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| GL_n(R) | |
|---|---|
| Name | General linear group |
| Type | Lie group, algebraic group |
| Field | Real numbers |
GL_n(R) GL_n(R) is the group of n×n invertible matrices with entries in the real numbers under matrix multiplication. It is a central example in the study of Élie Cartan, David Hilbert, Emmy Noether-inspired algebraic groups, Hermann Weyl-related representation theory, and John von Neumann-style operator theory; it connects to work by Sophus Lie, Évariste Galois, Bernhard Riemann, and Felix Klein. The group appears across research by Claude Chevalley, Armand Borel, Harish-Chandra, Alexander Grothendieck, and Jean-Pierre Serre in contexts ranging from Sophus Lie-type continuous symmetries to Andrey Kolmogorov-influenced analysis.
GL_n(R) consists of all n×n matrices with nonzero determinant and real entries; the identity element is the identity matrix, inverses are given by matrix inverses, and the operation is matrix multiplication. Foundational results due to Arthur Cayley and James Joseph Sylvester show GL_n(R) is a nonabelian group for n≥2 and contains important subgroups considered by Camille Jordan and Augustin-Louis Cauchy. Classical theorems by Frobenius and Jordan classify finite subgroups in low dimensions, while studies by Isaac Newton-era determinant properties and modern expositions by Hermann Weyl and Richard Brauer establish basic structural features. GL_n(R) acts naturally on R^n, linking it to linear actions studied by Poincaré and Henri Lebesgue in geometric and measure-theoretic contexts.
The determinant map from GL_n(R) to R^× (the multiplicative group of nonzero reals) is a surjective homomorphism studied by Carl Friedrich Gauss and applied by Pierre-Simon Laplace in expansion formulas; its kernel is SL_n(R), central to work by Élie Cartan and Harish-Chandra. Inversion uses adjugate matrices and minors, techniques dating to Arthur Cayley and refined by Gaspard Monge; explicit formulas connect to expansions by Jacques Charles François Sturm and algorithms developed later by John von Neumann and Alan Turing in numerical linear algebra. Matrix operations—multiplication, transpose, trace—feature in identities used by Augustin-Louis Cauchy and Adrien-Marie Legendre and underpin spectral results proved by David Hilbert and John von Neumann; eigenvalue problems relate GL_n(R) to studies by Sofia Kovalevskaya and Issai Schur.
Key subgroups include SL_n(R), O(n), SO(n), and the group of upper triangular invertible matrices B_n(R) (a Borel subgroup), subjects in papers by Armand Borel and Claude Chevalley. Compact subgroups like O(n) and SO(n) were analyzed by Hermann Weyl and Élie Cartan; discrete subgroups and arithmetic groups connect to work by G. A. Margulis, Gregory A. Moore, and Serge Lang. Parabolic subgroups, Levi decompositions, and unipotent radicals arise in investigations by Joseph Bernstein and Harish-Chandra; tori and Cartan subgroups relate to results of Élie Cartan and John Milnor. Extensions and coverings studied by Albrecht Dold and Michael Atiyah link GL_n(R) to spin groups and metaplectic groups examined by Raoul Bott and Isadore Singer.
As a real Lie group of dimension n^2, GL_n(R) carries a smooth manifold structure rooted in work by Sophus Lie and formalized by Élie Cartan and Hermann Weyl; local charts use open sets in R^{n^2} as in techniques by Bernhard Riemann. The group has two connected components distinguished by sign of determinant; connected component of the identity GL_n^+(R) is a subject in topology texts by Henri Poincaré and Lefschetz. GL_n(R) admits a topology making it a smooth manifold so that matrix multiplication and inversion are smooth maps, a framework used by André Weil and Jean-Pierre Serre in algebraic group theory. Deformation and homotopy-theoretic properties studied by Raoul Bott and John Milnor link GL_n(R) to K-theory developed by Daniel Quillen and Michael Atiyah.
As a linear algebraic group over R, GL_n(R) is reductive; representation theory for real reductive groups follows from work by Harish-Chandra, Hermann Weyl, and Bertram Kostant. Finite-dimensional representations relate to highest-weight theory established by Élie Cartan and Nicolas Bourbaki-style expositions; infinite-dimensional unitary representations were developed by George Mackey, Harish-Chandra, and Israel Gelfand. Group cohomology and algebraic K-theory of GL_n(R) connect to studies by Daniel Quillen, Charles Weibel, and Andrey Suslin; Bass–Serre theory and structure results reference work by Hyman Bass and Jean-Pierre Serre. The center, derived subgroup, and normal subgroups have been classified in special cases in literature by Claude Chevalley and Armand Borel.
GL_n(R) appears in differential equations studied by Sophus Lie and Sofia Kovalevskaya through linear systems, in mechanics originating with Isaac Newton and extended by Lagrange and Joseph-Louis Lagrange, and in control theory with connections to Norbert Wiener and Rudolf Kalman. In geometry and topology GL_n(R) governs frame bundles over manifolds studied by Élie Cartan and Marcel Berger; in numerical analysis and scientific computing, practitioners build on algorithms by John von Neumann and Gilbert Strang. Applications in statistics trace to Ronald Fisher and Karl Pearson via multivariate analysis, while cryptographic and coding contexts reference work by Claude Shannon and Donald Knuth for linear transformations over reals and reals-approximations. Concrete examples include GL_1(R) ≅ R^× relevant to number-theoretic investigations by Leonhard Euler and GL_2(R) used in modular forms studies related to Bernhard Riemann and Hecke.
Category:Linear algebraic groups