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| GL_n(C) | |
|---|---|
| Name | GL_n(C) |
| Caption | General linear group of degree n over the complex numbers |
| Type | Matrix group; complex Lie group; algebraic group |
| Base field | Complex numbers |
| Dimension | n^2 |
GL_n(C)
GL_n(C) is the group of invertible n×n matrices with entries in the complex numbers, notable in the work of Évariste Galois, Sophus Lie, Alexander Grothendieck, Hermann Weyl, and Emmy Noether. It appears across research in William Rowan Hamilton's quaternion studies, John von Neumann's operator theory, David Hilbert's linear algebra foundations, Felix Klein's Erlangen program, and modern developments by Pierre Deligne. GL_n(C) connects to topics studied at institutions like Princeton University, University of Göttingen, École Normale Supérieure, Harvard University, and the Institute for Advanced Study.
GL_n(C) consists of all n×n complex matrices with nonzero determinant; its elements were used in proofs by Arthur Cayley, James Joseph Sylvester, Carl Friedrich Gauss, Augustin-Louis Cauchy, and Niels Henrik Abel. As a group under matrix multiplication it is nonabelian for n≥2, arises in classification problems studied by Élie Cartan, Wilhelm Killing, Émile Picard, and is central to results linked to Noether's theorem and the Riemann–Hilbert correspondence. Basic properties (closure, associativity, identity, inverses) are illustrated in work by Camille Jordan, Leopold Kronecker, Hermann Minkowski, Srinivasa Ramanujan, and André Weil.
GL_n(C) is an affine algebraic group defined by polynomial equations studied by Alexander Grothendieck, Jean-Pierre Serre, Armand Borel, Claude Chevalley, and John Tate. Important algebraic subgroups include the Borel subgroup (upper triangular matrices) examined by Élie Cartan and Jacques Tits, the maximal torus of diagonal matrices central in work of Harish-Chandra, the parabolic subgroup family used in research by Robert Langlands and George Lusztig, and the special linear group SL_n studied by Henri Poincaré and Niels Abel. Other notable subgroups are the unitary group U(n) investigated by Eugene Wigner and Hermann Weyl, the orthogonal group O(n) in texts by Élie Cartan and Émile Picard, and finite subgroups connected to Klein four-group studies by Felix Klein and Évariste Galois.
As a complex Lie group GL_n(C) is a complex manifold of complex dimension n^2; foundational analytic structures were developed by Henri Poincaré, Bernhard Riemann, Kurt Gödel's contemporaries, Élie Cartan, and Hermann Weyl. The group is noncompact but has maximal compact subgroup U(n), a fact exploited by researchers at Institute for Advanced Study and by Harish-Chandra in harmonic analysis. Topological invariants and homotopy groups relate to work by Jean-Pierre Serre, Raoul Bott, Aleksandr Lyapunov, John Milnor, and Michel Kervaire in the study of characteristic classes and fibrations.
Representation theory of GL_n(C) is central in the contributions of Frobenius, Issai Schur, Hermann Weyl, Harish-Chandra, Roger Howe, and James Arthur. Polynomial representations, highest-weight theory, and the role of Young tableaux connect to studies by William Fulton, Joe Harris, André Weil, Igor Schur, and Alfred Young. GL_n(C) acts naturally on C^n, flag varieties studied by Élie Cartan and André Weil, and on tensor spaces used in quantum applications by Paul Dirac and Richard Feynman; these actions underpin advances by Robert Langlands and George Mackey.
The determinant homomorphism det: GL_n(C) → C^× echoes developments by Augustin-Louis Cauchy, Camille Jordan, Évariste Galois, David Hilbert, and Emmy Noether. Its kernel is SL_n(C), extensively studied by Élie Cartan, Claude Chevalley, Jean-Pierre Serre, Armand Borel, and Nicholas Katz. The image relates to the multiplicative group C^×, linking to number-theoretic perspectives advanced by Ernst Kummer, Carl Gauss, Peter Gustav Lejeune Dirichlet, and André Weil.
The Lie algebra gl_n(C) of all n×n complex matrices, with Lie bracket [X,Y]=XY−YX, was formalized in works by Sophus Lie, Wilhelm Killing, Élie Cartan, Hermann Weyl, and Nikolai Chebotarev. The exponential map exp: gl_n(C) → GL_n(C) features in analytic studies by Henri Poincaré, John von Neumann, Israel Gelfand, Francesco Severi, and Lars Hörmander, and relates to the Baker–Campbell–Hausdorff formula investigated by Felix Hausdorff and Élie Cartan.
GL_n(C) appears across geometry, number theory, and physics in the work of Bernhard Riemann, David Hilbert, Alexander Grothendieck, Robert Langlands, Edward Witten, and Pierre Deligne. Concrete examples include change-of-basis matrices used by Carl Friedrich Gauss and Arthur Cayley, monodromy groups in the Riemann–Hilbert problem studied by Hermann Weyl and George David Birkhoff, and symmetry groups in quantum mechanics developed by Paul Dirac, Eugene Wigner, and Richard Feynman. GL_n(C) underlies modern research at Stanford University, Massachusetts Institute of Technology, University of Cambridge, University of Oxford, and the Clay Mathematics Institute.
Category:Linear algebraic groups