
59:History
43:) is a
588:
580:
548:
540:
532:
498:
488:
466:
456:
428:
418:
226:
201:
546:JSTOR
400:, in
162:) of
71:or a
67:of a
39:, or
578:ISSN
530:ISSN
486:ISBN
454:ISBN
416:ISBN
274:and
118:If (
108:1984
568:doi
538:JFM
522:doi
446:doi
142:of
110:).
602::
586:MR
584:,
576:,
564:74
562:,
544:,
536:,
528:,
518:35
510:,
496:MR
494:,
464:MR
462:,
452:,
444:,
426:MR
424:,
268:tv
246:.
122:,
103:.
87:.
55:.
35:,
31:,
27:,
570::
524::
448::
360:)
354:(
345:)
339:(
332:)
328:1
322:(
319:=
316:)
310:,
304:(
288:v
286:(
284:â„“
280:u
278:(
276:â„“
272:t
264:u
260:v
256:u
238:.
236:u
232:v
228:v
223:R
218:u
213:.
211:u
207:v
203:v
198:L
193:u
184:u
180:v
176:v
172:u
164:w
160:w
158:(
156:â„“
152:S
148:w
144:W
140:w
136:W
132:S
124:S
120:W
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.