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Volumes of real and complex Stiefel manifolds
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Numbers
$\mathbb F$
$n$
$k$ 
$V_{\mathbb F}(n,k)$
$\mathbb R$
1
1:
2
comment: The two points of $O(1)=\{-1,1\}$ have total zero-dimensional volume $2$.
$\mathbb R$
2
1:
6.283185307179586476925286766559005768394338798750211641949889184615632812572417997256069650684234136
$\mathbb R$
2
2:
17.77153175263346498806352396024277479445848675750276089234158242782573917239789564230130773905906095
comment: Each component of $O(2)$ is a circle of Frobenius radius $\sqrt2$, so the total length is $4\sqrt2\pi$.
$\mathbb R$
3
1:
12.56637061435917295385057353311801153678867759750042328389977836923126562514483599451213930136846827
$\mathbb R$
3
2:
111.6618271942220725465209749787104846387178603351686778584559466803185242053292354346784213969729830
$\mathbb R$
3
3:
446.6473087768882901860838999148419385548714413406747114338237867212740968213169417387136855878919319
comment: The full orthogonal group $O(3)$ has volume $32\sqrt2\pi^2$ in this metric.
$\mathbb R$
4
1:
19.73920880217871723766898199975230227062739881448158125282669875244008964483841048600354680743710446
$\mathbb R$
4
2:
350.7959759997810578361195922781033018743809087272951348330102631032346530026103357559832960342247225
$\mathbb R$
4
3:
4408.232244439094382524383934799396466167437533524344172992751529550246481826436771789678495156353927
$\mathbb R$
4
4:
24936.72730470462393252872516830850847993026193220746795301577200383253899694776222467119801209676609
$\mathbb R$
5
1:
26.31894506957162298355864266633640302750319841930877500376893166992011952645121398133806240991613928
$\mathbb R$
5
2:
734.7053740731823970873973224665660776945729222540573621654585882583744136377394619649464158593923211
$\mathbb R$
5
3:
18465.16004593000611927380881656370822790048954023388121033100181046157036957359912213071265619088687
$\mathbb R$
5
4:
328154.1780737039145063149034333593254800102495849972449078873386612373889168555395410675743238539046
$\mathbb R$
5
5:
2625233.424589631316050519227466874603840081996679977959263098709289899111334844316328540594590831237
$\mathbb R$
6
1:
31.00627668029982017547631506710139520222528856588510769414453810380639491746570603756670103260288619
$\mathbb R$
6
2:
1154.072502870625382454613051035231764243780596264617575645687613153848148098349945133169541011930429
$\mathbb R$
6
3:
45560.95621403254293275096154028522869273764541484743002092202962963103589483152638457978980893394292
$\mathbb R$
6
4:
1619375.960076044630549943552666754695969651963160078112786788934838085785912554887856719245144561515
$\mathbb R$
6
5:
40699356.95659856099534083741182587612346785488116210799502835169669749532523609048340448038989967246
$\mathbb R$
6
6:
460460660.7030836713537177653297548112342119233286629955122778503066378449847656547383077298138336370
$\mathbb R$
7
1:
33.07336179231980818717473607157482154904030780361078154042084064406015457863008644007114776810974527
$\mathbb R$
7
2:
1450.250278691336907969392765508401082198829879150223486507778892289066996696585095744435534971693003
$\mathbb R$
7
3:
76338.11484401646735646210310167261346298206053375921001416480208800186636729086689261103254085966805
$\mathbb R$
7
4:
4262026.694022622821203817366638107108056093913338979857160075919050444709992093761928090117613551149
$\mathbb R$
7
5:
214232828.0215210463546541066064761593146497951622779888585656395649733913988705082419381541315124746
$\mathbb R$
7
6:
7614491011.281851220946934239109837508707815684033629839721255667575659032694926511230929218322872369
$\mathbb R$
7
7:
121831856180.5096195351509478257574001393250509445380774355400906812105445231188241796948674931659579
$\mathbb R$
8
1:
32.46969701133414574548011089623503708324252855756180723048928646332361848560906539670728907825099752
$\mathbb R$
8
2:
1518.698540467751431091698718009507623091254847161581000697400987654367863161050879485992993631131431
$\mathbb R$
8
3:
94178.37427942122790558765942988038752537451502349955551441468091490061643598883366033790775186467573
$\mathbb R$
8
4:
7010752.902814334568088304225533918679593128914034151774479244584868781618814952267163799151407255949
$\mathbb R$
8
5:
553546861.6365308234838091919064369697831564861924093953816663009543808349600443628346057513857884689
$\mathbb R$
8
6:
39349502512.57684307507363383666247629504013030542955258538686227699492290650528748794680251755519647
$\mathbb R$
8
7:
1977921728254.792333141949991505927093278999029349750245943630840645024667031096300242594412075521864
$\mathbb R$
8
8:
44755259734565.73940403375996461312512017104583420165462606123604144094759010438142874377413737362437
$\mathbb R$
9
1:
29.68658012464836182443895853370060533325031182405650946787591905218159404398543121984666430011519773
$\mathbb R$
9
2:
1363.180622214579774222537555387010954696108223817128750252942894428604756558765480225197009658208358
$\mathbb R$
9
3:
90169.93181356485090737966719791105507561253884509620132661227496942819234227094233277682903853736088
$\mathbb R$
9
4:
7907812.309093297478340131312949099568330806945605848505452375727919726213714919469065796448368406698
$\mathbb R$
9
5:
832501111.1340353311762232136274762857174673704278581589669128727946604834468820950287160367815034295
$\mathbb R$
9
6:
92958595210.44651816342107366354036883271840536963257316534602445828763729286451654617409456474412105
$\mathbb R$
9
7:
9345217273637.315879159741463464029262821356048534436220194470402338650511652993867863132523458439121
$\mathbb R$
9
8:
664315302054817.1985089473629134268325660621338607675816739597629463373055160770619240849941325939259
$\mathbb R$
9
9:
21258089665754150.35228631561322965864211398828354456261356671241428279377651446598157071981224300563
$\mathbb R$
10
1:
25.50164039877345443856177583695296720669172555233958254612437030718624934446783537381529170612542658
$\mathbb R$
10
2:
1070.639557066321780491345250487711149633119241370589182273711519500181001173655261203282481896004550
$\mathbb R$
10
3:
69526.68405258492379544090562833936582480649899660626194230260336188829788477774371522326920993571935
$\mathbb R$
10
4:
6503914.930734651024814243856464658482167713831467641866929006818708285026928766367411844037355495543
$\mathbb R$
10
5:
806648743.3899665210178075169760471396284112754019126042489334512689838416651727377925497841683187625
$\mathbb R$
10
6:
120095830121.1291012852377950739002815516614445581045443118981802293817851706197319760360017928539692
$\mathbb R$
10
7:
18964773336255.61181826181531267312999765034481535055581014156658133710697702998729190372713543176353
$\mathbb R$
10
8:
2696264572214104.785052039223129150030015580207770574085618192761813031966377120255256156943574131215
$\mathbb R$
10
9:
271058079110472260.7836762016129054905251031942736695629639829700811860980907745621171964588868183938
$\mathbb R$
10
10:
12266688373402534229.72445707539767453239682724224242113224918048275056202640212929911576565457554518
$\mathbb R$
11
1:
20.72514267328890265483115750558717334893041822666327978351694856711009470534528843078322119608923557
$\mathbb R$
11
2:
747.4474149160414913141877732530189486140834525674567897969419120230207653649907433360151408878148868
$\mathbb R$
11
3:
44378.31514373270966882846655618978324215159679481596180170313420440222732307022276618434038030834412
$\mathbb R$
11
4:
4075623.328551714461773542475377693802060547173455034999762981922630254517127012906756510709346600176
$\mathbb R$
11
5:
539178259.4976382139377044472362280848245985780575189414100269029183840093581937028023707460499739531
$\mathbb R$
11
6:
94570781889.16860904916107493034080670812086342871520422195625967842537783655514846639692921182363914
$\mathbb R$
11
7:
19912025709819.74000115973488857086804898354947559825772990258446366317970625362923827883034609512743
$\mathbb R$
11
8:
4446826347793597.822310556941503649321622437850380090934809720213884545955484659495665132550305009121
$\mathbb R$
11
9:
894087487105145458.6594457711011822703102568551131243978170825359321095610319531024869609501750430658
$\mathbb R$
11
10:
127114433333770849603.9340078123133977188428693624381296532875707651338843083668542858472219803332324
$\mathbb R$
11
11:
8135323733361334374651.776499988057454005943639196040297810404528968568595735478674294222206741326873
$\mathbb R$
12
1:
16.02315322625507395050365739420699831445362546801744359901793645806823188070583689165369015253192894
$\mathbb R$
12
2:
469.6350615289835984759357666557867451854819731515427152427722654657718351159944913165459845757154213
$\mathbb R$
12
3:
23952.92891553597021123431967485127637686349813491276630151695179585078002725142585200469500920761186
$\mathbb R$
12
4:
2011239.497032823103977225660637225295070049146200414935139125530141004851556436029840151451148968427
$\mathbb R$
12
5:
261217348.3435353873167045688330417238280843300248959578855671000950435545302734622295228933739198923
$\mathbb R$
12
6:
48871463818.79748692949971437445592654538337100133978631155399461971141650982060549839232199621554874
$\mathbb R$
12
7:
12122577031495.17532386150434611448266209057515739804550679088313723887293895405333384010524562198948
$\mathbb R$
12
8:
3609677604371892.944025432261777336096561612741133639020986130350086354528480258599725026512133792514
$\mathbb R$
12
9:
1140034879059920871.878999023730876647872428601415234889356301975597903615707224123435484828631940245
$\mathbb R$
12
10:
324162656837168573860.1735809428613518238010301386922576710086633861550530245694681659885815853054450
$\mathbb R$
12
11:
65176769362419068936080.47177060147954691895821349774623835745330928093824733531297984507678286319575
$\mathbb R$
12
12:
5899127755775761303220868.307227679875260540007430767036878106531416296365376521790896776376404993118
$\mathbb R$
13
1:
11.83817381218268089849620863756707523376660352038229599234802809023222903740460099354500628873931835
$\mathbb R$
13
2:
268.2549198486628006116879678167640942315106873715401024658293195478663248238145444128410647463358603
$\mathbb R$
13
3:
11119.24297335043133892156571170111868410225664512294764333516559829205261716391368568884985672707964
$\mathbb R$
13
4:
802025.7855175716740229630875181613851192669800397549233201072728181244646159651669778424533967094665
$\mathbb R$
13
5:
95237610.97520573284936305892369381392121297100173437627748697632952171316276050745698800129945122996
$\mathbb R$
13
6:
17492896149.34368733601948679092998941138611133823671790301714738418307783398853294926422113365366447
$\mathbb R$
13
7:
4628391065141.774448171669485112179957026361826102386365050461896556447115025239570935466291229949289
$\mathbb R$
13
8:
1623620961005074.995963155327346295299023753466869196551015533659583059083951402857092868748896047837
$\mathbb R$
13
9:
683711854183962542.5100440543751772714246989381685557388782330310433067690716663336189318941729371941
$\mathbb R$
13
10:
305378059650123139761.8825131673184748076497214894016891899304872956404071166015398987834149709919881
$\mathbb R$
13
11:
122799804001834562710746.4557069278119388380351648034864602100217138282417491275742259798587952593961
$\mathbb R$
13
12:
34917449856572303581460656.67727856869028350682861961442101771422169700393865532615425148491228705376
$\mathbb R$
13
13:
4469433581641254858426964054.691656792356288874063310645890267420377216504147881747744190068772742881
$\mathbb R$
14
1:
8.389703410491089076428351369644560644215090209153895174317318034397596298037810780384590619694471488
$\mathbb R$
14
2:
140.4579475809796440311937487441199309592050484779924483682880875997743391611671590206530691109812512
$\mathbb R$
14
3:
4501.158431870680086135739732734617205736062044343291494675101849690182286875654407396765681689888117
$\mathbb R$
14
4:
263855.9074177125124411157260939815392202872020453494524096637374697409641719933961914372660342625175
$\mathbb R$
14
5:
26915033.87223426316300563653012665444884206475372038699813209808029639483196425626033968847734287128
$\mathbb R$
14
6:
4519913149.552514109542645981554863239283556470803111684619591695855905385766835518491666300029247353
$\mathbb R$
14
7:
1174081683868.121384952118076809723438841780103659623909189327592699185886611808661881172578859691318
$\mathbb R$
14
8:
439320672209049.6432387323037208820859488908065752702058432377350362170168354325079227070198758481723
$\mathbb R$
14
9:
217947173022225555.8263602409232843402222810540671441041336770688460361401126023467490220944254130890
$\mathbb R$
14
10:
129794024381210500371.6998947176633185727125785112332360335618825518961967690027865511572886830297458
$\mathbb R$
14
11:
81985003153145258939367.46479493222524334248161772670934164419767532488210011588791143635349498357678
$\mathbb R$
14
12:
46623970776828022942496828.13107096572819924010924957626948057454935692421591487482688403860958179158
$\mathbb R$
14
13:
18748611081429519646094914986.86542692731464717623641340813812918119328678942374849819197372046224603
$\mathbb R$
14
14:
3393861128578064255621465312411.379631327415076261688380130022137311308847276724075184469130624941959
$\mathbb R$
15
1:
5.721649212349567242266155046630722677107061188260798307359929182299792613513631860881665530847245290
$\mathbb R$
15
2:
67.88640865518552615423935006410150408545917341864329821064727697351739319525602258813163956777937255
$\mathbb R$
15
3:
1607.302210289897967698734866662634383847555371039249783805961914651498184094741027520108217848781879
$\mathbb R$
15
4:
72843.45245044144776502856342502804823519348743576711472272823801427578585438412939236407442047187606
$\mathbb R$
15
5:
6038763.779401340534251574707487686971425428687486015882801051847293026854055233494610318493117434260
$\mathbb R$
15
6:
871146403.6422727696890110225362231283349475783423348801260548746646612456253219927053816874052799217
$\mathbb R$
15
7:
206890860096.2047526596413251782657953534974963522545767423836521115334908666376618571745467606070065
$\mathbb R$
15
8:
76001913379652.27893034644806929564001773434804778964921808050669062312727919046998254787102531615095
$\mathbb R$
15
9:
40218220449820660.90176647613902747670238490107823966339929100360296997961237612565255027827491503849
$\mathbb R$
15
10:
28216779791338384603.57501624193193458401455743261208330100111172560534603605431104816347224445410222
$\mathbb R$
15
11:
23764348075789874759149.70417791189225706684431521416331982774617046528845675932593678774181887618878
$\mathbb R$
15
12:
21228564225777542557637425.01977484986528275317719532244375692923798678560137389514039833964716272906
$\mathbb R$
15
13:
17073024362998227085850142245.96346101916341356184076857916770417357513370941715801720937933060623423
$\mathbb R$
15
14:
9709241426576247352006629072757.587247983952443520625039029830737255964903291251451070902040805389558
$\mathbb R$
15
15:
2485565805203519322113697042625942.335483891825541280009991636668737527015242560371474150922446179727
$\mathbb R$
16
1:
3.765290085742291272670069644407446744721102964936937576234163259947186072155308053982327113896394892
$\mathbb R$
16
2:
30.46734895853220058949114024357840533028489546278804326361484364237179836107073070774308971469006910
$\mathbb R$
16
3:
511.2240429320394682834474016518077671991746152922245050245944895133561385540069251932103605360712723
$\mathbb R$
16
4:
17117.52521179572232277725929053253225880382229537274418102855712931176663319708932456227881240143801
$\mathbb R$
16
5:
1097106.917291548784671309380439817665799634547181792837880448986032933833166996807290785340134217843
$\mathbb R$
16
6:
128623840.0970497928981570492249511994467338554813913126727597367427258759859424675803743918551995199
$\mathbb R$
16
7:
26240951334.91441535373678867768140255426915239718309817746194447957195052408047603562074014859738860
$\mathbb R$
16
8:
8813425356139.186519314882383580308187159553155176033533226566879176667846469543358916811726117562362
$\mathbb R$
16
9:
4578708015133585.977792485930364497635460868430029664770708861380076760783489336476848713459443075003
$\mathbb R$
16
10:
3426543673571854061.267241628731143923435206922105611028351385850532364858533629433248402165800013749
$\mathbb R$
16
11:
3399819558396795449911.082748812112954081553539022239422256897727098272491761375251281467767328223311
$\mathbb R$
16
12:
4049387349559886632516072.506977609545256000658954937051042141027183548274190976554652871741394516166
$\mathbb R$
16
13:
5115628954487274263299101516.037081063458106545393662567886906811643487831983928557273527056135917022
$\mathbb R$
16
14:
5818403993559150674223621735231.963821419088119439727970930267243515741356832293299372988295870798678
$\mathbb R$
16
15:
4679438141896433257755175744512834.090904398953009387824874799070585004021126308971011664089877382261
$\mathbb R$
16
16:
1694137570446308227701899413736860781.251230974666709851922506059445330780010806433603030309900256245
$\mathbb C$
1
1:
6.283185307179586476925286766559005768394338798750211641949889184615632812572417997256069650684234136
$\mathbb C$
2
1:
19.73920880217871723766898199975230227062739881448158125282669875244008964483841048600354680743710446
$\mathbb C$
2
2:
248.0502134423985614038105205368111616178023085270808615531563048304511593397256483005336082608230895
comment: The unitary group $U(2)$ has Frobenius volume $8\pi^3$.
$\mathbb C$
3
1:
31.00627668029982017547631506710139520222528856588510769414453810380639491746570603756670103260288619
$\mathbb C$
3
2:
1224.078739141125813050965240173742425921202826512299962213969774744939968534456097943134001894020476
$\mathbb C$
3
3:
30764.45419440974198496702219687743676375096089859349171011443799949100521095520683197508509286130356
$\mathbb C$
4
1:
32.46969701133414574548011089623503708324252855756180723048928646332361848560906539670728907825099752
$\mathbb C$
4
2:
2013.528818517861378342804328714694554611621650196934841836156328255423111529074587292301748726673157
$\mathbb C$
4
3:
158981.8631117131288880501461399448286072963081295394843555794988152245911142034784507987515230455956
$\mathbb C$
4
4:
7991300.051292417789757157562446450827605176087510587232175142348219583831235175803615168970525408405
$\mathbb C$
5
1:
25.50164039877345443856177583695296720669172555233958254612437030718624934446783537381529170612542658
$\mathbb C$
5
2:
1656.061074080345092583855688957758631326003209682702962037286445991922824106286233862486995031724954
$\mathbb C$
5
3:
205393.1514496387080494620378574390238162049486533442902679466186973275864887193047020644282638844516
$\mathbb C$
5
4:
32434386.42401547793850444483826779119823182536283648801064450982154575204844009938484018074526104310
$\mathbb C$
5
5:
3260660163.628145601806657077664488485886941433211708863622246240901268582009183403323294321777270580
$\mathbb C$
6
1:
16.02315322625507395050365739420699831445362546801744359901793645806823188070583689165369015253192894
$\mathbb C$
6
2:
817.2333832608072141824878944644018528543464476100081972448945601080833106951542215039755025158404336
$\mathbb C$
6
3:
106141.2813681036979754771184135575678999655414456936479043699948210048366539447722435968203703538600
$\mathbb C$
6
4:
26328367.49840780387171092136193695752014699175039556301116418497871009143632342810370538900617945329
$\mathbb C$
6
5:
8315218295.545078020505842340697028329305234438822340497465822113676716933153640210870376155844016340
$\mathbb C$
6
6:
1671873837457.910993061522618031710366147524809617416155044059982564895416517711567517663664900270797
$\mathbb C$
7
1:
8.389703410491089076428351369644560644215090209153895174317318034397596298037810780384590619694471488
$\mathbb C$
7
2:
268.8590065382669819433885369896114009694881357354545938240728890524773020387081298629441762404491055
$\mathbb C$
7
3:
27425.38281084146236636554308535099741681978307332871146996269268615634524617023760802644688143693051
$\mathbb C$
7
4:
7123950.962302991071811634407136110008325617895824312377366651087232766927219324386614099868336372599
$\mathbb C$
7
5:
3534195113.504875126268218106199686291954894289217306190632061642721859154543693987318082176535816410
$\mathbb C$
7
6:
2232390889379.598136013224890812691416051021972538683896694334070236919783327919330477718503012803582
$\mathbb C$
7
7:
897697640706013.4763972269948744702228556038324055267802052703617811972502298285694526627142415660198
$\mathbb C$
8
1:
3.765290085742291272670069644407446744721102964936937576234163259947186072155308053982327113896394892
$\mathbb C$
8
2:
63.17933414768077260445066572235569517855091752719859994307737786964573610353855490690286926806777383
$\mathbb C$
8
3:
4049.328607124234137305662652955337684625352670863114989856683598024629663177276048248929619926018601
$\mathbb C$
8
4:
826116.1759627872885208014229038453543158843320538046228684592047473964045391402632126488583749018607
$\mathbb C$
8
5:
429179870.8747793227965267936163296477609098945081344189380558418058525254763796692999378334016510726
$\mathbb C$
8
6:
425832634302.6802592245990299053663878003956891629875734005517986065382495339819507568632843149176529
$\mathbb C$
8
7:
537958354130074.6839801425709380692095186654095319510367177689767074121956161605192810657722514927152
$\mathbb C$
8
8:
432651779397708552.5436730861609613431401248328249649183397655276106980990370745687122653607669223175
$\mathbb C$
9
1:
1.478625959000308118378442579426570206541074294359556216104720041064216807237755565948649102706897280
$\mathbb C$
9
2:
11.13491132789009562996939410839741378866793298475959339636160081417273172980062582395513246612255026
$\mathbb C$
9
3:
373.6744141724615869284218342300058228894486379979552620204454900193837469057180341189235115631864915
$\mathbb C$
9
4:
47899.53916013162084782665107325107675903012215492571639933658948726032650406023094865377604929141435
$\mathbb C$
9
5:
19544269.16685829831005796054433438290255103009657908981751102181234494874558804899964344468803379296
$\mathbb C$
9
6:
20307087940.98716746525629199063474377153665201515532858484099245023755446689936999234348421423298272
$\mathbb C$
9
7:
40297419985243.39849133507908799497715784910284270169310643962427393342595109106023237235235248387120
$\mathbb C$
9
8:
101816215971559557.7852805687580882533377789888650308360085027598177626788989630091233704091050228017
$\mathbb C$
9
9:
163770918969632399071.2547391897082869861082618085978853526698651255244525713624409177361878042426779
$\mathbb C$
10
1:
0.5161378278002812002519658850579769931437288209629337198360474503593377867189551171109747650647301022
$\mathbb C$
10
2:
1.526349581215053363485577592728719142937303110535017278930371623699596650383155947247289067541453124
$\mathbb C$
10
3:
22.98859578210375462183078455302171236598839961702330013026696942598816162774938271432332766016469644
$\mathbb C$
10
4:
1542.940003484135483164760162322372068437479563996458173381638158884470092867332081081037527350596786
$\mathbb C$
10
5:
395564.2255159174481435411332164888018172877295997000181021625009505940821937458011629859808427588091
$\mathbb C$
10
6:
322801172.2792401181943019609399815695825634293782448590288044463498654909774157281072843559761264137
$\mathbb C$
10
7:
670800400563.8656999320264637325432310469435597149249111672569031337146420177361627171394898870128160
$\mathbb C$
10
8:
2662274920593813.426721396528410823561635036564850538428322046957534392901102976370389715886897687358
$\mathbb C$
10
9:
13453107339879692154.67742901517454656442176031842586306885665079491138285788662522482540534429603950
$\mathbb C$
10
10:
43278523583407069718947.38108487273738047257462848087052646651171435820395807291584198677168317807642
$\mathbb C$
11
1:
0.1621494808057157170996999790603971930266560411924619438977965397039847838680861550603125997055238955
$\mathbb C$
11
2:
0.1673829616040110010460343358529818818049500938683496606256896328548540700718120059626707018818625414
$\mathbb C$
11
3:
0.9899871684881700737398034696025768179363037813104686755815267337810087323094459108104699084261618377
$\mathbb C$
11
4:
29.82071096416472042936258643910909668916940568797117338960292600664567809590738003493876394119398406
$\mathbb C$
11
5:
4002.990727669148291099974714037469860894883796853234858988150376362419282954064088631998462005535885
$\mathbb C$
11
6:
2052497.081367073905918304856141668874802600271622033122936442932059860790728555520365020074553302728
$\mathbb C$
11
7:
3349890719.267531389691926277913406663806032737679941781464750974341688403830342329810849897448070866
$\mathbb C$
11
8:
13922551894489.21034302658163748272092137979493240482568934633425722585850587044392598166017359553734
$\mathbb C$
11
9:
110511743010909410.5914070417067266632867527455018728599866255099409602068138855190644605612569639101
$\mathbb C$
11
10:
1116884157637148350981.851167117284120830760946417833024794133201498884102173895270200365943640453580
$\mathbb C$
11
11:
7186012292185476758806700.407577824062153705342529836058805103525090972181444299431545073614853474084
$\mathbb C$
12
1:
0.04630978342569415314968029352846298336404332356000728045363154281595589522202365275569634475646745392
$\mathbb C$
12
2:
0.01501821467740289186765197405484668006162896740941971288521137796576518636103153356916284449088702538
$\mathbb C$
12
3:
0.03100587480413211791903815368570490929812014311182662110918212604141446868046021206147472017755165644
$\mathbb C$
12
4:
0.3667687309352267480048457373542147622478118930719075753297648317951964274713200658651256270802060181
$\mathbb C$
12
5:
22.09585066161106047281495987706907650202702960900045824153799990655639838430084852962541103394791790
$\mathbb C$
12
6:
5932.084276909443233177639469855910493939663881391942368632861493857251215806236671245845871129569062
$\mathbb C$
12
7:
6083244.500478626740658309612121508450350011162358671245416721599461070073879705380331623537411154989
$\mathbb C$
12
8:
19856987354.75484038206550070607656069757783763539242040859803490509908989545412738856285908742525389
$\mathbb C$
12
9:
165056092919496.4902998983622936455676051325358719898919950194383580654334208827817932456119586405047
$\mathbb C$
12
10:
2620300741033566357.130313318279014039955982293068829983245942510738294517323766213596752399708121119
$\mathbb C$
12
11:
52964007374607553049003.27592599808104393341435754351102138664056779282919681129253339116914492793480
$\mathbb C$
12
12:
681538914192354184595396836.8758978041467184852499127641673302550589721093745856288620959680029127590
$\mathbb C$
13
1:
0.01212387294995792667493635177622706533953319515401580960606389290460305294186093659194373089107364292
$\mathbb C$
13
2:
0.001122907861186366543545739665886742848660018505476382946915663377868357606096354495106417005007837800
$\mathbb C$
13
3:
0.0007283157067361041229520108941748503652810138003475011712894583379774115764623255976117097345920186043
$\mathbb C$
13
4:
0.003007290294620795299322559445520844303338258594383460819660011616344244216632357054543842292426668375
$\mathbb C$
13
5:
0.07114651993401588147540153882967339796954434677233849021868688964933182307147111955793237823584012100
$\mathbb C$
13
6:
8.572393156563721394551414834902468392561230556624660733192304743107794002919408059855845758662454609
$\mathbb C$
13
7:
4602.869510508360057868515717454287391155427263952188421189047309283349308407055800808555990409149291
$\mathbb C$
13
8:
9440317.881258642236275668156605389168140328669800120263024301092847081190514710105307704773862240091
$\mathbb C$
13
9:
61630359515.64001630871814053613516260300656833706375003910521842919780386965193649454763036338948719
$\mathbb C$
13
10:
1024572979288281.890670457569562243109842166097124802017471309055676799666572884613003207678587311159
$\mathbb C$
13
11:
32530629913570883004.82582702421747526300175047516717611974130189347492914840258733673388470335254999
$\mathbb C$
13
12:
1315079979684611394309233.697055577083157200820897760626900313639855402563512056410236724532462309158
$\mathbb C$
13
13:
33844802380269073969787279907.51719381714802005941106200833390063091399614081389704687527841280752820
$\mathbb C$
14
1:
0.002929866937895679799304468393862701173088325281745569969187759381600184397528840100224589207925776760
$\mathbb C$
14
2:
0.00007104266903065898599315510519140354001638712179450232257534646552143689137628279252324967049037264032
$\mathbb C$
14
3:
0.00001315988246717234727536979213802232613718345833785304210672865119530547130429140143294243871589756210
$\mathbb C$
14
4:
0.00001707094487612989856052932285480278478780254547915819114337937938673169508570251683215214615976586509
$\mathbb C$
14
5:
0.0001409753665098244202826626396551685168505346076683322041534891647116380006427013304619654891872935797
$\mathbb C$
14
6:
0.006670394768032289710828814465661046548485915981362883153813185214243656405234212599002099360073405330
$\mathbb C$
14
7:
1.607422162435790798694765731625883775039000934279225518044989244605145967604718919034282962891979181
$\mathbb C$
14
8:
1726.181785380673975684755904679619697747549602730499357416863960892667552453149410136354230793016915
$\mathbb C$
14
9:
7080672.062342423003860978462126675397729433942346869378635478705372272329299120239957302576275053610
$\mathbb C$
14
10:
92451201390.33502014417139125637069328421589655769219469336575672157564116672444803487212783831005387
$\mathbb C$
14
11:
3073907197417484.516836282339340159840620745995387651422918389271939422172711691907583696659293720439
$\mathbb C$
14
12:
195195734123912244826.0495494043190478388309641308461383077920916562431187739729431228845619203774257
$\mathbb C$
14
13:
15781926310569841394292043.23972293037364697248045530721832343994492094296450061370133236143410127452
$\mathbb C$
14
14:
812325007471111100753067103721.0238766267144920659642963576492522845952939053005304919708411983564796
$\mathbb C$
15
1:
0.0006574606034349064748215395617128256657774612629786734093310582945142954556229037099036678869887172601
$\mathbb C$
15
2:
0.000003852544169945750587204046904415324821697498184789082211643198918323674862361175096840306544724212492
$\mathbb C$
15
3:
1.868310242020935966738622800188959740532401202747829841566034360865223785756231284053853627114029711e-7
$\mathbb C$
15
4:
6.921683414399661789962992772125176150110097231614379276086941803514032978471000813519571145559600335e-8
$\mathbb C$
15
5:
1.795755795114302059501586899807578815716353512047260775908766350047342261255193220823284688386497171e-7
$\mathbb C$
15
6:
0.000002965943985120200590996066409992698016506093708144383895858644428986490387582781514677682437800985556
$\mathbb C$
15
7:
0.0002806733932377313402230510220052765752716602412448390407471538814201208219601279921472333380800542092
$\mathbb C$
15
8:
0.1352725433458786910088548118189383834001635296115576591349098896263347425702830178139015286739516365
$\mathbb C$
15
9:
290.5335086732088657359788157197808667217305400763161130758482592109440238057029183942785351314658209
$\mathbb C$
15
10:
2383494.618538154571059309475755277722516106413938763572994867049897461749413769138647175771999215295
$\mathbb C$
15
11:
62241815198.07664824902747196459311748742315680929862402451523989947116356561882299895455458444990253
$\mathbb C$
15
12:
4138952460118019.187383792246742945395968051029148052736736344194337678914309578715393230356642314840
$\mathbb C$
15
13:
525654037074030222665.5286812767680889976429708936420645617904437278336771715497078197984157514651977
$\mathbb C$
15
14:
85000149364838196300421881.71010676241627122412185674289681066888583532652641969137228377547935022209
$\mathbb C$
15
15:
8750230562360878936715214330410.557581595075747043931694091316347140973497793269156447359390665962077
$\mathbb C$
16
1:
0.0001376982267850543042335303588789415456286748849792269148098615408331861193626278973993807612251025042
$\mathbb C$
16
2:
1.810623185480368092987743736034201163767656414541281509678166746915767318465751225184154651762793105e-7
$\mathbb C$
16
3:
2.121954003250515019848642105433004984935875225488204487347958088887832073279239064766817448042951521e-9
$\mathbb C$
16
4:
2.058104059285108276973023723056641640167348649126038399140417404624267237647696004051674904566609297e-10
$\mathbb C$
16
5:
1.524965652048565824790691178461473558046738994026505945067013519137994386778038982251164711161583774e-10
$\mathbb C$
16
6:
7.912716439239189668105710699139527033863580188626187437957032228446439547748279248832498713932992768e-10
$\mathbb C$
16
7:
2.613793455967034302582229198073262589682864546619216494593403686745744253891808448109928039245901557e-8
$\mathbb C$
16
8:
0.000004946973254986221729078759952126176889654979087837505440768693262277617811277509556938165670982467754
$\mathbb C$
16
9:
0.004768458073966564495723604610931058583678923468115976716367278491722599813453530130904714669696960916
$\mathbb C$
16
10:
20.48304587056182069621187142758173592791595284282962380047113569464480980230695876476485459776206341
$\mathbb C$
16
11:
336079.8541050094665210804669205684361132277174301298756744678689190342894392874978976881087323607334
$\mathbb C$
16
12:
17552563373.37998950497306170381493370846864061836643428688991796194148770322376320622050666717942226
$\mathbb C$
16
13:
2334418593816123.132192031284975720357262916237763817819086104628035079440791471632420754931713812313
$\mathbb C$
16
14:
592950303191033091474.4521491877457654076053957134449379118578794317342623571865779878539997268296040
$\mathbb C$
16
15:
191764395524144816569828373.5980592654439731179370083699501843276891197087877665907225747565114102039
$\mathbb C$
16
16:
39481875903200676000682612310021.00765170997055751771440690610056298140219735490154785744692734083280
Definition
$V_{\mathbb F}(n,k)$ is the volume of the Stiefel manifold of ordered orthonormal $k$-frames in $\mathbb F^n$, where the scalar field $\mathbb F$ specifies the frame space and the ambient Frobenius metric (1) fixes its volume [1] [2].
Parameters
$\mathbb F$
—   scalar field ($\mathbb F\in\{\mathbb R,\mathbb C\}$, identified by real and complex, respectively.)
$n$
—   ambient dimension ($n\geq1$.)
$k$
—   number of columns ($1\leq k\leq n$.)
Formulas
(1)
$\displaystyle \operatorname{St}_{\mathbb F}(n,k)=\{Q\in\mathbb F^{n\times k}:Q^*Q=I_k\},\qquad g_Q(\Delta_1,\Delta_2)=\operatorname{Re}\operatorname{tr}(\Delta_1^*\Delta_2)$. Here $*$ is transpose over $\mathbb R$ and conjugate transpose over $\mathbb C$, and $\Delta_1,\Delta_2$ are tangent vectors to the frame space.
(2)
$S_d=2\pi^{(d+1)/2}/\Gamma((d+1)/2)$ is the unit $d$-sphere volume, with $\Gamma$ the Gamma function. The one-column cases are $V_{\mathbb R}(n,1)=S_{n-1}$ and $V_{\mathbb C}(n,1)=S_{2n-1}$.
(3)
$\displaystyle V_{\mathbb R}(n,k)=2^{k(k-1)/4}\prod_{j=0}^{k-1}S_{n-j-1}$. This follows from the canonical quotient metric in [1], equations (2.22) and (2.39), and the orthogonal-group volume in [3], equations (A24)–(A27), with the metric correction described in the comments.
(4)
$\displaystyle V_{\mathbb C}(n,k)=2^{k(k-1)/2}\prod_{j=0}^{k-1}S_{2(n-j)-1}=\frac{2^{k(k+1)/2}\pi^{nk-k(k-1)/2}}{\prod_{j=0}^{k-1}(n-j-1)!}$ [2], Theorem 1, equation (35).
(5)
$\displaystyle V_{\mathbb F}(n,k)=2^{\beta(k-1)/2}S_{\beta(n-k+1)-1}V_{\mathbb F}(n,k-1)\quad(2\leq k\leq n)$, where $\beta=1$ for $\mathbb F=\mathbb R$ and $\beta=2$ for $\mathbb F=\mathbb C$.
(6)
$\displaystyle G_{\mathbb F}(n,k)=\frac{V_{\mathbb F}(n,k)}{V_{\mathbb F}(k,k)}$, where $G_{\mathbb F}(n,k)$ is the volume of the Grassmann manifold with the quotient metric specified in the comments. For $1\leq k<n$, orthogonal complementation gives $G_{\mathbb F}(n,k)=G_{\mathbb F}(n,n-k)$. The complex quotient formula is [2], equation (91).
Comments
(7)
The columns are ordered and no volume is normalized to one. At $k=n$ the frame space is $O(n)$ over $\mathbb R$, including both components, or $U(n)$ over $\mathbb C$. At $k=1$ it is the unit sphere in $\mathbb R^n$ or $\mathbb R^{2n}$, respectively.
(8)
This is the embedded metric used for complex frames in [4]. For real frames, the canonical Stiefel metric of [1], equation (2.39), assigns half the Frobenius squared norm to tangent rotations within the frame. Passing from that metric to the metric here multiplies the volume by $2^{k(k-1)/4}$. Together with the half-trace orthogonal-group volumes in [3], equation (A26), this gives (3).
(9)
The full-frame values give the geometric total masses removed when normalizing Haar measure to one in orthogonal Weingarten functions and unitary Weingarten functions.
(10)
Forgetting the ordered basis gives the real or complex Grassmann manifold. The metric on this quotient is $ds^2=\tfrac12\operatorname{Re}\operatorname{tr}(dP^2)$ for the orthogonal projector $P=QQ^*$. The fibre is $O(k)$ or $U(k)$ with its Frobenius metric. The same complex subspace space occurs in chiral unitary Weingarten functions, where $W=I_n-2P$ has signature $(n-k,k)$.
Programs
(P1)
Sage
import numberdb.sage as numberdb
from sage.rings.rational_field import QQ
from sage.rings.real_arb import RealBallField

def stiefel_volume(field, n, k, digits=100):
    beta = {"real": 1, "complex": 2}[field]
    if n < 1 or not 1 <= k <= n:
        raise ValueError("need integers 1 <= k <= n")
    if field == "real" and n == k == 1:
        return QQ(2)
    R = RealBallField(numberdb.bits(digits, losing=96))
    v = R(2)**(QQ(beta*k*(k-1))/4)
    for j in range(k):
        a = QQ(beta*(n-j))/2
        v *= 2*R.pi()**a/R(a).gamma()
    return v

stiefel_volume("real", 3, 2)
# Table selection: both fields, every 1 <= k <= n <= 16.
References
[1]
A. Edelman, T. A. Arias and S. T. Smith, The Geometry of Algorithms with Orthogonality Constraints, SIAM Journal on Matrix Analysis and Applications 20 (1998), 303–353. Sections 2.2–2.4. https://www.cis.upenn.edu/~cis5150/Stiefel-Grassmann-manifolds-Edelman.pdf
[2]
R.-A. Pitaval, L. Wei, O. Tirkkonen and C. Hollanti, Density of Spherically-Embedded Stiefel and Grassmann Codes. Theorem 1 and Appendix B. (arXiv)
[3]
K. Życzkowski and H.-J. Sommers, Hilbert–Schmidt volume of the set of mixed quantum states, Journal of Physics A: Mathematical and General 36 (2003), 10115–10130. Appendix A, equations (A23)–(A27). (arXiv) (doi)
Links
Similar tables
Volume_of_the_unit_sphere —   Unit-sphere volumes are the one-column cases and the factors in the product formulas.
Values_of_the_Gamma_function_at_rational_numbers —   Integer and half-integer Gamma values determine the sphere factors.
Orthogonal_Weingarten_functions —   Normalized Haar integration on $O(n)$ divides the geometric measure by the full real frame volume $V_{\mathbb R}(n,n)$.
Unitary_Weingarten_functions —   Normalized Haar integration on $U(n)$ divides the geometric measure by the full complex frame volume $V_{\mathbb C}(n,n)$.
Chiral_unitary_Weingarten_functions —   Coefficients for integration over complex subspaces use the quotient obtained by forgetting the ordered frame basis.
Bures_volumes_of_quantum_states_of_fixed_rank —   The rank-one volumes equal $V_{\mathbb F}(n,1)/V_{\mathbb F}(1,1)$, the projective-space quotient volumes.
Hilbert_Schmidt_volumes_of_quantum_states_of_fixed_rank —   The rank-one volumes are $2^{\beta(n-1)/2}V_{\mathbb F}(n,1)/V_{\mathbb F}(1,1)$, where $\beta=1$ over $\mathbb R$ and $\beta=2$ over $\mathbb C$.
Volumes of real and complex Grassmann manifolds —   Forgetting the ordered frame basis gives the Grassmann manifold with metric $ds^2=\tfrac12\operatorname{Re}\operatorname{tr}(dP^2)$, where $P=QQ^*$. Its volume is $G_{\mathbb F}(n,k)=V_{\mathbb F}(n,k)/V_{\mathbb F}(k,k)$: the denominator is the Frobenius volume of the $O(k)$ or $U(k)$ fibre.
Data properties
Entries are of type: real number
How well the digits are known: proven
Table is complete: no (it holds both scalar fields and every $1\leq k\leq n\leq16$, covering vectors, all intermediate ordered frames and full orthogonal or unitary frames before extending ambient dimension. The cutoff $16$ is a noncanonical practical reference bound for geometric normalization constants in frame integration and packing; the finite choice is independent of numeral notation)
Repeats values from: Volume_of_the_unit_sphere
How they were obtained:

The attached generator reduces each sphere factor exactly to a rational coefficient times an integer power of $\pi$, then evaluates the remaining powers of $\pi$ and optional $\sqrt2$ with Arb balls at 397 bits. The exact case $V_{\mathbb R}(1,1)=2$ is stored as an integer. All other entries are serialized to 100 significant digits; the least retained relative accuracy before serialization is 389 bits.

more

Every written interval contains its generating ball and a separate 600-bit computation from Gaussian QR integration. The checks evaluate every tangent-coordinate Gram determinant, all complex factorial formulas, the low-dimensional group controls, the column recurrences, and complementary Grassmann quotient volumes. They also compare the one-column cases with the unit-sphere table and their quotients with the pure-state Bures and Hilbert–Schmidt volumes. Known-answer and deliberately corrupted controls test the checkers. The attached check_values.py repeats the enclosure and identity checks on stored entries.



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