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Mathematical space
In mathematics, a Grassmannian G r k ( V ) {\displaystyle \mathbf {Gr} _{k}(V)} , also known as a Grassmann manifold, is a differentiable manifold that
Grassmannian
Topics referred to by the same term
In mathematics, a Grassmannian may refer to: Affine Grassmannian Affine Grassmannian (manifold) Grassmannian, the classical parameter space for linear
Grassmannian_(disambiguation)
Type of vector space in mathematics
In mathematics, the Lagrangian Grassmannian is the smooth manifold of Lagrangian subspaces of a real symplectic vector space V. Its dimension is 1/2n(n
Lagrangian_Grassmannian
In mathematics, the affine Grassmannian of an algebraic group G over a field k is an ind-scheme—a colimit of finite-dimensional schemes—which can be thought
Affine_Grassmannian
Geometric structure used in certain particle interactions
amplituhedron is defined as a mathematical space known as the positive Grassmannian. Amplituhedron theory challenges the notion that spacetime locality and
Amplituhedron
Describes a periodicity in the homotopy groups of classical groups
space BU is the classifying space for stable complex vector bundles (a Grassmannian in infinite dimensions). One formulation of Bott periodicity describes
Bott_periodicity_theorem
Mathematical concept
In mathematics, there are two distinct meanings of the term affine Grassmannian. In one it is the manifold of all k-dimensional affine subspaces of Rn
Affine Grassmannian (manifold)
Affine_Grassmannian_(manifold)
Embedding of a Grassmannian into projective space
In mathematics, the Plücker map embeds the Grassmannian G r ( k , V ) {\displaystyle \mathrm {Gr} (k,V)} , whose elements are k-dimensional subspaces of
Plücker_embedding
German polymath, linguist and mathematician (1809–1877)
the concept which is now known as a vector space. He introduced the Grassmannian, the space which parameterizes all k-dimensional linear subspaces of
Hermann_Grassmann
(pseudo-)Riemannian manifold whose geodesics are reversible
either a compact simple Lie group, a Grassmannian, a Lagrangian Grassmannian, or a double Lagrangian Grassmannian of subspaces of ( A ⊗ B ) n , {\displaystyle
Symmetric_space
Vector bundle existing over a Grassmannian
bundle is a vector bundle occurring over a Grassmannian in a natural tautological way: for a Grassmannian of k {\displaystyle k} -dimensional subspaces
Tautological_bundle
Branch of algebraic geometry
space, which is roughly equivalent to describing the cohomology ring of Grassmannians. Sometimes it is used to mean the more general enumerative geometry
Schubert_calculus
algebraic geometry, a Schubert variety is a certain subvariety of a Grassmannian, G r k ( V ) {\displaystyle \mathbf {Gr} _{k}(V)} of k {\displaystyle
Schubert_variety
Mathematical object studied in the field of algebraic geometry
of algebraic curves). Let V be a finite-dimensional vector space. The Grassmannian variety Gn(V) is the set of all n-dimensional subspaces of V. It is a
Algebraic_variety
space, and let Λ ( V ) {\displaystyle \Lambda (V)} denote its Lagrangian Grassmannian, the manifold of all Lagrangian subspaces of V {\displaystyle V} . The
Maslov_index
Geometric space whose points represent algebro-geometric objects of some fixed kind
in C n + 1 {\displaystyle \mathbb {C} ^{n+1}} . More generally, the Grassmannian G r ( k , V ) {\displaystyle \mathbf {Gr} (k,V)} of a vector space V
Moduli_space
Type of topological space
R n + 1 ) {\displaystyle \mathbf {Gr} (1,\mathbb {R} ^{n+1})} of a Grassmannian space. Like all projective spaces, R P n {\displaystyle \mathbb {RP}
Real_projective_space
American mathematician
tropical geometry, algebraic combinatorics, amplituhedra, and the positive Grassmannian. She is Dwight Parker Robinson Professor of Mathematics at Harvard University
Lauren Williams (mathematician)
Lauren_Williams_(mathematician)
Family of polynomials
field with q elements; i.e. it is the number of points in the finite Grassmannian G r ( k , F q n ) {\displaystyle \mathrm {Gr} (k,\mathbb {F} _{q}^{n})}
Gaussian_binomial_coefficient
Topological space in group theory
point stabilizer general linear group): An = Aff(n, K) / GL(n, K). Grassmannian: Gr(r, n) = O(n) / (O(r) × O(n − r)) Topological vector spaces (in the
Homogeneous_space
Set of topological invariants
{\displaystyle Gr_{n}(V)} denote the Grassmannian, the space of n-dimensional linear subspaces of V, and denote the infinite Grassmannian G r n = G r n ( R ∞ ) {\displaystyle
Stiefel–Whitney_class
Exact homotopy case
the Grassmannian of n-planes in an infinite-dimensional complex Hilbert space; or, the direct limit, with the induced topology, of Grassmannians of n
Classifying_space_for_U(n)
Ordering obtained by a single shuffle
Schubert varieties in a Grassmannian space. A permutation π {\displaystyle \pi } which is both a riffle shuffle and Grassmannian (i.e. both π {\displaystyle
Riffle_shuffle_permutation
Special functions of several complex variables
parametrized by points in a tube domain inside a complex Lagrangian Grassmannian, namely the Siegel upper half space. One example of a theta function
Theta_function
American mathematician and politician (born 1977)
Mnev, N (2007). "On D.K. Biss' papers "The homotopy type of the matroid Grassmannian" and "Oriented matroids, complex manifolds, and a combinatorial model
Daniel_Biss
Differential geometry concept
{\displaystyle \mathrm {S} (\mathrm {U} (p)\times \mathrm {U} (2))} p Grassmannian of complex 2-dimensional subspaces of C p + 2 {\displaystyle \mathbb
Quaternion-Kähler symmetric space
Quaternion-Kähler_symmetric_space
1 ( x ) = G d ( E x ) {\displaystyle p^{-1}(x)=G_{d}(E_{x})} is the Grassmannian of the d-dimensional vector subspaces of E x {\displaystyle E_{x}} .
Grassmann_bundle
Class of commutative rings
homogeneous functions on the Grassmannians. The Plücker coordinates provide some of the distinguished elements. For the Grassmannian of planes in C n {\displaystyle
Cluster_algebra
Hodge theory, in Bhatt and Scholze's work on projectivity of the affine Grassmannian, Huber and Jörder's study of differential forms, etc. Voevodsky defined
H_topology
British geometer
Mathematician Workplaces University of Manchester University of Cambridge Thesis Grassmannian Varieties / The Conic as a Space Element (1932) Doctoral advisor H.F
J._A._Todd
American mathematician (born 1965)
Robin Hartshorne with a thesis on the surfaces in the four-dimensional Grassmannian. From 1990 to 1993 he was an assistant professor at the University of
Mark_Gross_(mathematician)
Connected non-abelian Lie group lacking nontrivial connected normal subgroups
SU(p,q), A III 2pq Hermitian. Grassmannian of p subspaces of Cp+q. If p or q is 2; quaternion-Kähler Hermitian. Grassmannian of maximal positive definite
Simple_Lie_group
Subspace defined by a polynomial of degree 2 over a field
projective homogeneous variety, known as the isotropic Grassmannian or orthogonal Grassmannian OGr(r + 1, n + 2). (The numbering refers to the dimensions
Quadric_(algebraic_geometry)
Hypergeometric function in mathematics
hypergeometric function is a function that is (more or less) defined on a Grassmannian, and depends on a choice of some complex numbers and signs. Gelfand,
General hypergeometric function
General_hypergeometric_function
Type of surface in algebraic geometry
remarkable geometric properties. The surface S is naturally embedded into the grassmannian of lines G(2,5) of P4. Let U be the restriction to S of the universal
Fano_surface
Process in algebraic geometry
X\times G_{r}(TY)} , where G r ( T Y ) {\displaystyle G_{r}(TY)} is the Grassmannian of r-planes in the tangent bundle of Y {\displaystyle Y} , by τ ( a )
Nash_blowing-up
Topics referred to by the same term
dynamics General relativity, Einstein's 1915 theory of gravity Galois ring Grassmannian, Gr k ( V ) {\displaystyle \operatorname {Gr} _{k}(V)} Associated graded
GR
Riemannian manifold equipped with a differential p-form
equality. For x in M, set Gx(φ) to be the subset of such planes in the Grassmannian of p-planes in TxM. In cases of interest, Gx(φ) is always nonempty. Let
Calibrated_geometry
{\text{Quot}}_{{\mathcal {E}}/X/S}^{\Phi }} over S {\displaystyle S} . The Grassmannian G ( n , k ) {\displaystyle G(n,k)} of k {\displaystyle k} -planes in
Quot_scheme
similar to the basis of standard monomials of the coordinate ring of a Grassmannian. Hodge algebras were introduced by Corrado De Concini, David Eisenbud
Hodge_algebra
direct generalization of the construction of a Grassmannian variety via the Plücker embedding, as Grassmannians are the d = 1 {\displaystyle d=1} case of Chow
Chow_variety
American mathematician
Peter G. Casazza discussing the core structures of Grassmannian frames in a classroom he and his wife, Janet Tremain, installed in the basement of their
Peter_G._Casazza
1960–67 foundational treatise on algebraic geometry by Alexander Grothendieck
Second edition brings in certain schemes representing functors such as Grassmannians, presumably from intended Chapter V of the first edition. In addition
Éléments de géométrie algébrique
Éléments_de_géométrie_algébrique
Grassmann number Grassmann variables Grassmannian Affine Grassmannian Affine Grassmannian (manifold) Lagrangian Grassmannian Grassmann–Cayley algebra Grassmann–Plücker
List of things named after Hermann Grassmann
List_of_things_named_after_Hermann_Grassmann
Property of certain dynamical systems
within the Grassmannian, and the Hirota equations as expressing the Plücker relations, characterizing the Plücker embedding of the Grassmannian in the projectivization
Integrable_system
Mathematical group of loops in a Lie group
defined by LG(R) = G(R((t))), together with their associated affine Grassmannians and affine flag varieties. Let G be a topological group. The set C(S1
Loop_group
Holland & Polo (1996) and a theorem relating D-modules on the affine Grassmannian to representations of the Kac–Moody algebra g ^ {\displaystyle {\widehat
Beilinson–Bernstein localization
Beilinson–Bernstein_localization
German mathematician and physicist (1801–1868)
underlying vector space of dimension 4. It is now part of the theory of Grassmannians G r ( k , V ) {\displaystyle \mathbf {Gr} (k,V)} ( k {\displaystyle
Julius_Plücker
Used to count, measure, and label
Pracna, Petr (2015). "From Cayley-Dickson Algebras to Combinatorial Grassmannians". Mathematics. 3 (4). MDPI AG: 1192–1221. arXiv:1405.6888. doi:10.3390/math3041192
Number
Type of geometric transformation
synthetic description as an incidence correspondence. Recall that the Grassmannian G r ( 1 , 2 ) {\displaystyle \mathbf {Gr} (1,2)} parametrizes the set
Blowing_up
Manifold of all orthonormal k-frames in n-dimensional Euclidean space
manifold V k ( F n ) {\displaystyle V_{k}(\mathbb {F} ^{n})} to the Grassmannian of k-planes in F n {\displaystyle \mathbb {F} ^{n}} which sends a k-frame
Stiefel_manifold
Skeletonized version of algebraic geometry
PMID 15534224. Zbl 1135.62302. Speyer, David E. (2003). "The Tropical Grassmannian". arXiv:math/0304218v3. Speyer, David; Sturmfels, Bernd (2009) [2004]
Tropical_geometry
Lagrangian derivative Lagrangian drifter Lagrangian foliation Lagrangian Grassmannian Lagrangian intersection Floer homology Lagrangian mechanics Relativistic
List of things named after Joseph-Louis Lagrange
List_of_things_named_after_Joseph-Louis_Lagrange
Space in mathematics and theoretical physics
stands for projective space, Gr {\displaystyle \operatorname {Gr} } a Grassmannian, and F {\displaystyle F} a flag manifold. The double fibration gives
Twistor_space
Superconformal Yang–Mills theory
a description (the amplituhedron formalism) in terms of the positive Grassmannian. N = 4 super Yang–Mills can be derived from a simpler 10-dimensional
N = 4 supersymmetric Yang–Mills theory
N_=_4_supersymmetric_Yang–Mills_theory
Completion of the usual space with "points at infinity"
through the origin of V. That is, if V is n-dimensional, then P(V∗) is the Grassmannian of n − 1 planes in V. In algebraic geometry, this construction allows
Projective_space
Type of group in mathematics
thinking of it as the fundamental group π1(U/O) of the stable Lagrangian Grassmannian as U/O ≅ Ω7(KO), so π1(U/O) = π1+7(KO). The orthogonal group anchors
Orthogonal_group
Indian American mathematician (1944/1945–2023)
and Readings in Mathematics 53, Hindustan Book Agency, 2009) and The Grassmannian Variety: Geometric and Representation-Theoretic Aspects (Developments
V._Lakshmibai
Theory proposed by Roger Penrose
polytopes. These ideas have evolved more recently into the positive Grassmannian and amplituhedron. Twistor string theory was extended first by generalising
Twistor_theory
}^{2k+1}{\mathbb {C} }\,} is simply connected, such a structure has to be unique. Grassmannian G r ( 2 , 4 ) , {\displaystyle Gr(2,4)\,,} etc. Metaplectic group Symplectic
Metaplectic_structure
Typeface style used in mathematics
if infinite). G {\displaystyle \mathbb {G} } U+1D53E 𝔾 Represents a Grassmannian or a group, especially an algebraic group or group scheme. H {\displaystyle
Blackboard_bold
Group of 𝑛 × 𝑛 invertible matrices
to the Schubert decomposition of the Grassmannian, and are q-analogs of the Betti numbers of complex Grassmannians. This was one of the clues leading to
General_linear_group
Algebra associated to any vector space
k-dimensional linear subspaces of V {\displaystyle V} . In particular, the Grassmannian of k-dimensional subspaces of V {\displaystyle V} , denoted Gr k
Exterior_algebra
Branch of mathematics
important role played by its analytic methods. In wireless communications, Grassmannian manifolds are used for beamforming techniques in multiple antenna systems
Differential_geometry
Riemannian manifold which satisfies vacuum Einstein equations
Einstein constant k {\displaystyle k} . Examples of these include the Grassmannians G r ( k , R ℓ ) {\displaystyle Gr(k,\mathbb {R} ^{\ell })} , G r ( k
Einstein_manifold
of the Grassmannian on which the complex symplectic group and the unitary symplectic group act transitively. This is the Lagrangian Grassmannian. The subspace
Invariant_convex_cone
Algebraic structure in linear algebra
used to formalize the idea of parallel lines intersecting at infinity. Grassmannians and flag manifolds generalize this by parametrizing linear subspaces
Vector_space
Natural number
Pracna, Petr (2015). "From Cayley-Dickson Algebras to Combinatorial Grassmannians". Mathematics. 3 (4). MDPI AG: 1192–1221. arXiv:1405.6888. doi:10.3390/math3041192
32_(number)
Discrete dynamical system on polygons in the projective plane and on their moduli space
m\geq 1} be integers. The pentagram map can also be generalized to the Grassmannian space G r ( m , m d ) {\displaystyle \mathrm {Gr} (m,md)} , which consists
Pentagram_map
{\displaystyle {\mathfrak {p}}} , and the space of possible choices is the Grassmannian G r ( k , n ) {\displaystyle \mathrm {Gr} (k,n)} . In general, for a
Parabolic_Lie_algebra
Plücker embedding The Plücker embedding is the closed embedding of the Grassmannian variety into a projective space. plurigenus The n-th plurigenus of a
Glossary of algebraic geometry
Glossary_of_algebraic_geometry
Subgroup of a root system's isometry group
to the decomposition of the flag variety G/B into Schubert cells (see Grassmannian). The structure of the Hasse diagram of the group is related geometrically
Weyl_group
Differential geometry topic
Gauss map can also be defined, and its target space is the oriented Grassmannian G ~ k , n {\displaystyle {\tilde {G}}_{k,n}} , i.e. the set of all oriented
Gauss_map
Monoidal category
reductive group G and certain equivariant perverse sheaves on the affine Grassmannian associated to G. This equivalence provides a non-combinatorial construction
Tannakian_formalism
Association of cohomology classes to principal bundles
itself was not so new, having been reflected in the Schubert calculus on Grassmannians, and the work of the Italian school of algebraic geometry. On the other
Characteristic_class
Class of error-correcting code
ISBN 978-0-471-06259-2. Etzion, Tuvi; Raviv, Netanel (2013). "Equidistant codes in the Grassmannian". arXiv:1308.6231 [math.CO]. Bonisoli, A. (1984). "Every equidistant
Linear_code
classifying space for the orthogonal group O(n) may be constructed as the Grassmannian of n-planes in an infinite-dimensional real space R ∞ {\displaystyle
Classifying_space_for_O(n)
Quotient of a weakly contractible space by a free action
BS1 for the circle S1 thought of as a compact topological group. The Grassmannian G r ( n , R ∞ ) {\displaystyle Gr(n,\mathbb {R} ^{\infty })} of n-planes
Classifying_space
American professor of engineering (born 1979)
beamforming is related to the famous applied mathematics problem of Grassmannian line packing. They also showed how MIMO precoding also can be understood
David_J._Love
the Mumford conjecture) Group scheme Abelian variety Theta function Grassmannian Flag manifold Weil restriction Differential Galois theory Prime ideal
List of algebraic geometry topics
List_of_algebraic_geometry_topics
Manifold with inversion symmetry
{\displaystyle \mathrm {S} (\mathrm {U} (p)\times \mathrm {U} (q))} pq min(p,q) Grassmannian of complex p-dimensional subspaces of C p + q {\displaystyle \mathbb
Hermitian_symmetric_space
pullback is always globally generated. This class of examples includes Grassmannians, projective spaces, and flag varieties. Also, products of convex spaces
Convexity (algebraic geometry)
Convexity_(algebraic_geometry)
Every rigid motion is a screw displacement
_{i}^{2}=1} representing orthogonal planes through the origin, and one Grassmannian basis vector e 0 {\displaystyle \mathbf {e} _{0}} satisfying e 0 2 =
Chasles'_theorem_(kinematics)
Bundle of linear subspaces of the tangent bundle
k-dimensional submanifolds. Since the contact bundle is obtained by combining Grassmannians of the tangent spaces at each point, it is a special case of the Grassmann
Contact_bundle
Aspect of theoretical physics
in turn led to new insights in pure mathematics. Such topics include Grassmannian residue formulae, the amplituhedron and holomorphic linking. BCFW recursion
Twistor_string_theory
Description in Riemannian geometry
{\displaystyle p} ). The sectional curvature is a real-valued function on the 2-Grassmannian bundle over the manifold. The sectional curvature determines the Riemann
Sectional_curvature
Smooth manifold with an inner product on each tangent space
Cayley hyperbolic space, which are instead analogues of hyperbolic space. Grassmannian manifolds also carry natural Riemannian metrics making them into symmetric
Riemannian_manifold
Lets one glue 2 sheaves over an infinitesimal neighborhood of an algebraic curve point
τ. The importance of this corollary is that it shows that the affine Grassmannian may be formed either from the data of bundles over an infinitesimal disk
Beauville–Laszlo_theorem
Class of spinors constructed using Clifford algebras
it, up to multiplication by a complex number, as follows. Denote the Grassmannian of maximal isotropic ( n {\displaystyle n} -dimensional) subspaces of
Pure_spinor
Anticommutating number
_{2})^{2}(\eta _{1})^{2}(\eta _{2})^{2}(\psi _{1})^{2}(\psi _{2})^{2}.} Grassmannian Hermann Grassmann (linguist and mathematician) Superspace Exterior algebra
Grassmann_number
Mathematical concept
even-dimensional ones cannot. Complex projective space is a special case of a Grassmannian, and is a homogeneous space for various Lie groups. It is a Kähler manifold
Complex_projective_space
Notation used to describe juggling patterns
patterns naturally label strata in the positroid stratification of the Grassmannian. Number: Relative duration (height) of a toss. 1, 2, 3... Brackets []:
Siteswap
Venezuelan-born theoretical physicist
Alexander; Trnka, Jaroslav (2012). "Scattering Amplitudes and the Positive Grassmannian". arXiv:1212.5605 [hep-th]. Arkani-Hamed, Nima; Cachazo, Freddy; Cheung
Freddy_Cachazo
Manifold with Riemannian, complex and symplectic structure
provided by the Hermitian symmetric spaces of compact type, such as Grassmannians. The natural Kähler metric on a Hermitian symmetric space of compact
Kähler_manifold
Mathematics concept
to see that G r = G ( K ) / G ( O ) {\displaystyle Gr=G(K)/G(O)} is a Grassmannian. For simplicity, we can think that K = Z / p Z ( ( x ) ) {\displaystyle
Satake_isomorphism
Generalization of an orientation of a vector space
bundle is classified by the real infinite Grassmannian, oriented bundles are classified by the infinite Grassmannian of oriented real vector spaces. From the
Orientation of a vector bundle
Orientation_of_a_vector_bundle
Sequence of spaces in linear algebra
{\displaystyle (0,1,2)} . Filtration (mathematics) Flag (geometry) Flag manifold Grassmannian Matroid Kostrikin, Alexei I. and Manin, Yuri I. (1997). Linear Algebra
Flag_(linear_algebra)
German mathematician (1804–1851)
formula for determinants, which underlie the Plücker relations for Grassmannians. Students of vector fields, Lie theory, Hamiltonian mechanics and operator
Carl_Gustav_Jacob_Jacobi
Data visualisation technique
time, in the space of all 2-dimensional subspaces of Rp (known as the Grassmannian G(2,p)). To display these views on a computer screen, it is necessary
Grand Tour (data visualisation)
Grand_Tour_(data_visualisation)
jumping lines of a vector bundle form a proper closed subset of the Grassmannian of all lines of projective space. The Birkhoff–Grothendieck theorem classifies
Jumping_line
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GRASSMANNIAN
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