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{"Title": "Volumes of real and complex Stiefel manifolds", "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 CITE{metric-definition} fixes its volume CITE{EAS} CITE{PWTH}.", "Parameters": {"field": {"type": "Symbolic", "title": "scalar field", "display": "$\\mathbb F$", "constraints": "$\\mathbb F\\in\\{\\mathbb R,\\mathbb C\\}$, identified by real and complex, respectively.", "values": {"real": "$\\mathbb R$", "complex": "$\\mathbb C$"}}, "n": {"type": "Z", "title": "ambient dimension", "display": "$n$", "constraints": "$n\\geq1$."}, "k": {"type": "Z", "title": "number of columns", "display": "$k$", "constraints": "$1\\leq k\\leq n$."}}, "Tags": ["volume", "groups", "integral"], "Links": {"Paper": {"title": "Pitaval, Wei, Tirkkonen and Hollanti: Density of Spherically-Embedded Stiefel and Grassmann Codes", "url": "https://arxiv.org/abs/1607.03132v4"}}, "References": {"EAS": {"bib": "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\u2013353. Sections 2.2\u20132.4.", "url": "https://www.cis.upenn.edu/~cis5150/Stiefel-Grassmann-manifolds-Edelman.pdf"}, "PWTH": {"bib": "R.-A. Pitaval, L. Wei, O. Tirkkonen and C. Hollanti, Density of Spherically-Embedded Stiefel and Grassmann Codes. Theorem 1 and Appendix B.", "arxiv": "1607.03132v4"}, "ZS": {"bib": "K. \u017byczkowski and H.-J. Sommers, Hilbert\u2013Schmidt volume of the set of mixed quantum states, Journal of Physics A: Mathematical and General 36 (2003), 10115\u201310130. Appendix A, equations (A23)\u2013(A27).", "arxiv": "quant-ph/0302197v3", "doi": "10.1088/0305-4470/36/39/310"}}, "Data properties": {"type": "R", "digits": "100", "rigour": "proven", "complete": "no", "complete-note": "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": "HREF{Volume_of_the_unit_sphere}", "rigour details": "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.\n\nEvery 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\u2013Schmidt volumes. Known-answer and deliberately corrupted controls test the checkers. The attached check_values.py repeats the enclosure and identity checks on stored entries."}, "Display properties": {"number-header": "$V_{\\mathbb F}(n,k)$"}, "Numbers": {"real": {"1": {"1": {"number": "2", "comment": "The two points of $O(1)=\\{-1,1\\}$ have total zero-dimensional volume $2$."}}, "2": {"1": "6.283185307179586476925286766559005768394338798750211641949889184615632812572417997256069650684234136", "2": {"number": "17.77153175263346498806352396024277479445848675750276089234158242782573917239789564230130773905906095", "comment": "Each component of $O(2)$ is a circle of Frobenius radius $\\sqrt2$, so the total length is $4\\sqrt2\\pi$."}}, "3": {"1": "12.56637061435917295385057353311801153678867759750042328389977836923126562514483599451213930136846827", "2": "111.6618271942220725465209749787104846387178603351686778584559466803185242053292354346784213969729830", "3": {"number": 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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.", "metric": "This is the embedded metric used for complex frames in CITE{Paper}. For real frames, the canonical Stiefel metric of CITE{EAS}, 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 CITE{ZS}, equation (A26), this gives CITE{real-volume}.", "haar": "The full-frame values give the geometric total masses removed when normalizing Haar measure to one in HREF{Orthogonal_Weingarten_functions}[orthogonal Weingarten functions] and HREF{Unitary_Weingarten_functions}[unitary Weingarten functions].", "subspaces": "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 HREF{Chiral_unitary_Weingarten_functions}[chiral unitary Weingarten functions], where $W=I_n-2P$ has signature $(n-k,k)$."}, "Formulas": {"metric-definition": "$\\displaystyle \\operatorname{St}_{\\mathbb F}(n,k)=\\{Q\\in\\mathbb F^{n\\times k}:Q^*Q=I_k\\},\\qquad g_Q(\\Delta_1,\\Delta_2)=\\operatorname{Re}\\operatorname{tr}(\\Delta_1^*\\Delta_2)$. Here $*$ is transpose over $\\mathbb R$ and conjugate transpose over $\\mathbb C$, and $\\Delta_1,\\Delta_2$ are tangent vectors to the frame space.", "spheres": "$S_d=2\\pi^{(d+1)/2}/\\Gamma((d+1)/2)$ is the HREF{Volume_of_the_unit_sphere}[unit $d$-sphere volume], with $\\Gamma$ the HREF{Values_of_the_Gamma_function_at_rational_numbers}[Gamma function]. The one-column cases are $V_{\\mathbb R}(n,1)=S_{n-1}$ and $V_{\\mathbb C}(n,1)=S_{2n-1}$.", "real-volume": "$\\displaystyle V_{\\mathbb R}(n,k)=2^{k(k-1)/4}\\prod_{j=0}^{k-1}S_{n-j-1}$. This follows from the canonical quotient metric in CITE{EAS}, equations (2.22) and (2.39), and the orthogonal-group volume in CITE{ZS}, equations (A24)\u2013(A27), with the metric correction described in the comments.", "complex-volume": "$\\displaystyle V_{\\mathbb C}(n,k)=2^{k(k-1)/2}\\prod_{j=0}^{k-1}S_{2(n-j)-1}=\\frac{2^{k(k+1)/2}\\pi^{nk-k(k-1)/2}}{\\prod_{j=0}^{k-1}(n-j-1)!}$ CITE{PWTH}, Theorem 1, equation (35).", "recurrence": "$\\displaystyle V_{\\mathbb F}(n,k)=2^{\\beta(k-1)/2}S_{\\beta(n-k+1)-1}V_{\\mathbb F}(n,k-1)\\quad(2\\leq k\\leq n)$, where $\\beta=1$ for $\\mathbb F=\\mathbb R$ and $\\beta=2$ for $\\mathbb F=\\mathbb C$.", "quotient": "$\\displaystyle G_{\\mathbb F}(n,k)=\\frac{V_{\\mathbb F}(n,k)}{V_{\\mathbb F}(k,k)}$, 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 CITE{PWTH}, equation (91)."}, "Similar tables": [{"table": "HREF{Volume_of_the_unit_sphere}", "relation": "Unit-sphere volumes are the one-column cases and the factors in the product formulas."}, {"table": "HREF{Values_of_the_Gamma_function_at_rational_numbers}", "relation": "Integer and half-integer Gamma values determine the sphere factors."}, {"table": "HREF{Orthogonal_Weingarten_functions}", "relation": "Normalized Haar integration on $O(n)$ divides the geometric measure by the full real frame volume $V_{\\mathbb R}(n,n)$."}, {"table": "HREF{Unitary_Weingarten_functions}", "relation": "Normalized Haar integration on $U(n)$ divides the geometric measure by the full complex frame volume $V_{\\mathbb C}(n,n)$."}, {"table": "HREF{Chiral_unitary_Weingarten_functions}", "relation": "Coefficients for integration over complex subspaces use the quotient obtained by forgetting the ordered frame basis."}, {"table": "HREF{Bures_volumes_of_quantum_states_of_fixed_rank}", "relation": "The rank-one volumes equal $V_{\\mathbb F}(n,1)/V_{\\mathbb F}(1,1)$, the projective-space quotient volumes."}, {"table": "HREF{Hilbert_Schmidt_volumes_of_quantum_states_of_fixed_rank}", "relation": "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 $\\mathbb C$."}], "Keywords": ["orthonormal frames", "orthogonal group volume", "unitary group volume", "Frobenius metric", "Stiefel manifold volume", "Haar measure normalization", "Euclidean Stiefel metric"], "Programs": {"program-sage": {"language": "Sage", "code": "import numberdb.sage as numberdb\nfrom sage.rings.rational_field import QQ\nfrom sage.rings.real_arb import RealBallField\n\ndef stiefel_volume(field, n, k, digits=100):\n beta = {\"real\": 1, \"complex\": 2}[field]\n if n < 1 or not 1 <= k <= n:\n raise ValueError(\"need integers 1 <= k <= n\")\n if field == \"real\" and n == k == 1:\n return QQ(2)\n R = RealBallField(numberdb.bits(digits, losing=96))\n v = R(2)**(QQ(beta*k*(k-1))/4)\n for j in range(k):\n a = QQ(beta*(n-j))/2\n v *= 2*R.pi()**a/R(a).gamma()\n return v\n\nstiefel_volume(\"real\", 3, 2)\n# Table selection: both fields, every 1 <= k <= n <= 16.\n"}}}