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Volumes of real and complex Grassmann manifolds
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Numbers
$\mathbb F$
$n$
$k$ 
$G_{\mathbb F}(n,k)$
$\mathbb R$
2
1:
3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068
$\mathbb R$
3
1:
6.283185307179586476925286766559005768394338798750211641949889184615632812572417997256069650684234136
$\mathbb R$
4
1:
9.869604401089358618834490999876151135313699407240790626413349376220044822419205243001773403718552232
$\mathbb R$
4
2:
19.73920880217871723766898199975230227062739881448158125282669875244008964483841048600354680743710446
$\mathbb R$
5
1:
13.15947253478581149177932133316820151375159920965438750188446583496005976322560699066903120495806964
$\mathbb R$
5
2:
41.34170224039976023396842008946852693630038475451347692552605080507519322328760805008893471013718159
$\mathbb R$
6
1:
15.50313834014991008773815753355069760111264428294255384707226905190319745873285301878335051630144310
$\mathbb R$
6
2:
64.93939402266829149096022179247007416648505711512361446097857292664723697121813079341457815650199503
$\mathbb R$
6
3:
102.0065615950938177542471033478118688267669022093583301844974812287449973778713414952611668245017063
$\mathbb R$
7
1:
16.53668089615990409358736803578741077452015390180539077021042032203007728931504322003557388405487264
$\mathbb R$
7
2:
81.60524927607505420339768267824949506141352176748666414759798498299599790229707319620893345960136504
$\mathbb R$
7
3:
170.9136344133874554720390122048746486875053383255193983895246555527278067275289268443060282936739086
$\mathbb R$
8
1:
16.23484850566707287274005544811751854162126427878090361524464323166180924280453269835364453912549876
$\mathbb R$
8
2:
85.45681720669372773601950610243732434375266916275969919476232777636390336376446342215301414683695432
$\mathbb R$
8
3:
210.8562448015683112695239000868170177043817660364685042691131425570424200406972510230103183781981139
$\mathbb R$
8
4:
281.1416597354244150260318667824226902725090213819580056921508567427232267209296680306804245042641519
$\mathbb R$
9
1:
14.84329006232418091221947926685030266662515591202825473393795952609079702199271560992333215005759886
$\mathbb R$
9
2:
76.70585975306138584163064109389312588996653905512132730804405060020659472491712713494482852292088217
$\mathbb R$
9
3:
201.8817309355087350959366935110410521997413436565580753721644429399677347481948932708555574895817086
$\mathbb R$
9
4:
317.1150814004927694348078397796210645875069875996264774672675535007771361601260239529828956557701748
$\mathbb R$
10
1:
12.75082019938672721928088791847648360334586277616979127306218515359312467223391768690764585306271329
$\mathbb R$
10
2:
60.24464137187666036272111431051914791553764743899100121974661215915497715448492886371723382234231827
$\mathbb R$
10
3:
155.6635015734870884157119798979813053055897995447634661418846781158253925133627088579000957173029397
$\mathbb R$
10
4:
260.8167002535094705389994131523035223062919982899610035148528491394636018904372123200818136684247005
$\mathbb R$
10
5:
307.2674360437336936495868994166987439651292979833766786560833017742597737585235769862219157033704062
$\mathbb R$
11
1:
10.36257133664445132741557875279358667446520911333163989175847428355504735267264421539161059804461778
$\mathbb R$
11
2:
42.05869394489765314498681114813355254161277992844959903586759227915994473920726943142824114366285597
$\mathbb R$
11
3:
99.35874295371789353866644310563943706906361839617562038661060919598613405349989040765021473082845733
$\mathbb R$
11
4:
163.4385811237867369035245393714678316867312943453589720868959930538964577900735670689855416742808924
$\mathbb R$
11
5:
205.3829783086511464982777196815626408344004430090519566403531267889435992699158102370429781920679378
$\mathbb R$
12
1:
8.011576613127536975251828697103499157226812734008721799508968229034115940352918445826845076265964468
$\mathbb R$
12
2:
26.42625678337439745290065331496842204895891563330220645560562945839809468001050199608190797131418123
$\mathbb R$
12
3:
53.62828443124252304646898948126288227220870595707091271601274772080977521236788919451088086187341782
$\mathbb R$
12
4:
80.65370697835628751009727004544467363799248302911262235775058058914043932203331308794449451085683957
$\mathbb R$
12
5:
99.50252266971959135189312684027572758936882747692956542390092584546519817204621354386012473990723086
$\mathbb R$
12
6:
106.1360241810342307753526686296274427619934159753915364521609875684962113835159611134507997225677129
$\mathbb R$
13
1:
5.919086906091340449248104318783537616883301760191147996174014045116114518702300496772503144369659174
$\mathbb R$
13
2:
15.09464257682299039182661623253152051448143510737100277739487787789821988008365661652364564531422446
$\mathbb R$
13
3:
24.89490646165468442403366299170456252538187187988508565337613658047801203271706790752036099297793185
$\mathbb R$
13
4:
32.16243157001037296222808140291740689757376241678531407641242047530188223742907912528812112805082210
$\mathbb R$
13
5:
36.27776870549825223624144409518804644787245934704933144190341865333702834697756186580359760369712170
$\mathbb R$
13
6:
37.98999055127433753957716444834985491244747827448021942429505031610637692199574587972151383040603814
$\mathbb R$
14
1:
4.194851705245544538214175684822280322107545104576947587158659017198798149018905390192295309847235744
$\mathbb R$
14
2:
7.903536371318468804212103428857682494130060554241242530753116640589806117892036736972176171770445469
$\mathbb R$
14
3:
10.07765712099952089338644153648897247944087672156019765973736800959471477767814215986242291883414649
$\mathbb R$
14
4:
10.58101587243699023557042119442984688062946730955605500388849710722329993453512221085938263441166049
$\mathbb R$
14
5:
10.25243455310704084831708095477533358873870841816385732963763402462903388913086024988183517244554266
$\mathbb R$
14
6:
9.816067984289900016410555286234432262659494123710917128335704518963577536160468608312398289965123419
$\mathbb R$
14
7:
9.636902208307224949713685025941540150328722625284554296425167457759375295290923453621852300501550463
$\mathbb R$
15
1:
2.860824606174783621133077523315361338553530594130399153679964591149896306756815930440832765423622645
$\mathbb R$
15
2:
3.819952584848282127733792067340466125440612873142212011264438872096469603835568944661655517679818570
$\mathbb R$
15
3:
3.598593742099062709060484632494255317211050340339615560997193899334476334257799062969300266075616458
$\mathbb R$
15
4:
2.921131211820992402472913343233396653260733464597009481789434997152259850728311359913470474201301961
$\mathbb R$
15
5:
2.300276890747458822495760714816815178073802323611045128097755588984988415472705271913467775224612511
$\mathbb R$
15
6:
1.891901910387105444997044002061743704792968701531227108489307820459840249187989979755150386898107472
$\mathbb R$
15
7:
1.698167183709893087577109056600449628849585545175867478276642572352357800912409655214511786944587104
$\mathbb R$
16
1:
1.882645042871145636335034822203723372360551482468468788117081629973593036077654026991163556948197446
$\mathbb R$
16
2:
1.714390711088672065421586077323089143567111479147122084827580346719193613182589864359988199380568801
$\mathbb R$
16
3:
1.144581043893424424520838831128228628914245261851015565324476850540981964072308893001615909972415198
$\mathbb R$
16
4:
0.6864383205797132878608779920443658924936709177420221768066926237037466808503824201617061741234352042
$\mathbb R$
16
5:
0.4179083303661065020209291818977669008497026927581236372321425080398380525682883135879462600683944827
$\mathbb R$
16
6:
0.2793373051688113699746147893700889250666848770869281565482530873573981607591356940363338465434554873
$\mathbb R$
16
7:
0.2153866169126985891264946165049505395384205323387370652295059412502243021256250129884913056206955608
$\mathbb R$
16
8:
0.1969249068916101386299379350902404932922702009954167453526911462859193619434285833037634794246359413
$\mathbb C$
2
1:
3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068
$\mathbb C$
3
1:
4.934802200544679309417245499938075567656849703620395313206674688110022411209602621500886701859276116
$\mathbb C$
4
1:
5.167712780049970029246052511183565867037548094314184615690756350634399152910951006261116838767147699
$\mathbb C$
4
2:
8.117424252833536436370027724058759270810632139390451807622321615830904621402266349176822269562749379
$\mathbb C$
5
1:
4.058712126416768218185013862029379635405316069695225903811160807915452310701133174588411134781374690
$\mathbb C$
5
2:
6.676313844272947479376523914252915964355677278340601499590806857528429950294098704855704230221637057
$\mathbb C$
6
1:
2.550164039877345443856177583695296720669172555233958254612437030718624934446783537381529170612542658
$\mathbb C$
6
2:
3.294628825024504863586310938856515901630965094319820379204892852453787813135894547234536224659345530
$\mathbb C$
6
3:
3.450127237667385609549699351995330481929173353505631170911013429149839216888096320546847906316093653
$\mathbb C$
7
1:
1.335262768854589495875304782850583192871135455668120299918161371505685990058819740971140846044327411
$\mathbb C$
7
2:
1.083889438380590520529128358621751685601830524022160811655699645754609352109805745933047006635933215
$\mathbb C$
7
3:
0.8914633309446124481313456504229124297578339785301401487324071992071509830239553155471546365619984880
$\mathbb C$
8
1:
0.5992645293207920768877393835460400460153635863681353695940941453141140212884150557417564728353193920
$\mathbb C$
8
2:
0.2547038088413178423232416144065464085022382795800400424949734854877574237211300901563298961605709966
$\mathbb C$
8
3:
0.1316236128076682762592722202549076982886477448483304165480778705617619502157053227227143109751411955
$\mathbb C$
8
4:
0.1033769437588795189033229169283653653807276698981546062935035517805144791114928939725851064897457050
$\mathbb C$
9
1:
0.2353306303588932045418793527754654215450689353085585985146352037466991295097067533738954446185246807
$\mathbb C$
9
2:
0.04488974701275880835362917646311067154481755520960176845648388481094568237642401439272674179286279393
$\mathbb C$
9
3:
0.01214630403683100480369789145999459565299912444786737859178478098699526344873611005174073788328945689
$\mathbb C$
9
4:
0.005993960788943836402825193585864380881399062711597725555715338975611343537439215272188638005104566892
$\mathbb C$
10
1:
0.08214588661112822879880236552369834480783746079775312311692889117023426706876419810825828348371651556
$\mathbb C$
10
2:
0.006153389509457114220435887912402700320281408922509202755565687605983004709017434165034827767246768155
$\mathbb C$
10
3:
0.0007472453643036205403840818564865262580578129190551324135075430038163128128205893123333513476868863455
$\mathbb C$
10
4:
0.0001930774709472457769760777499948360113309403098121048600882758194957960001786324167694503924660975175
$\mathbb C$
10
5:
0.0001213141528603128140794950889038356113284484278237878005149957756447064913729097267034611228624539052
$\mathbb C$
11
1:
0.02580689139001406001259829425289884965718644104814668599180237251796688933594775585554873825323650511
$\mathbb C$
11
2:
0.0006747946687128336002054384144441442029443958026592988106547100548330701915964505584300409935160809385
$\mathbb C$
11
3:
0.00003217957849120762949601295833247266855515671830201747668137930324929933336310540279490839874434958625
$\mathbb C$
11
4:
0.000003731647012721274737397384661295228543943350290732886004255840694033902353842584957814440653934845023
$\mathbb C$
11
5:
0.000001227662659335528365252732477185735141290059252617148655235028191692719162090086694953459707869744342
$\mathbb C$
12
1:
0.007370430945714350777259089957290781501211638236020997449899842713817490175822097957286936350251086161
$\mathbb C$
12
2:
0.00006054505847418016568510296551378162506463699708684571268399332298597316704484055830493573742529865548
$\mathbb C$
12
3:
0.000001007847388034150297323529766108204982375333176251372234947871485075940026469091203454831634932865830
$\mathbb C$
12
4:
4.589600297587498760981668555767525308429321069186836529943250059899996996516641506158449754679123341e-8
$\mathbb C$
12
5:
6.776496032332589421844935018010841874821457688358891582328938082445228283362319184140505680513739885e-9
$\mathbb C$
12
6:
3.548165025376074153412042869509705400586004731709942285016316662127766856702443653330856310459274309e-9
$\mathbb C$
13
1:
0.001929574309403923047903345563685957640168471815000303352234647617331495634250985531487347698186143913
$\mathbb C$
13
2:
0.000004526937693795311597841724570040006219433490638704433830173332090732977158959871253833188553406825721
$\mathbb C$
13
3:
2.367393557947299877897831223223180713231519219922876194668098441280725656923032194125128913247440881e-8
$\mathbb C$
13
4:
3.763205329944321071800651528267869364257883806359029696229426762862783029835925086866059723214381843e-10
$\mathbb C$
13
5:
2.181966729548747316997123270928283634369069241161355930399563406858332732608773737357495171603516356e-11
$\mathbb C$
13
6:
5.127416294520207332658818419240239036590806045546661935869085820649758801198793516534559672626401337e-12
$\mathbb C$
14
1:
0.0004663028057676125644206289144702717438281998136159926771563035732539635746869590996901434958105247278
$\mathbb C$
14
2:
2.864043858085866579428998198694325236949739709016177199868469456390277603836463058669533263959912317e-7
$\mathbb C$
14
3:
4.277625854829451353437973717411344280065755775608276425363519101832141989229628571206315804674760117e-10
$\mathbb C$
14
4:
2.136191203754018352304875929580137823857830026311816994796775363357808269686911351259086181989456573e-12
$\mathbb C$
14
5:
4.323522214377616685407268950354188879896832960883325597533334744189376210264653283487559663378409614e-14
$\mathbb C$
14
6:
3.989771607512349289753355442922992231966969611981121641738001043129867765428091479431340407611694349e-15
$\mathbb C$
14
7:
1.790605310237419493225703527026725590006685343732661431344647854986636081653942113268671027954895010e-15
$\mathbb C$
15
1:
0.0001046381049248457071180167283522393276102973314909132131852771207714351570546014321508781859973491700
$\mathbb C$
15
2:
1.553130761905341496245671245854284155383676401293568623620823952213228777726098324347121842751181310e-8
$\mathbb C$
15
3:
6.072951043481959771598684553417755832583997535813388428557439638831817692042912996556309397333590924e-12
$\mathbb C$
15
4:
8.661523619401865743673728379601945215211161204637231390565825735976994975242776264415941141876258666e-15
$\mathbb C$
15
5:
5.507338100258076600323260354193869794008380283569083376430588321187797315822855746732956274593908944e-17
$\mathbb C$
15
6:
1.774023804110677993186973820291764385994554628513798937023298822382362483869280342433207933577818511e-18
$\mathbb C$
15
7:
3.126591632980006023136936542006070578463818380025318660875644312492542883293545801144029958651363307e-19
$\mathbb C$
16
1:
0.00002191535344783021582738465205709418885924870876595578031103527648380984852076345699678892956629057534
$\mathbb C$
16
2:
7.299422001508679617384487576498075297458751607869110478274420644596610795330011164975188329864237906e-10
$\mathbb C$
16
3:
6.897421257147148297228360844657745504887075384379026382939365043360570688240589146162084723258711636e-14
$\mathbb C$
16
4:
2.575430838630887115494366073968580246562863654856022576700162050433481075047719083011085792971507918e-17
$\mathbb C$
16
5:
4.676861664576943574921316812191195538528979040172186430074989250202690450752121129659990694197403630e-20
$\mathbb C$
16
6:
4.732843030351200367562954157363532098974723225138834195318768418015926873417603174344514665535958509e-22
$\mathbb C$
16
7:
2.911663501656705515766595348662992608226553671128383964055494537627792780431738019357459934527097441e-23
$\mathbb C$
16
8:
1.143407583316279852809062521764746745611903772225020770694622238774381963133267926077921349190421831e-23
Definition
$G_{\mathbb F}(n,k)$ is the Riemannian volume of the Grassmann manifold of $k$-dimensional subspaces of $\mathbb F^n$, where the scalar field $\mathbb F$ specifies the subspace space and the projector 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 ($n\geq2$.)
$k$
—   subspace dimension ($1\leq k<n$.)
Formulas
(1)
$\displaystyle ds^2=\tfrac12\operatorname{Re}\operatorname{tr}(dP^2),\qquad P=P^*=P^2,\quad\operatorname{tr}P=k$. This fixes the metric scale on the space of rank-$k$ orthogonal projectors [1] [2].
(2)
$\displaystyle G_{\mathbb F}(n,k)=\pi^{\beta k(n-k)/2}\prod_{i=1}^{k}\frac{\Gamma(\beta i/2)}{\Gamma(\beta(n-k+i)/2)}$, where $\beta=1$ over $\mathbb R$, $\beta=2$ over $\mathbb C$, and $\Gamma$ is the Gamma function.
(3)
$\displaystyle G_{\mathbb C}(n,k)=\pi^{k(n-k)}\prod_{i=1}^{k}\frac{(i-1)!}{(n-k+i-1)!}$ [2], Theorem 1, equation (36).
(4)
$\displaystyle G_{\mathbb F}(n,k)=\frac{V_{\mathbb F}(n,k)}{V_{\mathbb F}(k,k)}$, where $V_{\mathbb F}(n,k)$ is the Frobenius Stiefel volume. The fibre is $O(k)$ or $U(k)$ with its full-frame Frobenius metric.
(5)
$\displaystyle G_{\mathbb F}(n,k)=G_{\mathbb F}(n,n-k)$. The isometry is orthogonal complementation $P\mapsto I_n-P$.
(6)
$\displaystyle G_{\mathbb R}(n,1)=\frac{\pi^{n/2}}{\Gamma(n/2)}=\tfrac12 S_{n-1},\qquad G_{\mathbb C}(n,1)=\frac{\pi^{n-1}}{(n-1)!}=B_{2n-2}$. Here $S_d$ is the unit $d$-sphere volume and $B_d$ is the unit $d$-ball volume.
(7)
$\displaystyle \frac{G_{\mathbb F}(n,k)}{G_{\mathbb F}(n-1,k)}=\pi^{\beta k/2}\frac{\Gamma(\beta(n-k)/2)}{\Gamma(\beta n/2)}\qquad(1\leq k<n-1)$, with $\beta$ as in (2).
Comments
(8)
Real subspaces are unoriented. No volume is normalized to one. A point is represented by its orthogonal projector $P=QQ^*$, where $Q$ is an $n$ by $k$ matrix with $Q^*Q=I_k$; $*$ is transpose over $\mathbb R$ and conjugate transpose over $\mathbb C$. Write $\beta=1$ over $\mathbb R$ and $\beta=2$ over $\mathbb C$, so the real dimension is $D=\beta k(n-k)$.
(9)
Forgetting the ordered basis of a Stiefel frame gives the Grassmann manifold. The metric here is the Riemannian quotient of the ambient Frobenius frame metric by $O(k)$ or $U(k)$. On horizontal frame tangents $Q^*\Delta=0$, it is $\operatorname{Re}\operatorname{tr}(\Delta^*\Delta)$. The complex quotient is described in [5], Appendix B-C.
(10)
The quotient of the full Frobenius metric on $O(n)$ or $U(n)$ by $O(k)\times O(n-k)$ or $U(k)\times U(n-k)$ has twice the metric used here. Its volume is consequently $2^{D/2}G_{\mathbb F}(n,k)$. Real projective spaces here have the unit-sphere quotient metric, and complex projective spaces have the Fubini–Study metric of holomorphic sectional curvature four.
(11)
For real frames, changing the canonical metric of [1], equation (2.39), to the Frobenius metric doubles the squared norm of each of the $k(k-1)/2$ skew-block coordinates. The resulting volume factor $2^{k(k-1)/4}$ multiplies both the frame and its fibre, so it cancels in (4). Together with the half-trace orthogonal-group volume in [3], equation (A26), this gives the real case of (2).
(12)
The $k=1$ slices are projective spaces, whose volumes equal the rank-one Bures volumes. The rank-one Hilbert–Schmidt volumes use twice the metric, and are $2^{\beta(n-1)/2}G_{\mathbb F}(n,1)$. These equalities concern pure states; higher-rank density matrices also vary their eigenvalues.
(13)
The chiral unitary Weingarten coefficients integrate over the same complex subspace manifold with measure normalized to one. The involution $W=I_n-2P$ has eigenvalue multiplicities $(n-k,k)$, so its parameters are $a=n-k$ and $b=k$.
Programs
(P1)
Sage
import numberdb.sage as numberdb
from sage.rings.rational_field import QQ
from sage.rings.real_arb import RealBallField

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

grassmann_volume("real", 4, 2)
# Table selection: both fields, 2 <= n <= 16, 1 <= k <= n//2.
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)
[4]
A. S. Buch, Notes on Grassmannians (2017). Exercise 3.12 and Example 3.11. https://sites.math.rutgers.edu/~asbuch/notes/grass.pdf
Links
Similar tables
Volumes of real and complex Stiefel manifolds —   Dividing each frame volume by the full-frame fibre volume gives the corresponding subspace volume.
Unit-sphere volumes —   The real projective slice is half the unit $(n-1)$-sphere volume.
Unit-ball volumes —   The complex projective slice equals the unit $2(n-1)$-ball volume.
Bures volumes of fixed-rank states —   The pure-state volumes equal the projective slices in the same metric.
Hilbert–Schmidt volumes of fixed-rank states —   The pure-state metric is twice the Grassmann metric, giving the factor $2^{\beta(n-1)/2}$ in volume.
Chiral unitary Weingarten functions —   Moment coefficients for complex subspaces use a measure of total mass one.
Gamma values —   Integer and half-integer Gamma factors determine the volume products.
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 $2\leq n\leq16$, $1\leq k\leq\lfloor n/2\rfloor$, representing all positive-dimensional subspace shapes through ambient dimension 16 up to orthogonal complementation. Projective spaces, intermediate ranks and balanced subspaces are covered before extending dimension. The cutoff 16 is a noncanonical, notation-independent practical reference bound for geometric normalization constants in subspace integration and packing)
How they were obtained:

The attached generator reduces the integer and half-integer Gamma factors exactly to a rational coefficient times an integer power of $\pi$. Arb evaluates this expression at 397 bits, retaining at least 390 bits of relative accuracy before serialization to 100 significant digits. Every serialized interval contains the generating ball and independent 600-bit evaluations using unreduced Gamma functions and a sphere-recurrence group quotient.

more

For every complex entry, a recursive count of standard Young tableaux gives an independent exact coefficient through the Plücker degree in [4], Exercise 3.12. The checks verify metric-coordinate Gram matrices, complement symmetry, the ambient-dimension recurrence, and all projective cases against the sphere, ball, Bures and Hilbert–Schmidt tables. They repeat the numerical identities and all frame/fibre comparisons on stored values. Known-answer and deliberately corrupted controls exercise the comparisons. The attached mathematical-checks.md gives the derivations and limitations.



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