GL(n)

GL(n)
Original: Jakob.scholbach Vector: Pbroks13 · CC BY-SA 3.0 · source
Name	GL(n)
Caption	General linear group depiction
Type	Matrix group
Dimension	n^2
Base field	Field or ring

GL(n)

GL(n) is the group of invertible n×n matrices over a ring or field, a central object connecting algebra, geometry, and analysis. It appears throughout the work of Carl Friedrich Gauss, Évariste Galois, Arthur Cayley, Sophie Germain, and Niels Henrik Abel and underpins structures studied by Emmy Noether, Hermann Weyl, Élie Cartan, and Alexander Grothendieck. GL(n) serves as a standard example in contexts involving Pierre-Simon Laplace, Joseph-Louis Lagrange, Bernhard Riemann, David Hilbert, and Felix Klein.

Definition and basic properties

Over a field such as the real numbers, the complex numbers, Finite fields like Évariste Galois’s constructions, or rings considered by Jean-Pierre Serre, GL(n) consists of n×n matrices with entries in that ring whose determinant is invertible. Early matrix theory developments by Arthur Cayley and linear algebra foundations laid by Gottfried Wilhelm Leibniz and Carl Gustav Jacobi inform canonical forms such as the Jordan normal form and the Smith normal form. Over the Cayley–Hamilton context, GL(n) interacts with invariant theory studied by David Hilbert and Emmy Noether. The group operation is matrix multiplication, associativity follows from Augustin-Louis Cauchy’s algebraic manipulations, and identity/inverses reflect work by James Joseph Sylvester and Arthur Cayley.

Determinant and invertibility

The determinant map, developed by Gottfried Wilhelm Leibniz and used by René Descartes in analytic geometry, gives a homomorphism from GL(n) to the multiplicative group of the base field, linking to the study of Pierre-Simon Laplace’s expansions and Alfred Cayley’s determinants. Invertibility criteria trace to Carl Friedrich Gauss’s elimination methods and Gabriel Cramer’s rule. The subgroup with determinant 1, motivated by Élie Cartan and Sophus Lie’s studies, plays a key role in classification tasks pursued by Hermann Weyl and Élie Cartan. Determinant properties connect to volume forms in the work of Bernhard Riemann and characteristic polynomials used by Ferdinand Frobenius.

Matrix groups and subgroup structure

GL(n) contains many important subgroups studied by Sophus Lie, Élie Cartan, Hermann Weyl, and Claude Chevalley, including the finite general linear groups analyzed by Évariste Galois-inspired algebraists such as Richard Brauer and Emil Artin. Classical subgroups like SL(n), O(n), SO(n), U(n), Sp(2n), and Borel subgroups are central in the work of Armand Borel, Jean-Pierre Serre, and Claude Chevalley. Parabolic subgroups, Levi decompositions, and Weyl groups arise in studies by Élie Cartan, Harish-Chandra, and George Mackey, while finite subgroup classification relates to results from William Burnside, Issai Schur, and John G. Thompson. Maximal tori and Cartan subalgebras were developed by Élie Cartan and Hermann Weyl and later used by Harish-Chandra and Nicholas Bourbaki.

Representations and Lie group/Lie algebra aspects

Representations of GL(n) have been central to works by Ferdinand Frobenius, Issai Schur, Hermann Weyl, Harish-Chandra, Paul Dirac, and Roger Howe. Polynomial and rational representations, highest-weight theory, and Schur–Weyl duality connect to Schur functor developments and to combinatorial objects studied by Alfred Young and Issai Schur. The Lie algebra gl(n) is a basic example in Sophus Lie’s theory; structure and representation theory were advanced by Élie Cartan, Nathan Jacobson, and Israel Gelfand. Connections to number theory arise via automorphic representations influenced by Robert Langlands and Harish-Chandra; categorical approaches echo insights from Alexander Grothendieck and Pierre Deligne. Branching rules, tensor products, and symmetric group interactions bring in work by William Fulton and Joe Harris.

Topology and homotopy groups

As a real Lie group, GL(n,R) and GL(n,C) exhibit topological features studied by Hassler Whitney, John Milnor, Raoul Bott, Michael Atiyah, and Isadore Singer. Homotopy groups and K-theory connections were developed by Raoul Bott and Michael Atiyah, while stability phenomena relate to Daniel Quillen’s algebraic K-theory and to Friedhelm Waldhausen’s work. The deformation retract of GL(n,R) to O(n) involves ideas used by Marston Morse and Stephen Smale; classifying spaces and characteristic classes connect to Jean Leray and Shoshichi Kobayashi. Real and complex forms tie into studies by Hermann Weyl and Élie Cartan.

Algebraic group structure and schemes

GL(n) is a prototypical linear algebraic group treated in the frameworks of Claude Chevalley, Alexander Grothendieck, Serge Lang, and Armand Borel. As a smooth affine algebraic group scheme it is described via coordinate rings in the style of Grothendieck’s schemes, with structure sheaves and functor-of-points perspectives used by Pierre Deligne and Jean-Pierre Serre. Over finite fields, counting points in GL(n,q) links to work by Évariste Galois and modern enumerative techniques by George Lusztig and Robert Steinberg. Reductive group theory, root data, and classification draw on Claude Chevalley, Armand Borel, and Jacques Tits. Applications in arithmetic geometry and motives relate to Alexander Grothendieck, Pierre Deligne, and Jean-Pierre Serre.

Category:Linear algebraic groups