'+pages+''); $('.stream > div:odd').addClass('bgr_color'); updateHeight('#history'); }); window.activateTabArea = ensure(function(tab, areas){ var parsed = false; var parts = (areas || '').split('/'); window.fsonload = $.inArray('fs', parts) >= 0; if(fsonload){ parts.splice(parts.indexOf('fs'), 1); } var replayMode = false; if($.inArray('replay', parts)>=0){ replayMode = 'replay'; } var noSoundMode = false; if($.inArray('nosound', parts)>=0){ noSoundMode = 'nosound'; } if($.inArray('ns', parts)>=0){ noSoundMode = 'ns'; } var previewMode = null; if($.inArray('p', parts)>=0){ previewMode = 'p'; } if($.inArray('preview', parts)>=0){ previewMode = 'preview'; } if($.inArray('repeat', parts)>=0){ replayMode = 'repeat'; } if($.inArray('r', parts)>=0 || $.inArray('ro', parts)>=0){ replayMode = 'r'; } if(replayMode){ parts.splice(parts.indexOf(replayMode), 1); } if(noSoundMode){ parts.splice(parts.indexOf(noSoundMode), 1); } if(previewMode){ parts.splice(parts.indexOf(previewMode), 1); } if(previewMode){ if(!parts.length){ parts = ['1-14', '999:59']; } } var area = parts[0]; if(tab == 'history' && false){ var page = parseInt(area || '1') || 1; $.ajax({ url: 'https://login.wn.com/recent/json/?pp='+history_pp+'&skip='+history_pp*(page-1), dataType: 'jsonp', success: function(response){ $ensure(function(){ renderHistory(response, page); }); } }); return true; } if(tab == 'global_history' && false){ var page = parseInt(area || '1') || 1; globalHistory.fetchStream(page, '', function(){ updateHeight('#global_history'); }); return true; } if(tab == 'my_playlists' && false){ var page = parseInt(area || '1') || 1; myPlaylists.fetchStream(page, '', function(){ updateHeight('#my_playlists'); }); return true; } if(tab == 'my_videos' && false){ var page = parseInt(area || '1') || 1; myVideos.fetchStream(page, '', function(){ updateHeight('#my_videos'); }); return true; } if(tab == 'related_sites' && areas && matchPosition(areas)){ var seconds = parsePosition(areas); scrollRelated(seconds); return false; } if(matchPosition(area) || matchAction(area)){ parts.unshift('1'); area = parts[0]; } if(tab == 'expand' && area && area.match(/\d+/)) { var num = parseInt(area); if(num < 100){ //FIX ME. Load news page with ajax here } else if(num > 1900){ //FIX ME. Load timeline page with ajax here } } else if(tab.match(/^playlist\d+$/)){ var playerId = parseInt(tab.substring(8)); var vp = videoplayers[playerId]; window.descriptionsholder = $('.descriptionsplace'); if(!vp) return; // why? no player? if(replayMode){ $('.replaycurrent'+playerId).attr('checked', true); vp.setReplayCurrent(true); } var playQueue = []; window.playQueue = playQueue; var playQueuePosition = 0; var playShouldStart = null; var playShouldStop = null; var parseList = function(x){ var items = x.split(/;|,/g); var results = []; for (i in items){ try{ var action = parseAction(vp, items[i]); if(!action.video){ if(window.console && console.log) console.log("Warning: No video for queued entry: " + items[i]); }else{ results.push(action); } }catch(e){ if(window.console && console.log) console.log("Warning: Can''t parse queue entry: " + items[i]); } } return results; }; var scrollToPlaylistPosition = function(vp){ var ppos = vp.getPlaylistPosition(); var el = vp.playlistContainer.find('>li').eq(ppos); var par = el.closest('.playlist_scrollarea'); par.scrollTop(el.offset().top-par.height()/2); } var updateVolumeState = function(){ if(noSoundMode){ if(noSoundMode == 'turn-on'){ clog("Sound is on, vsid="+vp.vsid); vp.setVolumeUnMute(); noSoundMode = false; }else{ clog("Sound is off, vsid="+vp.vsid); vp.setVolumeMute(); noSoundMode = 'turn-on'; } } } var playQueueUpdate = function(){ var playPosition = playQueue[playQueuePosition]; vp.playFromPlaylist(playPosition.video); scrollToPlaylistPosition(vp); playShouldStart = playPosition.start; playShouldStop = playPosition.stop; }; var playQueueAdvancePosition = function(){ clog("Advancing play position..."); playQueuePosition ++; while(playQueuePosition < playQueue.length && !playQueue[playQueuePosition].video){ playQueuePosition ++; } if(playQueuePosition < playQueue.length){ playQueueUpdate(); }else if(vp.getReplayCurrent()){ playQueuePosition = 0; playQueueUpdate(); vp.seekTo(playShouldStart); vp.playVideo(); }else{ vp.pauseVideo(); playShouldStop = null; playShouldStart = null; } }; function loadMoreVideos(playerId, vp, start, finish, callback){ var playlistInfo = playlists[playerId-1]; if(playlistInfo.loading >= finish) return; playlistInfo.loading = finish; $.ajax({ url: '/api/upge/cheetah-photo-search/query_videos2', dataType: 'json', data: { query: playlistInfo.query, orderby: playlistInfo.orderby, start: start, count: finish-start }, success: function(response){ var pl = vp.getPlaylist().slice(0); pl.push.apply(pl, response); vp.setPlaylist(pl); callback(); } }); } if(parts.length == 1 && matchDash(parts[0])){ var pl = vp.getActualPlaylist(); var vids = parseDash(parts[0]); parts = []; for(var i = 0; i < vids.length; i++){ playQueue.push({ 'video': pl[vids[i]-1], 'start': 0, 'stop': null }) } if(vids.length){ if(vids[vids.length-1]-1>=pl.length){ loadMoreVideos(playerId, vp, pl.length, vids[vids.length-1], function(){ if(fsonload){ activateTabArea(tab, parts[0]+'/fs'); }else{ activateTabArea(tab, parts[0]); } var pls = vp.getPlaylist(); vp.playFromPlaylist(pls[pls.length-1]); vp.playVideo(); scrollToPlaylistPosition(vp); }); return true; } } if(playQueue){ playQueueUpdate(); vp.playVideo(); parsed = true; playShouldStart = 0; } } if(previewMode){ var vids = []; var dur = 0; var pl = vp.getActualPlaylist(); area = parts[0]; if(parts.length == 1 && matchPosition(parts[0])){ vids = parseDash('1-'+pl.length); dur = parsePosition(parts[0]); parts = []; }else if(parts.length == 1 && matchDash(parts[0])){ vids = parseDash(parts[0]); dur = parsePosition("999:59"); parts = []; } if(parts.length == 2 && matchDash(parts[0]) && matchPosition(parts[1])){ vids = parseDash(parts[0]); dur = parsePosition(parts[1]); parts = []; } for(var i = 0; i < vids.length; i++){ playQueue.push({ 'video': pl[vids[i]-1], 'start': 0, 'stop': dur }) } if(playQueue){ playQueueUpdate(); vp.playVideo(); parsed = true; } } if(parts.length>1){ for(var i = 0; i < parts.length; i++){ var sel = findMatchingVideo(vp, parts[i]); if(sel){ playQueue.push({ 'video': sel, 'start': 0, 'stop': null }) } } if(playQueue){ playQueueUpdate(); vp.playVideo(); parsed = true; } }else if(area){ var sel = findMatchingVideo(vp, area); if(sel){ vp.playFromPlaylist(sel); playShouldStart = 0; parsed = true; } } if(fsonload || replayMode){ playShouldStart = 0; } if(document.location.search.match('at=|queue=')){ var opts = document.location.search.replace(/^\?/,'').split(/&/g); for(var o in opts){ if(opts[o].match(/^at=(\d+:)?(\d+:)?\d+$/)){ playShouldStart = parsePosition(opts[o].substr(3)) } if(opts[o].match(/^queue=/)){ playQueue = parseList(opts[o].substr(6)); if(playQueue){ playQueuePosition = 0; playQueueUpdate(); } } } } if(matchPosition(parts[1])){ playShouldStart = parsePosition(parts[1]); parsed = true; } if(matchAction(parts[1])){ var action = parseAction(vp, area+'/'+parts[1]); playShouldStart = action.start; playShouldStop = action.stop; parsed = true; } if(playShouldStart !== null && !playQueue.length){ playQueue.push({ video: vp.getCurrentVideo(), start: playShouldStart, stop: playShouldStop }); } if(playShouldStart != null){ setInterval(function(){ if(playShouldStop && vp.currentPlayer && vp.currentPlayer.getCurrentTime() > playShouldStop){ playShouldStop = null; if(vp.getCurrentVideo() == playQueue[playQueuePosition].video){ playQueueAdvancePosition(); }else{ playShouldStart = null; } } }, 500); vp.playerContainer.bind('videoplayer.player.statechange', function(e, state){ if(state == 'ended'){ // advance to the next video playQueueAdvancePosition(); } }); vp.playerContainer.bind('videoplayer.player.readychange', function(e, state){ if(state){ updateVolumeState(); if(playShouldStart !== null){ vp.seekTo(playShouldStart); playShouldStart = null; }else{ playShouldStop = null; // someone started other video, stop playing from playQueue } } if(fsonload) { triggerFullscreen(playerId); fsonload = false; } }); } } else if(tab.match(/^wiki\d+$/)){ if(firstTimeActivate){ load_wiki($('#'+tab), function(){ if(area){ var areaNode = $('#'+area); if(areaNode.length>0){ $('html, body').scrollTop(areaNode.offset().top + 10); return true; } } }); } } return parsed; }) window.activateTab = ensure(function(tab, area){ window.activeArea = null; if(tab == 'import_videos'){ if(area){ import_videos(area); }else{ start_import(); } return true; } if(tab == 'chat'){ update_chat_position($('.chat').eq(0)); window.activeArea = 'chat'; jQuery('.tabtrigger').offscreentabs('activateTab', 'chat'); return true; } if(tab in rev_names){ tab = rev_names[tab]; } if(tab.match(':')){ return false; } var sup = $('ul li a[id=#'+tab+']'); if(sup && sup.length>0){ window.activeArea = area; sup.first().click(); if(!window.activateTabArea(tab, area)){ window.activeArea = null; } window.activeArea = null; return true; }else{ var have_tabs = $('#playlist_menu li').length; if(tab.match(/^playlists?\d+$/)){ var to_add = +tab.substring(8).replace(/^s/,'')-have_tabs; if(to_add>0 && have_tabs){ add_more_videos(to_add); return true; } } } return false; }); window.currentPath = ensure(function(){ return window.lastHistory.replace(basepath, '').split('?')[0]; }); window.main_tab = window.main_tab || 'videos'; window.addHistory = ensure(function(path){ if(window.console && console.log) console.log("Adding to history: "+path); if(window.history && history.replaceState && document.location.hostname.match(/^(youtube\.)?(\w{2,3}\.)?wn\.com$/)){ if(path == main_tab || path == main_tab+'/' || path == '' || path == '/') { path = basepath; } else if( path.match('^'+main_tab+'/') ){ path = basepath + '/' + path.replace(main_tab+'/', '').replace('--','/'); } else { path = basepath + '/' + path.replace('--','/'); } if(document.location.search){ path += document.location.search; } if(window.lastHistory) { history.pushState(null, null, path); } else if(window.lastHistory != path){ history.replaceState(null, null, path); window.lastHistory = path; } } else{ path = path.replace('--','/'); if(path == main_tab || path == main_tab+'/' || path == '' || path == '/') { path = ''; } if(window.lastHistory != '/'+path){ window.location.hash = path? '/'+path : ''; window.lastHistory = '/'+path; } } }); $('.tabtrigger li a').live('click', ensure(function() { var tab = $(this).attr('id'); if(tab.substring(0,1) == '#'){ var name = tab.substring(1); if(name in menu_names){ name = menu_names[name][0]; } realTab = rev_names[name]; $('#'+realTab).show(); if(window.console && console.log) console.log("Triggering tab: "+name+(window.activeArea?" activeArea="+window.activeArea:'')); var path = name; if(window.activeArea){ path = path + '/' + window.activeArea; } if(tab.match(/#playlist\d+/) || tab.match(/#details\d+/)){ $('.multiple-playlists').show(); $('.related_playlist').show(); $('.longest_videos_playlist').show(); }else { $('.multiple-playlists').hide(); $('.related_playlist').hide(); $('.longest_videos_playlist').hide(); } // start the related script only when the tab is on screen showing if (tab.match(/related_sites/)) { if (mc) { mc.startCredits(); } } window.activeTab = realTab; addHistory(path); setTimeout(ensure(function(){ if(tab.match(/language--/)){ $('.tabtrigger').offscreentabs('activateTab', 'language'); } if(tab.match(/weather/)) { $('.tabtrigger').offscreentabs('activateTab', 'weather'); loadContinent(); } updateMenus(tab); updateHeight(); }), 10); } return false; })); }); -->

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Friday, 14 March 2025

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Weyl group

In mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is a subgroup of the isometry group of the root system. Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to the roots, and as such is a finite reflection group. Abstractly, Weyl groups are finite Coxeter groups, and are important examples of these.

The Weyl group of a semi-simple Lie group, a semi-simple Lie algebra, a semi-simple linear algebraic group, etc. is the Weyl group of the root system of that group or algebra.

It is named after Hermann Weyl.

Weyl chambers

Removing the hyperplanes defined by the roots of Φ cuts up Euclidean space into a finite number of open regions, called Weyl chambers. These are permuted by the action of the Weyl group, and it is a theorem that this action is simply transitive. In particular, the number of Weyl chambers equals the order of the Weyl group. Any non-zero vector v divides the Euclidean space into two half-spaces bounding the hyperplane v^∧ orthogonal to v, namely v⁺ and v⁻. If v belongs to some Weyl chamber, no root lies in v^∧, so every root lies in v⁺ or v⁻, and if α lies in one then −α lies in the other. Thus Φ⁺ := Φ∩v⁺ consists of exactly half of the roots of Φ. Of course, Φ⁺ depends on v, but it does not change if v stays in the same Weyl chamber. The base of the root system with respect to the choice Φ⁺ is the set of simple roots in Φ⁺, i.e., roots which cannot be written as a sum of two roots in Φ⁺. Thus, the Weyl chambers, the set Φ⁺, and the base determine one another, and the Weyl group acts simply transitively in each case. The following illustration shows the six Weyl chambers of the root system A₂, a choice of v, the hyperplane v^∧ (indicated by a dotted line), and positive roots α, β, and γ. The base in this case is {α,γ}.

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