外尔群

在數學裡，尤其是在李群的理論中，一根系的外尔群是指經由正交於根之超平面的鏡面而產生之根系的等距同構群之子群。例如，根系A₂包含中心為原點之正六邊形的角。根系的對稱之整個群因此是有12階的二面體群。外尔群產生於將六邊形平分成兩半的線之鏡射；其為6階的二面體群。

半單李群、半單李代數和半單線性代數群等之外尔群為群或代數之根系的外尔群。

除去由Φ的根所定義之超平面會將歐幾里得空間切成有限個開領域，此領域稱為外尔腔。這些領域可以被外尔群的群作用置換，且此一群作用為簡單傳遞的。特別地是，外尔腔的數量是和外尔群的階相同的。任一非零向量都可以以正交於v之超平面v^∧將歐幾里得空間分成兩個半空間－v⁺和v⁻。若v在某一外尔腔裡，則沒有根會在v^∧，所以每一個根都會在v⁺或v⁻裡，且若其一根α在一邊，則其另外一根−α會在另外一邊。因此，Φ⁺ := Φ∩v⁺包含著Φ正好一半的根。當然Φ⁺和v有關，但只要v待在同一個外尔腔裡，Φ⁺就不會改變。根據上述選擇的根系之基為在Φ⁺內的簡單根，即其不能被寫成於Φ⁺內另外兩個根之和的根。因此，外尔腔、Φ⁺和其基決定了另一個，且外尔群在每一狀況下都為簡單傳遞。下面的圖示描繪了根系A₂的六個外尔腔，一選定的v及其超平面v^∧(以虛線表示)及正根α、β和γ。此例中的基為{α,γ}。

外尔群為考克斯特群的一特例。這表示其有一特殊種類的展現，其中每一產生子x_i均為二階，且有異於x_i²的關係(x_ix_j)^m_ij。產生子是由簡單根所給出的鏡射，且m_ij依據根i和j之間的角度為90度、120度、135度或150度等不同(即根據其在鄧肯圖內為不相連、以單邊相連、以雙邊相連、以三邊相連)而分別為2、3、4及6。一外尔群元素的長度為可以以最少字展現其以標準產生子表示之元素的長度。

若G為一在代數閉體上的半單線代數群，且T為一極大環面，則T的正規化子N包含著T，為一有限指數之子群，且G的外尔群W會同構於N/T。若B為G的波萊爾子群且將T選定放在B內，即可得到布吕阿分解

G=\bigcup _{w\in W}BwB

其將旗流形G/B的分解映射至舒伯特細胞內。(詳見格拉斯曼空間)

定义与样例

令 $\Phi$ 为欧式空间 $V$ 中的根系。对每个根 $\alpha \in \Phi$ ，令 $s_{\alpha }$ 表示关于垂直于 $\alpha$ 的超平面的反射，它可以被显式地写成

s_{\alpha }(v)=v-2{\frac {(\alpha ,v)}{(\alpha ,\alpha )}}\alpha

，其中

(\cdot ,\cdot )

为

V

上的内积。

\Phi

的外尔群

W

是正交群

O(V)

由所有

s_{\alpha }

生成的子群。由根系的定义，每个

s_{\alpha }

保持了

\Phi

，因此

W

为有限群。

Read other articles:

FC Baku

Football clubBakuFull nameBakı Futbol KlubuNickname(s)Zolaqlılar (The Stripes)Founded1997 as Dinamo BakuGroundFC Baku Training Base,Baku, AzerbaijanCapacity2,500WebsiteClub website Home colours Away colours Third colours Current season FC Baku (Azerbaijani: Bakı Futbol Klubu) was an Azerbaijani football club based in Baku that has been an amateur club since 2016. Prior to that, Baku played 18 seasons in the Azerbaijan Premier League, winning the championship twice and the National Cup thr...

Bus sekolah

Bus sekolah di IndonesiaBeberapa siswa sekolah dan bus sekolah yang dimiliki persatuan pegawai perkebunan di Pangalengan, Bandung pada masa Hindia Belanda, 1938Bus sekolah di Jakarta, 2014 Bus sekolah adalah jenis bus angkutan untuk siswa sekolah yang dimiliki, disewa, dikontrak, atau dioperasikan oleh sekolah atau perusahaan otobus. Bus jenis ini biasanya digunakan untuk mengangkut siswa ke dan dari sekolah atau kegiatan terkait sekolah. Ciri-ciri pelayanan bus sekolah di Indonesia diatur be...

Davion Davis

American football player (born 1996) American football player Davion DavisDavis with the Cleveland Browns in 2021No. 84 – Washington CommandersPosition:Wide receiverPersonal informationBorn: (1996-10-23) October 23, 1996 (age 27)Hutto, Texas, U.S.Height:5 ft 11 in (1.80 m)Weight:199 lb (90 kg)Career informationHigh school:HuttoCollege:Sam Houston StateUndrafted:2019Career history Minnesota Vikings (2019) Cleveland Browns (2021) Houston Texans (2021–2022) Hou...

Haida ermine

Species of carnivore Haida ermine In winter coat Conservation status Threatened (COSEWIC)(listed as Mustela erminea haidarum)[1] Vulnerable (NatureServe)(for both celenda and seclusa subspecies; species as whole unranked)[2][3][4] Scientific classification Domain: Eukaryota Kingdom: Animalia Phylum: Chordata Class: Mammalia Order: Carnivora Family: Mustelidae Genus: Mustela Species: M. haidarum Binomial name Mustela haidarumPreble, 1898 Subspeci...

Supercoppa d'Irlanda 2015

Supercoppa d'Irlanda 20152015 President's Cup Competizione Supercoppa d'Irlanda Sport Calcio Edizione 2ª Organizzatore FAI Date 28 febbraio 2015 Luogo Dublino Partecipanti 2 Formula Gara unica Impianto/i Oriel Park Risultati Vincitore Dundalk(1º titolo) Secondo St Patrick's Statistiche Gol segnati 3 Pubblico 1 800 spettatori Cronologia della competizione 2014 2016 Manuale La Supercoppa d'Irlanda 2015 è stata la seconda edizione del trofeo. La partita si è disputata il 28 febbraio 2015 al...

Commuter Line Dhoho, Penataran, dan Tumapel

Commuter Line Dhoho, Penataran & TumapelKereta api Commuter Line Penataran sedang berhenti di Stasiun Porong.Informasi umumJenis layananKereta api lokalStatusBeroperasiDaerah operasiKAI CommuterWilayah VIII SurabayaOperator saat iniKAI Commuter Wilayah VIII SurabayaOperator sebelumnyaKereta Api IndonesiaLintas pelayananStasiun awalSurabaya KotaStasiun akhir Blitar Kertosono (CL Dhoho Malam Terakhir) Malang (CL Tumapel Malam Terakhir) Jarak tempuh Dhoho: 179,5 kmLintas Blitar–Surabaya Ko...

B Complete

B CompleteAlbum mini karya AB6IXDirilis22 Mei 2019 (2019-05-22)Direkam2019StudioBRANDNEW MUSIC StudioGenreK-popDeep houseR&Bhip hopBahasaKoreaLabelBrand NewProduserRhymer (exec.)AB6IXBOOMBASTICOUOWNomadElapseJohn NapoleonLishBeatsKronologi AB6IX B Complete(2019) 6ixense(2019) Singel dalam album B Complete HollywoodDirilis: 25 April 2019 BreatheDirilis: 22 Mei 2019 Video musikBreathe di YouTube B Complete (ditulis sebagai B:COMPLETE) adalah album mini debut dari grup vokal pria ya...

コルク

この記事は検証可能な参考文献や出典が全く示されていないか、不十分です。出典を追加して記事の信頼性向上にご協力ください。（このテンプレートの使い方）出典検索?: コルク – ニュース · 書籍 · スカラー · CiNii · J-STAGE · NDL · dlib.jp · ジャパンサーチ · TWL（2017年4月）コルクを打ち抜いて作った瓶の栓コルク（木栓、�...

Valprato Soana

Valprato Soana komune di Italia Tempat Negara berdaulatItaliaRegion di ItaliaPiedmontKota metropolitan di ItaliaKota Metropolitan Turin NegaraItalia PendudukTotal89 (2023 )GeografiLuas wilayah71,85 km² [convert: unit tak dikenal]Ketinggian1.113 m Berbatasan denganChamporcher Traversella Cogne Ronco Canavese Valchiusa SejarahSanto pelindungPaus Silverius Informasi tambahanKode pos10080 Zona waktuUTC+1 UTC+2 Kode telepon0124 ID ISTAT001288 Kode kadaster ItaliaB510 Lain-lainSitus web...

STS-126

Cet article est une ébauche concernant l’astronautique. Vous pouvez partager vos connaissances en l’améliorant (comment ?) selon les recommandations des projets correspondants. STS-126 Données de la mission Vaisseau Navette spatiale Endeavour Équipage 5 hommes et 2 femmes Date de lancement 14 novembre 2008 19:55 (heure locale)15 novembre 2008, 00:55 UTC Site de lancement Kennedy Space CenterPas de tir 39A Date d'atterrissage 30 novembre 2008 à 21h25 UTC...

Canadian corporate law

This article is part of a series onCorporate law By jurisdiction Anguilla Australia BVI Canada Cayman Islands India South Africa UK United States Vietnam European Union France Germany General corporate forms Company Conglomerate Cooperative Corporation Holding company Joint-stock Partnership General Limited Limited liability Private limited Shell corporation Shelf corporation Sole proprietorship Corporate formsby jurisdiction European Union Societas (SE) Societas cooperativa (SCE) Societ...

Mattapoisett, Massachusetts

For the river, see Mattapoisett River.Town in Massachusetts, United StatesMattapoisett, MassachusettsTownNed's Point Light SealLocation in Plymouth County in MassachusettsCoordinates: 41°39′30″N 70°49′00″W / 41.65833°N 70.81667°W / 41.65833; -70.81667CountryUnited StatesStateMassachusettsCountyPlymouthSettled1750IncorporatedMay 21, 1857Government • TypeOpen town meetingArea • Total24.2 sq mi (62.6 km2) • L...

Bandiera del Ghana

Flag of GhanaProporzioni2:3 Simbolo FIAV ColoriRGB (R:206 G:17 B:38) (R:252 G:209 B:22) (R:0 G:107 B:63) (R:0 G:0 B:0) UsoBandiera civile e di stato Adozione6 marzo 1957 Nazione Ghana Altre bandiere ufficialiNavale Presidenziale Civile Aeronautica Aeronautica civile Fotografia Manuale La bandiera del Ghana venne adottata nel 1957. Fu rimpiazzata da una variante con una ban...

نوكيا 150

نوكيا 150معلومات عامةالنوع هاتف محمولالصانع إتش إم دي العالميةالسعر المبدئي 350٬000 روبية إندونيسية[1] موقع الويب nokia.com… أهم التواريختاريخ الإصدار 13 ديسمبر، 2016الوظائفالشاشة 2.4 بوصة، 320x240 بكسلالكاميرا VGAالخصائصشريحة ذاكرة حتى 32 GB، microSDHCنظام التشغيل نوكيا سيريس 30+البطارية ...

2012 Grand Prix Cycliste de Montréal

Cycling race 2012 Grand Prix Cycliste de Montréal2012 UCI World Tour, race 26 of 28Race detailsDates9 September 2011Stages1Distance205.7 km (127.8 mi)Winning time5h 28' 29Results Winner Lars Petter Nordhaug (Norway) (Team Sky) Second Moreno Moser (Italy) (Liquigas–Cannondale) Third Alexandr Kolobnev (Russia) (Team Katusha)← 2011 2013 → The 2012 Grand Prix Cycliste de Montréal was the third edition of the Gr...

Казанский военный округ

Казанский военный округ(КазВО) Годы существования 6 августа 1864 (1864-08-06)—30 марта 1918 (1918-03-30), 9 июля 1945 (1945-07-09)—6 мая 1946 (1946-05-06) Страна Российская империя 1864—1918, СССР 1945—1946 Подчинение Военное министерство Российской империиНародный комиссариат обороны СССР Входит в сух...

Alma-Ata Protocol

Founding declarations & principles of Commonwealth of Independent States (CIS) Alma-Ata ProtocolL–R: Ukrainian President Leonid Kravchuk, Kazakh President Nursultan Nazarbayev, Russian President Boris Yeltsin, and Belarusian leader Stanislav Shushkevich after signing the treatyTypeTreaty establishing a founding declarations and principles of the Commonwealth of Independent States (CIS).Signed21 December 1991LocationAlma-Ata, Kazakhstan, Soviet UnionEffective21 December 1991Signatories ...

List of palaces and mansions in Somogy County

This is a list of palaces and mansions in Somogy County in Hungary.[1] List of palaces and mansions in Somogy County Name Location Established Architect Style Family Picture Present function Kemény Palace Kaposvár Festetics Mansion Alsóbogát 1830 Classicism Festetics–Inkey Mansion Alsóbogát Csapody Mansion Ádánd 1820–1827 Classicism / late baroque / Zopf Csapody Somssich Mansion Babócsa Ágoston-Madách Mansion Balatonboglár Bárány Mansion Balatonboglár Gaál Mansion...

Carib language

Endangered South American Indigenous language This article is about the language of South America. For the language of the Caribs in the Caribbean, see Island Carib language. CaribKari'njaNative toBrazil, French Guiana, Guyana, Suriname, and VenezuelaEthnicityKali'naNative speakers8,600 (2001–2012)[1]Language familyCariban Guianan CaribCaribDialects Tyrewuju (Suriname) Aretyry (Suriname) Writing systemLatin scriptLanguage codesISO 639-2carISO 639-3carGlottologgali1262ELPKa...

Satsuma Rebellion

1877 Japanese samurai revolt Satsuma RebellionPart of the Shizoku rebellionsMap of the campaignDate29 January – 24 September 1877LocationKyūshū, JapanResult Imperial victoryBelligerents Empire of Japan Satsuma DomainCommanders and leaders Emperor Meiji Imperial Army: Prince Arisugawa Yamagata Aritomo Kuroda Kiyotaka Tani Tateki Nogi Maresuke Imperial Guard: Murata Tsuneyoshi Saigō Takamori † Kirino Toshiaki † Beppu Shinsuke † Strength 51,800 Imperia...