Grassmannian

Summary

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 .

Motivation edit

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 , 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.

Duality edit

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

 

otherwise.

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]

Applications edit

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

Notes 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

References edit