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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)
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
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
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
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
(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
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)
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
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
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
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
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
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)
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
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
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
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
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
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
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
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)
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
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
British geometer
Institutions University of Manchester University of Cambridge Thesis Grassmannian Varieties / The Conic as a Space Element (1932) Doctoral advisor H.F
J._A._Todd
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
{\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
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)
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
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
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
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
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
Topics referred to by the same term
Gradian, in geometry Gradshteyn and Ryzhik, a calculus reference work the Grassmannian, Gr k ( V ) {\displaystyle \operatorname {Gr} _{k}(V)} the associated
GR
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
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
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
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)
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
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
{\displaystyle G(n,m)} is the Grassmannian of all m-dimensional linear subspaces of an n-dimensional vector space. The Grassmannian is used to allow the construction
Varifold
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
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
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
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
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
American mathematician and professor
Skovsted; Tamvakis, Harry (2017). A Giambelli formula for isotropic Grassmannians. Selecta Mathematica, 23(2):869-914. with Buch, Anders Skovsted; Purbhoo
Andrew_Kresch
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)
{\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
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
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
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
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
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
Maurer–Cartan form Examples hyperbolic space Gauss–Bolyai–Lobachevsky space Grassmannian Complex projective space Real projective space Euclidean space Stiefel
List of differential geometry topics
List_of_differential_geometry_topics
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
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)
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
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
Indian-American mathematician (born 1983)
Bhargav; Scholze, Peter (2017). "Projectivity of the Witt vector affine Grassmannian". Inventiones Mathematicae. 209 (2): 329–423. arXiv:1507.06490. Bibcode:2017InMat
Bhargav_Bhatt_(mathematician)
Relation between Lie algebras depicted as a square
either a compact simple Lie group, a Grassmannian, a Lagrangian Grassmannian, or a double Lagrangian Grassmannian of subspaces of ( A ⊗ B ) n , {\displaystyle
Freudenthal_magic_square
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
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
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
Discrete dynamical system on polygons in the projective plane and on their moduli space
integers. The pentagram map can also be generalized to the space of Grassmannians G r ( m , m d ) {\displaystyle \mathrm {Gr} (m,md)} , which consists
Pentagram_map
}^{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
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
Mathematics timeline
calculus, a branch of intersection theory taking place on the complex Grassmannian manifolds. 1902 David Hilbert Tentative axiomatisation (topological spaces
Timeline_of_manifolds
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
^{mn}} where G r ( r , m ) {\displaystyle \mathbf {Gr} (r,m)} is the Grassmannian of r-planes in an m-dimensional vector space, and consider the subspace
Determinantal_variety
3-space CP3, which is also the Grassmannian Gr1(C4) of lines in 4-dimensional complex space. X = Gr2(C4), the Grassmannian of 2-planes in 4-dimensional
Penrose_transform
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
Romanian-American mathematician
Grojnowski, "The strong Macdonald conjecture and Hodge theory on the loop Grassmannian", Ann. of Math., vol. 168, 2008, p. 175–220, Arxiv "The quantization
Constantin_Teleman
Function that can be used to build the wave function of a multi-fermionic system
projective algebraic variety which is naturally identified with the Grassmannian G r N ( H ) {\displaystyle \mathbf {Gr} _{N}({\mathcal {H}})} . Its embedding
Slater_determinant
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
Exterior product of vectors
according to the Jacobian determinant of a change-of-coordinate function. Grassmannian Multivector Exterior algebra Differential form Geometric algebra Clifford
Blade_(geometry)
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)
Moduli scheme of subschemes of a scheme, represents the flat-family-of-subschemes functor
projective P n {\displaystyle \mathbb {P} ^{n}} space as a subscheme of a Grassmannian defined by the vanishing of various determinants. Its fundamental property
Hilbert_scheme
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
GRASSMANNIAN
GRASSMANNIAN
GRASSMANNIAN
GRASSMANNIAN
Girl/Female
Swedish Dutch Greek
Prophetess.
Female
English
English variant spelling of French Lorraine, LAURAINE means "land of the people of Lothar."
Girl/Female
Tamil
One who is outstanding example of peace & humility
Male
Japanese
(1-泰裕, 2-泰弘, 3-æ弘, 4-æ³°åš) Japanese name YASUHIRO means 1) "calm and leisurely," 2) "most calm," 3) "most respectful, and 4) "abundant tranquility."
Girl/Female
Muslim
Reveler of secrets
Girl/Female
Arabic, Muslim
Generous; Liberal; Open Handed
Girl/Female
Indian, Tamil
Accomplished Girl
Boy/Male
Indian, Punjabi, Sikh
Blessing of Guru
Girl/Female
Bengali, Indian, Telugu
Good Place
Boy/Male
Gujarati, Hindu, Indian, Sanskrit
Fighter; Stronger; Strength; One who has Strength in his Arms
GRASSMANNIAN
GRASSMANNIAN
GRASSMANNIAN
GRASSMANNIAN
GRASSMANNIAN