General linear group GL_n

General linear group GL_n
Name	General linear group
Type	Group, Lie group, Algebraic group
Elements	Invertible n×n matrices
Operation	Matrix multiplication

General linear group GL_n is the group of invertible n×n matrices over a field or ring, central to Évariste Galois-inspired linear algebra and to the development of Arthur Cayley's matrix theory. It underpins structures studied by Felix Klein in the Erlangen Program and appears throughout work of Hermann Weyl, Emmy Noether, Sophus Lie, and Alexander Grothendieck. GL_n connects to concrete computations in the tradition of Carl Friedrich Gauss and to categorical viewpoints advanced by Samuel Eilenberg and Saunders Mac Lane.

Definition and basic properties

For a field K or a commutative ring R, GL_n(K) (or GL_n(R)) consists of all n×n matrices with entries in K (or R) that have inverses under matrix multiplication, reflecting principles from Augustin-Louis Cauchy's determinant theory and James Joseph Sylvester's invariant theory. The group law is associative as in Galois theory and has identity the identity matrix, with inverses given by adjugate and determinant formulas linked to Bézout-type results and to Pierre-Simon Laplace expansions. Over finite fields like Galois fields GF(q), GL_n is finite and its order is given by q^{n^2} times a product ∏_{i=1}^n (1 − q^{-i}), reminiscent of counting arguments used by Richard Dedekind.

Matrix representations and examples

Typical elements are matrices studied by Arthur Cayley and represented concretely in linear algebra textbooks following David Hilbert's school. Examples include diagonal matrices, permutation matrices related to Augustin-Louis Cauchy’s permutation theory and to Émile Borel's consideration of matrix decompositions, and companion matrices tied to Joseph-Louis Lagrange's polynomial studies. Over the real numbers R, GL_n(R) contains orthogonal matrices studied by Gauss and Wilhelm Killing and upper triangular matrices related to Hermann Weyl's representation work. Over the complex numbers C, GL_n(C) features prominently in lectures of Bernhard Riemann and in later developments by Henri Poincaré.

Algebraic structure and subgroups

GL_n contains many classical subgroups investigated by Élie Cartan and Claude Chevalley, including the Borel subgroup of invertible upper triangular matrices, the maximal torus of invertible diagonal matrices, and parabolic subgroups studied in the work of Armand Borel and Harish-Chandra. Important discrete subgroups include GL_n(Z) appearing in Carl Ludwig Siegel’s arithmetic studies and in John Tate's number-theoretic frameworks. Normal subgroups and centers link to analyses by Emmy Noether and Richard Brauer, while finite subgroup classifications echo themes from William Burnside and Frobenius.

Topological and Lie group aspects

When K = R or C, GL_n(R) and GL_n(C) are real and complex Lie groups respectively; these were central to the programs of Sophus Lie and Élie Cartan and later to Weyl's representation theory. The groups carry natural topologies making them noncompact (GL_n(R)) or complex reductive (GL_n(C)), with deformation-retract behavior to compact subgroups such as the orthogonal group studied by Hermann Weyl and the unitary group related to John von Neumann and Harish-Chandra. Homotopy and cohomology computations for GL_n appear in work by Alexander Grothendieck and Daniel Quillen within algebraic K-theory contexts.

Representations and modules

GL_n plays a foundational role in representation theory explored by Weyl and extended by Harish-Chandra, with polynomial representations, rational representations, and highest-weight theory connected to Élie Cartan and Armand Borel's classification results. Its actions on vector spaces produce modules central to Emmy Noether's module theory and to Claude Chevalley's study of algebraic groups. Induced and restricted representations relate to Mackey theory developed by George Mackey, while connections to automorphic representations invoke work by Robert Langlands and André Weil.

Determinant, special linear group, and normal subgroups

The determinant map det: GL_n → K^×, formalized in expansion methods of Laplace and systematized in invariant theory by Sylvester, is a surjective homomorphism whose kernel is the special linear group SL_n studied by Klein and Hilbert. SL_n is a normal subgroup of GL_n; quotienting by the determinant recovers the multiplicative group K^× encountered in Richard Dedekind's arithmetic treatments. The center of GL_n consists of scalar matrices linked to scalar multiplication concepts used by David Hilbert and Jordan in canonical form theory.

Applications and occurrences in mathematics and physics

GL_n appears throughout number theory via adelic and arithmetic groups in research of Claude Chevalley, Yuri Manin, and Andrew Wiles, and in algebraic geometry through moduli problems addressed by Grothendieck and Alexander Grothendieck's school. In mathematical physics GL_n(C) and GL_n(R) enter gauge theory contexts studied by Chen Ning Yang and Robert Mills, and in quantum mechanics via unitary subgroups related to Paul Dirac and Eugene Wigner. Control theory and signal processing trace linear transformations back to applied work influenced by Norbert Wiener and Rudolf E. Kálmán, while crystallography and materials science employ matrix groups in analyses initiated by William Hallowes Miller.

Category:Matrix groups Category:Linear algebra Category:Lie groups