Grassmannian

Grassmannian
Name	Grassmannian
Field	Algebraic geometry, Differential geometry, Representation theory

Grassmannian.

The Grassmannian is a classical parameter space that classifies linear subspaces of a fixed dimension inside a finite-dimensional vector space. It plays a central role in algebraic geometry, differential geometry, representation theory, and mathematical physics, appearing in the work of Hermann Grassmann, Bernhard Riemann, David Hilbert, Élie Cartan, and Alexander Grothendieck. The theory connects to objects studied by Isaac Newton, Carl Friedrich Gauss, Niels Henrik Abel, Sophus Lie, and Hermann Weyl through enumerative problems, symmetry, and moduli constructions.

Definition and basic properties

The Grassmannian G(k,n) is defined as the set of k-dimensional linear subspaces of an n-dimensional vector space over a field used by Galois and Évariste Galois in the context of algebraic structures; classical treatments reference Gauss and Leopold Kronecker. It is a homogeneous space for the action of the general linear group GL(n), and it appears in the classification problems studied by Felix Klein and Sophus Lie via quotient constructions like GL(n)/P where P is a parabolic subgroup studied by Armand Borel and Jacques Tits. Basic properties were clarified in the works of Hermann Grassmann, Bernhard Riemann, and modern expositions by Jean-Pierre Serre and Alexander Grothendieck.

Coordinates and embeddings

Local coordinates on the Grassmannian are provided by Plücker coordinates introduced in correspondence with the classical invariant theory of Arthur Cayley and James Joseph Sylvester. The Plücker embedding realizes the Grassmannian as a projective variety inside a projective space studied by Oscar Zariski and André Weil; the coordinate ring relates to determinantal ideals analyzed by David Hilbert and Emmy Noether. Schubert varieties give affine charts akin to constructions in the work of Hermann Schubert and are used in concrete computations by researchers influenced by Paul Erdős and John Milnor.

Topology and manifold structure

Over the real numbers the Grassmannian carries the structure of a smooth manifold and a compact symmetric space appearing in classification results by Élie Cartan and Hermann Weyl. It admits metrics invariant under the orthogonal group O(n) and the unitary group U(n), studied by Elon Lages Lima and in contexts related to Richard Hamilton and Shing-Tung Yau via curvature calculations. The cohomology ring over integers and fields was computed with techniques developed by Raoul Bott, Jean-Louis Koszul, and Armand Borel.

Algebraic and geometric interpretations

Algebraically the Grassmannian is a projective variety with coordinate rings explored by Alexander Grothendieck in the context of schemes and by Michael Atiyah and Isadore Singer in index theory contexts. Geometrically it parametrizes linear subspaces that appear in moduli problems treated by David Mumford and Pierre Deligne; it also features in geometric representation theory via correspondences used by George Lusztig and Anthony Knapp and in the geometric Langlands program referenced by Edward Witten and Vladimir Drinfeld.

Schubert calculus and cohomology

Schubert calculus on the Grassmannian uses Schubert cycles introduced by Hermann Schubert and developed with modern rigor by William Fulton and Andrei Zelevinsky. The intersection theory computed there uses tools from Jean-Pierre Serre, Alexander Grothendieck, and William Thurston for enumerative geometry, and is intertwined with representation-theoretic multiplicities studied by Harish-Chandra, Weyl, and Roger Howe. Quantum cohomology enhancements tie to ideas from Maxim Kontsevich, Edward Witten, and Alexander Givental.

Applications and examples

Examples include the projective line as a special case linked to Bernhard Riemann's work, and the space of lines in projective three-space relevant to Hermann Grassmann and Cayley. Applications span to control theory referenced in engineering literature influenced by Norbert Wiener, to signal processing and coding theory traced to Claude Shannon, and to string theory and gauge theory explored by Edward Witten and Nathan Seiberg. Concrete enumerative problems studied by David Hilbert and Hermann Schubert give classical counts recovered via Grassmannian methods.

Generalizations and related spaces

Generalizations include flag varieties studied by Hermann Schubert and Armand Borel, partial flag varieties tied to Jacques Tits and Robert Steinberg, and orthogonal and symplectic Grassmannians related to Élie Cartan and André Weil. Related constructions appear in the theory of moduli spaces treated by David Mumford and in the study of toric varieties in work by Victor Guillemin and Tadeusz Oda. Infinite-dimensional analogues connect to loop groups investigated by Igor Frenkel and Vladimir Drinfeld and to Kac–Moody theory developed by Victor Kac.

Category:Algebraic geometry