Jump to content


From Wikipedia, the free encyclopedia

In mathematics, the Grassmannian (named in honour of Hermann Grassmann) is a differentiable manifold that parameterizes the set of all -dimensional linear subspaces of an -dimensional vector space over a field . For example, the Grassmannian is the space of lines through the origin in , so it is the same as the projective space of one dimension lower than .[1][2] When is a real or complex vector space, Grassmannians are compact smooth manifolds, of dimension .[3] In general they have the structure of a nonsingular projective algebraic variety.

The earliest work on a non-trivial Grassmannian is due to Julius Plücker, who studied the set of projective lines in real projective 3-space, which is equivalent to , parameterizing them by what are now called Plücker coordinates. (See § Plücker coordinates and Plücker relations below.) Hermann Grassmann later introduced the concept in general.

Notations for Grassmannians vary between authors, and include , ,, to denote the Grassmannian of -dimensional subspaces of an -dimensional vector space .


By giving a collection of subspaces of a vector space a topological structure, it is possible to talk about a continuous choice of subspaces or open and closed collections of subspaces. Giving them the further structure of a differential manifold, one can talk about smooth choices of subspace.

A natural example comes from tangent bundles of smooth manifolds embedded in a Euclidean space. Suppose we have a manifold of dimension embedded in . At each point , the tangent space to can be considered as a subspace of the tangent space of , which is also just . The map assigning to its tangent space defines a map from M to . (In order to do this, we have to translate the tangent space at each so that it passes through the origin rather than , and hence defines a -dimensional vector subspace. This idea is very similar to the Gauss map for surfaces in a 3-dimensional space.)

This can with some effort be extended to all vector bundles over a manifold , so that every vector bundle generates a continuous map from to a suitably generalised Grassmannian—although various embedding theorems must be proved to show this. We then find that the properties of our vector bundles are related to the properties of the corresponding maps. In particular we find that vector bundles inducing homotopic maps to the Grassmannian are isomorphic. Here the definition of homotopy relies on a notion of continuity, and hence a topology.

Low dimensions[edit]

For k = 1, the Grassmannian Gr(1, n) is the space of lines through the origin in n-space, so it is the same as the projective space of n − 1 dimensions.

For k = 2, the Grassmannian is the space of all 2-dimensional planes containing the origin. In Euclidean 3-space, a plane containing the origin is completely characterized by the one and only line through the origin that is perpendicular to that plane (and vice versa); hence the spaces Gr(2, 3), Gr(1, 3), and P2 (the projective plane) may all be identified with each other.

The simplest Grassmannian that is not a projective space is Gr(2, 4).

The Grassmannian as a differentiable manifold[edit]

To endow with the structure of a differentiable manifold, choose a basis for . This is equivalent to identifying with , with the standard basis denoted , viewed as column vectors. Then for any -dimensional subspace , viewed as an element of , we may choose a basis consisting of linearly independent column vectors . The homogeneous coordinates of the element consist of the elements of the maximal rank rectangular matrix whose -th column vector is , . Since the choice of basis is arbitrary, two such maximal rank rectangular matrices and represent the same element if and only if

for some element of the general linear group of invertible matrices with entries in . This defines an equivalence relation between matrices of rank , for which the equivalence classes are denoted .

We now define a coordinate atlas. For any homogeneous coordinate matrix , we can apply elementary column operations (which amounts to multiplying by a sequence of elements ) to obtain its reduced column echelon form. If the first rows of are linearly independent, the result will have the form

and the affine coordinate matrix with entries determines . In general, the first rows need not be independent, but since has maximal rank , there exists an ordered set of integers such that the submatrix whose rows are the -th rows of is nonsingular. We may apply column operations to reduce this submatrix to the identity matrix, and the remaining entries uniquely determine . Hence we have the following definition:

For each ordered set of integers , let be the set of elements for which, for any choice of homogeneous coordinate matrix , the submatrix whose -th row is the -th row of is nonsingular. The affine coordinate functions on are then defined as the entries of the matrix whose rows are those of the matrix complementary to , written in the same order. The choice of homogeneous coordinate matrix in representing the element does not affect the values of the affine coordinate matrix representing w on the coordinate neighbourhood . Moreover, the coordinate matrices may take arbitrary values, and they define a diffeomorphism from to the space of -valued matrices. Denote by

the homogeneous coordinate matrix having the identity matrix as the submatrix with rows and the affine coordinate matrix in the consecutive complementary rows. On the overlap between any two such coordinate neighborhoods, the affine coordinate matrix values and are related by the transition relations

where both and are invertible. This may equivalently be written as

where is the invertible matrix whose th row is the th row of . The transition functions are therefore rational in the matrix elements of , and gives an atlas for as a differentiable manifold and also as an algebraic variety.

The Grassmannian as a set of orthogonal projections[edit]

An alternative way to define a real or complex Grassmannian as a manifold is to view it as a set of orthogonal projection operators (Milnor & Stasheff (1974) problem 5-C). For this, choose a positive definite real or Hermitian inner product on , depending on whether is real or complex. A -dimensional subspace determines a unique orthogonal projection operator whose image is by splitting into the orthogonal direct sum

of and its orthogonal complement and defining

Conversely, every projection operator of rank defines a subspace as its image. Since the rank of an orthogonal projection operator equals its trace, we can identify the Grassmann manifold with the set of rank orthogonal projection operators :

In particular, taking or this gives completely explicit equations for embedding the Grassmannians , in the space of real or complex matrices , , respectively.

Since this defines the Grassmannian as a closed subset of the sphere this is one way to see that the Grassmannian is a compact Hausdorff space. This construction also turns the Grassmannian into a metric space with metric

for any pair of -dimensional subspaces, where denotes the operator norm. The exact inner product used does not matter, because a different inner product will give an equivalent norm on , and hence an equivalent metric.

For the case of real or complex Grassmannians, the following is an equivalent way to express the above construction in terms of matrices.

Grassmannians Gr(k,Rn) and Gr(k,Cn) as affine algebraic varieties[edit]

Let denote the space of real matrices and the subset of matrices that satisfy the three conditions:

  • is a projection operator: .
  • is symmetric: .
  • has trace .

There is a bijective correspondence between and the Grassmannian of -dimensional subspaces of given by sending to the -dimensional subspace of spanned by its columns and, conversely, sending any element to the projection matrix

where is any orthonormal basis for , viewed as real component column vectors.

An analogous construction applies to the complex Grassmannian , identifying it bijectively with the subset of complex matrices satisfying

  • is a projection operator: .
  • is self-adjoint (Hermitian): .
  • has trace ,

where the self-adjointness is with respect to the Hermitian inner product in which the standard basis vectors are orthonomal. The formula for the orthogonal projection matrix onto the complex -dimensional subspace spanned by the orthonormal (unitary) basis vectors is

The Grassmannian as a homogeneous space[edit]

The quickest way of giving the Grassmannian a geometric structure is to express it as a homogeneous space. First, recall that the general linear group acts transitively on the -dimensional subspaces of . Therefore, if we choose a subspace of dimension , any element can be expressed as

for some group element , where is determined only up to right multiplication by elements of the stabilizer of :

under the -action.

We may therefore identify with the quotient space

of left cosets of .

If the underlying field is or and is considered as a Lie group, this construction makes the Grassmannian a smooth manifold under the quotient structure. More generally, over a ground field , the group is an algebraic group, and this construction shows that the Grassmannian is a non-singular algebraic variety. It follows from the existence of the Plücker embedding that the Grassmannian is complete as an algebraic variety. In particular, is a parabolic subgroup of .

Over or it also becomes possible to use smaller groups in this construction. To do this over , fix a Euclidean inner product on . The real orthogonal group acts transitively on the set of -dimensional subspaces and the stabiliser of a -space is


where is the orthogonal complement of in . This gives an identification as the homogeneous space


If we take and (the first components) we get the isomorphism

Over C, if we choose an Hermitian inner product , the unitary group acts transitively, and we find analogously

or, for and ,

In particular, this shows that the Grassmannian is compact, and of (real or complex) dimension k(nk).

The Grassmannian as a scheme[edit]

In the realm of algebraic geometry, the Grassmannian can be constructed as a scheme by expressing it as a representable functor.[4]

Representable functor[edit]

Let be a quasi-coherent sheaf on a scheme . Fix a positive integer . Then to each -scheme , the Grassmannian functor associates the set of quotient modules of

locally free of rank on . We denote this set by .

This functor is representable by a separated -scheme . The latter is projective if is finitely generated. When is the spectrum of a field , then the sheaf is given by a vector space and we recover the usual Grassmannian variety of the dual space of , namely: . By construction, the Grassmannian scheme is compatible with base changes: for any -scheme , we have a canonical isomorphism

In particular, for any point of , the canonical morphism induces an isomorphism from the fiber to the usual Grassmannian over the residue field .

Universal family[edit]

Since the Grassmannian scheme represents a functor, it comes with a universal object, , which is an object of and therefore a quotient module of , locally free of rank over . The quotient homomorphism induces a closed immersion from the projective bundle:

For any morphism of S-schemes:

this closed immersion induces a closed immersion

Conversely, any such closed immersion comes from a surjective homomorphism of -modules from to a locally free module of rank .[5] Therefore, the elements of are exactly the projective subbundles of rank in

Under this identification, when is the spectrum of a field and is given by a vector space , the set of rational points correspond to the projective linear subspaces of dimension in , and the image of in

is the set

The Plücker embedding[edit]

The Plücker embedding[6] is a natural embedding of the Grassmannian into the projectivization of the th Exterior power of .

Suppose that is a -dimensional subspace of the -dimensional vector space . To define , choose a basis for , and let be the projectivization of the wedge product of these basis elements: where denotes the projective equivalence class.

A different basis for will give a different wedge product, but the two will differ only by a non-zero scalar multiple (the determinant of the change of basis matrix). Since the right-hand side takes values in the projectivized space, is well-defined. To see that it is an embedding, notice that it is possible to recover from as the span of the set of all vectors such that


Plücker coordinates and Plücker relations[edit]

The Plücker embedding of the Grassmannian satisfies a set of simple quadratic relations called the Plücker relations. These show that the Grassmannian embeds as a nonsingular projective algebraic subvariety of the projectivization of the th exterior power of and give another method for constructing the Grassmannian. To state the Plücker relations, fix a basis for , and let be a -dimensional subspace of with basis . Let be the components of with respect to the chosen basis of , and the -component column vectors forming the transpose of the corresponding homogeneous coordinate matrix:

For any ordered sequence of positive integers, let be the determinant of the matrix with columns . The elements are called the Plücker coordinates of the element of the Grassmannian (with respect to the basis of ). These are the linear coordinates of the image of under the Plücker map, relative to the basis of the exterior power space generated by the basis of . Since a change of basis for gives rise to multiplication of the Plücker coordinates by a nonzero constant (the determinant of the change of basis matrix), these are only defined up to projective equivalence, and hence determine a point in .

For any two ordered sequences and of and positive integers, respectively, the following homogeneous quadratic equations, known as the Plücker relations, or the Plücker-Grassmann relations, are valid and determine the image of under the Plücker map embedding:

where denotes the sequence with the term omitted. These are consistent, determining a nonsingular projective algebraic variety, but they are not algebraically independent. They are equivalent to the statement that is the projectivization of a completely decomposable element of .

When , and (the simplest Grassmannian that is not a projective space), the above reduces to a single equation. Denoting the homogeneous coordinates of the image under the Plücker map as , this single Plücker relation is

In general, many more equations are needed to define the image of the Grassmannian in under the Plücker embedding.


Every -dimensional subspace determines an -dimensional quotient space of . This gives the natural short exact sequence:

Taking the dual to each of these three spaces and the dual linear transformations yields an inclusion of in with quotient

Using the natural isomorphism of a finite-dimensional vector space with its double dual shows that taking the dual again recovers the original short exact sequence. Consequently there is a one-to-one correspondence between -dimensional subspaces of and -dimensional subspaces of . In terms of the Grassmannian, this gives a canonical isomorphism

that associates to each subspace its annihilator . Choosing an isomorphism of with therefore determines a (non-canonical) isomorphism between and . An isomorphism of with is equivalent to the choice of an inner product, so with respect to the chosen inner product, this isomorphism of Grassmannians sends any -dimensional subspace into its }-dimensional orthogonal complement.

Schubert cells[edit]

The detailed study of Grassmannians makes use of a decomposition into affine subpaces called Schubert cells, which were first applied in enumerative geometry. The Schubert cells for are defined in terms of a specified complete flag of subspaces of dimension . For any integer partition

of weight

consisting of weakly decreasing non-negative integers

whose Young diagram fits within the rectangular one , the Schubert cell consists of those elements whose intersections with the subspaces have the following dimensions

These are affine spaces, and their closures (within the Zariski topology) are known as Schubert varieties.

As an example of the technique, consider the problem of determining the Euler characteristic of the Grassmannian of k-dimensional subspaces of Rn. Fix a -dimensional subspace and consider the partition of into those k-dimensional subspaces of Rn that contain R and those that do not. The former is and the latter is a rank vector bundle over . This gives recursive formulae:

Solving these recursion relations gives the formula: if is even and is odd and


Cohomology ring of the complex Grassmannian[edit]

Every point in the complex Grassmann manifold defines a -plane in -space. Fibering these planes over the Grassmannian one arrives at the vector bundle which generalizes the tautological bundle of a projective space. Similarly the -dimensional orthogonal complements of these planes yield an orthogonal vector bundle . The integral cohomology of the Grassmannians is generated, as a ring, by the Chern classes of . In particular, all of the integral cohomology is at even degree as in the case of a projective space.

These generators are subject to a set of relations, which defines the ring. The defining relations are easy to express for a larger set of generators, which consists of the Chern classes of and . Then the relations merely state that the direct sum of the bundles and is trivial. Functoriality of the total Chern classes allows one to write this relation as

The quantum cohomology ring was calculated by Edward Witten.[7] The generators are identical to those of the classical cohomology ring, but the top relation is changed to

reflecting the existence in the corresponding quantum field theory of an instanton with fermionic zero-modes which violates the degree of the cohomology corresponding to a state by units.

Associated measure[edit]

When is an -dimensional Euclidean space, we may define a uniform measure on in the following way. Let be the unit Haar measure on the orthogonal group and fix . Then for a set , define

This measure is invariant under the action of the group ; that is,

for all . Since , we have . Moreover, is a Radon measure with respect to the metric space topology and is uniform in the sense that every ball of the same radius (with respect to this metric) is of the same measure.

Oriented Grassmannian[edit]

This is the manifold consisting of all oriented -dimensional subspaces of . It is a double cover of and is denoted by .

As a homogeneous space it can be expressed as:

Orthogonal isotropic Grassmannians[edit]

Given a real or complex nondegenerate symmetric bilinear form on the -dimensional space (i.e., a scalar product), the totally isotropic Grassmannian is defined as the subvariety consisting of all -dimensional subspaces for which

Maximal isotropic Grassmannians with respect to a real or complex scalar product are closely related to Cartan's theory of spinors.[8] Under the Cartan embedding, their connected components are equivariantly diffeomorphic to the projectivized minimal spinor orbit, under the spin representation, the so-called projective pure spinor variety which, similarly to the image of the Plücker map embedding, is cut out as the intersection of a number of quadrics, the Cartan quadrics.[8][9][10]


A key application of Grassmannians is as the "universal" embedding space for bundles with connections on compact manifolds.[11][12]

Another important application is Schubert calculus, which is the enumerative geometry involved in calculating the number of points, lines, planes, etc. in a projective space that intersect a given set of points, lines, etc., using the intersection theory of Schubert varieties. Subvarieties of Schubert cells can also be used to parametrize simultaneous eigenvectors of complete sets of commuting operators in quantum integrable spin systems, such as the Gaudin model, using the Bethe ansatz method.[13]

A further application is to the solution of hierarchies of classical completely integrable systems of partial differential equations, such as the Kadomtsev–Petviashvili equation and the associated KP hierarchy. These can be expressed in terms of abelian group flows on an infinite-dimensional Grassmann manifold.[14][15][16][17] The KP equations, expressed in Hirota bilinear form in terms of the KP Tau function are equivalent to the Plücker relations.[18][17] A similar construction holds for solutions of the BKP integrable hierarchy, in terms of abelian group flows on an infinite dimensional maximal isotropic Grassmann manifold.[15][16][19]

Finite dimensional positive Grassmann manifolds can be used to express soliton solutions of KP equations which are nonsingular for real values of the KP flow parameters.[20][21][22]

The scattering amplitudes of subatomic particles in maximally supersymmetric super Yang-Mills theory may be calculated in the planar limit via a positive Grassmannian construct called the amplituhedron.[23]

Grassmann manifolds have also found applications in computer vision tasks of video-based face recognition and shape recognition,[24] and are used in the data-visualization technique known as the grand tour.

See also[edit]


  1. ^ Lee 2012, p. 22, Example 1.36.
  2. ^ Shafarevich 2013, p. 42, Example 1.24.
  3. ^ Milnor & Stasheff (1974), pp. 57–59.
  4. ^ Grothendieck, Alexander (1971). Éléments de géométrie algébrique. Vol. 1 (2nd ed.). Berlin, New York: Springer-Verlag. ISBN 978-3-540-05113-8., Chapter I.9
  5. ^ EGA, II.3.6.3.
  6. ^ Griffiths, Phillip; Harris, Joseph (1994), Principles of algebraic geometry, Wiley Classics Library (2nd ed.), New York: John Wiley & Sons, p. 211, ISBN 0-471-05059-8, MR 1288523, Zbl 0836.14001
  7. ^ Witten, Edward (1993). "The Verlinde algebra and the cohomology of the Grassmannian". arXiv:hep-th/9312104.
  8. ^ a b Cartan, Élie (1981) [1938]. The theory of spinors. New York: Dover Publications. ISBN 978-0-486-64070-9. MR 0631850.
  9. ^ Harnad, J.; Shnider, S. (1992). "Isotropic geometry and twistors in higher dimensions. I. The generalized Klein correspondence and spinor flags in even dimensions". Journal of Mathematical Physics. 33 (9). American Institute of Physics: 3197–3208. Bibcode:1992JMP....33.3197H. doi:10.1063/1.529538.
  10. ^ Harnad, J.; Shnider, S. (1995). "Isotropic geometry and twistors in higher dimensions. II. Odd dimensions, reality conditions, and twistor superspaces". Journal of Mathematical Physics. 36 (9). American Institute of Physics: 1945–1970. Bibcode:1995JMP....36.1945H. doi:10.1063/1.531096.
  11. ^ Narasimhan, M. S.; Ramanan, S. (1961). "Existence of Universal Connections". American Journal of Mathematics. 83 (3): 563–572. doi:10.2307/2372896. hdl:10338.dmlcz/700905. JSTOR 2372896. S2CID 123324468.
  12. ^ Narasimhan, M. S.; Ramanan, S. (1963). "Existence of Universal Connections II". American Journal of Mathematics. 85 (2): 223–231. doi:10.2307/2373211. JSTOR 2373211.
  13. ^ Mukhin, E.; Tarasov, V.; Varchenko, A. (2009). "Schubert Calculus and representations of the general linear group". J. Amer. Math. Soc. 22 (4). American Mathematical Society: 909–940. arXiv:0711.4079. doi:10.1090/S0894-0347-09-00640-7.
  14. ^ M. Sato, "Soliton equations as dynamical systems on infinite dimensional Grassmann manifolds", Kokyuroku, RIMS, Kyoto Univ., 30–46 (1981).
  15. ^ a b Date, Etsuro; Jimbo, Michio; Kashiwara, Masaki; Miwa, Tetsuji (1981). "Operator Approach to the Kadomtsev-Petviashvili Equation–Transformation Groups for Soliton Equations III–". Journal of the Physical Society of Japan. 50 (11). Physical Society of Japan: 3806–3812. Bibcode:1981JPSJ...50.3806D. doi:10.1143/jpsj.50.3806. ISSN 0031-9015.
  16. ^ a b Jimbo, Michio; Miwa, Tetsuji (1983). "Solitons and infinite-dimensional Lie algebras". Publications of the Research Institute for Mathematical Sciences. 19 (3). European Mathematical Society Publishing House: 943–1001. doi:10.2977/prims/1195182017. ISSN 0034-5318.
  17. ^ a b Harnad, J.; Balogh, F. (2021). Tau functions and Their Applications, Chapts. 4 and 5. Cambridge Monographs on Mathematical Physics. Cambridge, U.K.: Cambridge University Press. doi:10.1017/9781108610902. ISBN 9781108610902. S2CID 222379146.
  18. ^ Sato, Mikio (October 1981). "Soliton Equations as Dynamical Systems on Infinite Dimensional Grassmann Manifolds (Random Systems and Dynamical Systems)". 数理解析研究所講究録. 439: 30–46. hdl:2433/102800.
  19. ^ Harnad, J.; Balogh, F. (2021). Tau functions and Their Applications, Chapt. 7. Cambridge Monographs on Mathematical Physics. Cambridge, U.K.: Cambridge University Press. doi:10.1017/9781108610902. ISBN 9781108610902. S2CID 222379146.
  20. ^ Chakravarty, S.; Kodama, Y. (July 2009). "Soliton Solutions of the KP Equation and Application to Shallow Water Waves". Studies in Applied Mathematics. 123: 83–151. arXiv:0902.4433. doi:10.1111/j.1467-9590.2009.00448.x. S2CID 18390193.
  21. ^ Kodama, Yuji; Williams, Lauren (December 2014). "KP solitons and total positivity for the Grassmannian". Inventiones Mathematicae. 198 (3): 637–699. arXiv:1106.0023. Bibcode:2014InMat.198..637K. doi:10.1007/s00222-014-0506-3. S2CID 51759294.
  22. ^ Hartnett, Kevin (16 December 2020). "A Mathematician's Unanticipated Journey Through the Physical World". Quanta Magazine. Retrieved 17 December 2020.
  23. ^ Arkani-Hamed, Nima; Trnka, Jaroslav (2013). "The Amplituhedron". Journal of High Energy Physics. 2014 (10): 30. arXiv:1312.2007. Bibcode:2014JHEP...10..030A. doi:10.1007/JHEP10(2014)030. S2CID 7717260.
  24. ^ Pavan Turaga, Ashok Veeraraghavan, Rama Chellappa: Statistical analysis on Stiefel and Grassmann manifolds with applications in computer vision, CVPR 23–28 June 2008, IEEE Conference on Computer Vision and Pattern Recognition, 2008, ISBN 978-1-4244-2242-5, pp. 1–8 (abstract, full text)
  25. ^ Morel, Fabien; Voevodsky, Vladimir (1999). "A1-homotopy theory of schemes" (PDF). Publications Mathématiques de l'IHÉS. 90 (90): 45–143. doi:10.1007/BF02698831. ISSN 1618-1913. MR 1813224. S2CID 14420180. Retrieved 2008-09-05., see section 4.3., pp. 137–140