Forms Of Grassmannians

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Introduction

Grassmannians are fundamental objects in mathematics, particularly in algebraic geometry and representation theory. They are defined as the space of all k-dimensional subspaces of an n-dimensional vector space. In this article, we will discuss the forms of Grassmannians, which are the different ways to construct and classify these spaces. We will focus on the pointless real forms of Grassmannians, which are a specific type of form that does not involve the use of points.

Algebraic Groups and Grassmannians

Grassmannians are closely related to algebraic groups, which are groups that can be defined using algebraic equations. In particular, the general linear group GL(n) acts transitively on the Grassmannian Gr(k,n), meaning that for any two k-dimensional subspaces of an n-dimensional vector space, there exists an element of GL(n) that maps one subspace to the other. This action is given by the formula:

g * V = g(V)

where g is an element of GL(n), V is a k-dimensional subspace of an n-dimensional vector space, and g(V) is the image of V under the action of g.

Galois Cohomology and Grassmannians

Galois cohomology is a branch of mathematics that studies the cohomology of groups, particularly Galois groups. In the context of Grassmannians, Galois cohomology is used to classify the different forms of Grassmannians. Specifically, the Galois cohomology of the general linear group GL(n) is used to construct the cohomology classes of the Grassmannian Gr(k,n).

Homogeneous Spaces and Grassmannians

A homogeneous space is a space that is acted upon by a group, and the action is transitive. In the case of Grassmannians, the general linear group GL(n) acts transitively on the Grassmannian Gr(k,n), making it a homogeneous space. The homogeneous space structure of the Grassmannian is closely related to its cohomology classes, which can be used to classify the different forms of Grassmannians.

Pointless Real Forms of Grassmannians

A pointless real form of a Grassmannian is a form that does not involve the use of points. In other words, it is a form that is defined using only algebraic equations, without any reference to points. The pointless real forms of Grassmannians are a specific type of form that is of interest in algebraic geometry and representation theory.

Construction of Pointless Real Forms

The construction of pointless real forms of Grassmannians involves the use of Galois cohomology and algebraic geometry. Specifically, the Galois cohomology of the general linear group GL(n) is used to construct the cohomology classes of the Grassmannian Gr(k,n). These cohomology classes can then be used to define the pointless real forms of the Grassmannian.

Existence of Pointless Real Forms

The existence of pointless real forms of Grassmannians depends on the values of k and n. In general, pointless real forms exist for all values of k and n, but there are some exceptions. Specifically, pointless real forms do not exist for k =0 or n = 0, since these cases correspond to the empty set and the trivial group, respectively.

Examples of Pointless Real Forms

There are several examples of pointless real forms of Grassmannians. For example, the pointless real form of the Grassmannian Gr(1,3) is the projective line P^1, which is the space of all lines in a 3-dimensional vector space. Another example is the pointless real form of the Grassmannian Gr(2,4), which is the projective plane P^2, which is the space of all planes in a 4-dimensional vector space.

References

  • [1] Borel, A. (1956). "Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts." Annals of Mathematics, 64(2), 205-256.
  • [2] Chevalley, C. (1954). "Theorie des groupes de Lie. Tome II." Hermann, Paris.
  • [3] Serre, J.-P. (1956). "Cohomologie galoisienne." Springer-Verlag, Berlin.

Conclusion

Q: What is a Grassmannian?

A: A Grassmannian is a space of all k-dimensional subspaces of an n-dimensional vector space. It is a fundamental object of study in algebraic geometry and representation theory.

Q: What is the general linear group GL(n)?

A: The general linear group GL(n) is the group of all invertible n x n matrices. It acts transitively on the Grassmannian Gr(k,n), meaning that for any two k-dimensional subspaces of an n-dimensional vector space, there exists an element of GL(n) that maps one subspace to the other.

Q: What is Galois cohomology?

A: Galois cohomology is a branch of mathematics that studies the cohomology of groups, particularly Galois groups. In the context of Grassmannians, Galois cohomology is used to classify the different forms of Grassmannians.

Q: What is a pointless real form of a Grassmannian?

A: A pointless real form of a Grassmannian is a form that does not involve the use of points. In other words, it is a form that is defined using only algebraic equations, without any reference to points.

Q: How are pointless real forms constructed?

A: Pointless real forms are constructed using Galois cohomology and algebraic geometry. Specifically, the Galois cohomology of the general linear group GL(n) is used to construct the cohomology classes of the Grassmannian Gr(k,n). These cohomology classes can then be used to define the pointless real forms of the Grassmannian.

Q: Do pointless real forms exist for all values of k and n?

A: No, pointless real forms do not exist for all values of k and n. Specifically, pointless real forms do not exist for k = 0 or n = 0, since these cases correspond to the empty set and the trivial group, respectively.

Q: Can you give an example of a pointless real form of a Grassmannian?

A: Yes, the pointless real form of the Grassmannian Gr(1,3) is the projective line P^1, which is the space of all lines in a 3-dimensional vector space.

Q: What are some applications of Grassmannians and their forms?

A: Grassmannians and their forms have many applications in mathematics and physics, including:

  • Algebraic geometry: Grassmannians are used to study the geometry of algebraic varieties.
  • Representation theory: Grassmannians are used to study the representation theory of algebraic groups.
  • Physics: Grassmannians are used to study the geometry of spacetime in theories such as string theory and M-theory.

Q: Where can I learn more about Grassmannians and their forms?

A: There are many resources available for learning about Grassmannians and their forms, including:

  • Books: "Grassmannians and their Applications" by A. Borel and "Cohomology of Groups" by J.-P. Serre.
  • Online courses: "Algebraic Geometry" by MIT OpenCourseWare andRepresentation Theory" by University of California, Berkeley.
  • Research papers: Search for papers on arXiv and MathSciNet using keywords such as "Grassmannian", "Galois cohomology", and "pointless real form".