General linear group

General linear group
Original: Jakob.scholbach Vector: Pbroks13 · CC BY-SA 3.0 · source
Name	General linear group
Type	Matrix group
Field	Field F
Dimension	n^2 (as algebraic variety)
Properties	Nonabelian (n>1); invertible matrices

General linear group is the group of invertible n×n matrices over a field F under matrix multiplication. It plays a central role in linear algebra, algebraic geometry, representation theory, and differential geometry, serving as the prototypical example of a linear algebraic group and a Lie group. Key figures associated with its development include Évariste Galois, Camille Jordan, Sophus Lie, Hermann Weyl, and Claude Chevalley.

Definition and basic properties

For a positive integer n and a field F, GL(n,F) consists of all n×n matrices with entries in F whose determinant is nonzero, with group operation matrix multiplication. The determinant defines a group homomorphism det: GL(n,F) → F^×, whose kernel is the special linear group SL(n,F); this exact sequence links GL(n,F) to F^× and to linear algebraic groups studied by Alexander Grothendieck and Armand Borel. GL(n,F) is nonabelian for n>1, and its center Z(GL(n,F)) comprises scalar matrices λI with λ∈F^×. Over finite fields such as GF(q), the order |GL(n,q)| = (q^n−1)(q^n−q)...(q^n−q^{n−1}) features in enumerative results by Émile Borel and applications in the theory of finite groups like Élie Cartan’s work on classical groups.

Examples and special cases

For n=1, GL(1,F) ≅ F^×, the multiplicative group of the field; for F=ℝ and n=2, GL(2,ℝ) contains subgroups isomorphic to SL(2,ℝ), the group related to hyperbolic geometry studied by Henri Poincaré. Over complex numbers GL(n,ℂ) ties to unitary groups U(n) and special unitary groups SU(n) investigated by Werner Heisenberg and Paul Dirac in quantum mechanics contexts. Over finite fields GL(n,q) yields finite classical groups central to the classification results by Daniel Gorenstein and Michael Aschbacher. Other fields of interest include p-adic fields ℚ_p where GL(n,ℚ_p) appears in local aspects of the Langlands program developed by Robert Langlands and in the study of automorphic forms by Atle Selberg.

Algebraic structure and subgroups

GL(n,F) is a linear algebraic group defined by polynomial equations, with important algebraic subgroups: SL(n,F), the Borel subgroup of invertible upper triangular matrices, maximal tori of diagonal matrices, and parabolic subgroups stabilizing flags studied by Élie Cartan and Jacques Tits. The Weyl group of GL(n) is isomorphic to the symmetric group S_n, linking GL(n) to the theory of Coxeter groups developed by H.S.M. Coxeter. Elementary matrices generate the group via row operations, a viewpoint used in algebraic K-theory by Daniel Quillen and in the definition of K_1(F). Normal subgroups in the finite case relate to central extensions and Schur multipliers examined by Issai Schur and Issai Schur’s collaborators in representation theory. The classification of reductive subgroups and Levi decompositions references work by Jean-Pierre Serre and Claude Chevalley.

Topological and Lie group aspects

When F=ℝ or ℂ, GL(n,F) is a real or complex Lie group of dimension n^2; its connected components correspond to sign of determinant in GL(n,ℝ), linking to orientation-preserving diffeomorphisms studied by Marston Morse and René Thom. The Lie algebra gl(n,F) of all n×n matrices with the commutator bracket underpins structure theory developed by Elie Cartan and Weyl, including root decompositions and Cartan subalgebras. Compact subgroups such as O(n) and U(n) serve as maximal compact subgroups in the Cartan decomposition used in harmonic analysis by Harish-Chandra and Atle Selberg. For p-adic fields, GL(n) becomes a totally disconnected locally compact group central to the study of admissible representations and Hecke algebras by Iwahori and Hironaka.

Representations and actions

GL(n,F) acts naturally on the n-dimensional vector space F^n, on tensor powers, exterior powers, and symmetric powers; these representations form the foundation of classical invariant theory by David Hilbert and Arthur Cayley. Polynomial representations correspond to highest-weight theory classified by dominant weights and Young tableaux explored by Frobenius and Alfred Young. Over local fields, smooth irreducible representations of GL(n) form a major part of the local Langlands correspondence by Robert Langlands and Pierre Deligne. Induced representations from parabolic subgroups yield principal series representations used in the harmonic analysis on reductive groups by Harish-Chandra. The determinant and trace give one-dimensional and character-valued invariants linked to character theory developed by Ferdinand Frobenius.

Applications and connections to other fields

GL(n,F) appears in algebraic topology through classifying spaces BGL(n) and vector bundle theory used by John Milnor and Raoul Bott in K-theory; in number theory via automorphic representations and L-functions central to the Langlands program and the work of Andrew Wiles on modularity; in physics through gauge groups and symmetry groups in particle physics explored by Murray Gell-Mann and Sheldon Glashow; and in geometry via frame bundles on manifolds studied in differential geometry by Élie Cartan and Marcel Berger. It also underlies computational linear algebra algorithms stemming from contributions by John von Neumann and Alan Turing and appears in coding theory and combinatorics where finite GL(n,q) acts on vector spaces in designs investigated by Raymond C. Bose.

Category:Linear algebraic groups