GL(n,C)

GL(n,C)
Original: Jakob.scholbach Vector: Pbroks13 · CC BY-SA 3.0 · source
Name	General linear group over the complex numbers
Type	Complex Lie group, linear algebraic group
Dimension	n^2 (real dimension 2n^2)
Field	Complex numbers

GL(n,C)

GL(n,C) is the group of all invertible n×n matrices with entries in the complex numbers, equipped with matrix multiplication. It is a central object in linear algebra, complex analysis, differential geometry, and representation theory, connecting figures such as David Hilbert, Élie Cartan, Hermann Weyl, Emmy Noether, and Claude Chevalley. GL(n,C) serves as a prototypical example of a complex Lie group, an algebraic group studied by researchers at institutions like Princeton University, École Normale Supérieure, Institute for Advanced Study, and linked to applications in physics through work at CERN and Max Planck Society.

Definition and basic properties

The set consists of all n×n complex matrices with nonzero determinant; it is an open subset of the vector space of all n×n matrices studied by Augustin-Louis Cauchy and Arthur Cayley. As an algebraic variety it is defined by the nonvanishing of the determinant polynomial, a perspective developed in the tradition of David Hilbert and Alexander Grothendieck. GL(n,C) is noncompact and reductive; its structure theory was shaped by contributions from Élie Cartan, Hermann Weyl, Claude Chevalley, and Armand Borel.

Matrix representation and topology

Elements are represented by complex matrices with the usual operations from linear algebra introduced by Carl Friedrich Gauss and Gottfried Leibniz. The topology is the subspace topology from the complex vector space of matrices studied in functional analysis at places like University of Göttingen and Cambridge University. With this topology GL(n,C) is a complex manifold of complex dimension n^2, a fact used in the work of Bernhard Riemann and later in complex geometry by Henri Poincaré.

Group structure and subgroups

GL(n,C) contains notable subgroups such as the special linear group SL(n,C) connected to Felix Klein's Erlangen program, the unitary group U(n) relevant to Paul Dirac and Weyl's quantum mechanics, and the Borel and parabolic subgroups central to studies by Armand Borel and Jean-Pierre Serre. Finite subgroups appear in the classification programs related to William Burnside and Évariste Galois. Parabolic subgroups, Levi decompositions, and maximal tori are structures investigated by Claude Chevalley and Robert Steinberg.

Lie algebra and exponential map

The Lie algebra gl(n,C) consists of all n×n complex matrices with the commutator bracket, a viewpoint originating in the work of Sophus Lie and Wilhelm Killing. The exponential map exp: gl(n,C) → GL(n,C) relates linear flows studied by Henri Poincaré and spectral theory developed by John von Neumann and Issai Schur. Cartan subalgebras, root systems, and representation-theoretic methods were formalized by Élie Cartan, Hermann Weyl, and Harish-Chandra.

Representations and characters

Finite-dimensional representations of GL(n,C) were classified using highest-weight theory by Hermann Weyl and expanded by Harish-Chandra and George Mackey; they play a role in the theory of symmetric functions developed by Alfred Young and Isaiah Schur. Characters and trace functions relate to the work of Frobenius and Issai Schur in character theory, while modern geometric representation theory connects GL(n,C) to the geometric Langlands program influenced by Alexander Beilinson, Vladimir Drinfeld, and institutions like Harvard University and IHÉS.

Determinant, connectedness, and homotopy

The determinant map to C× links GL(n,C) to the multiplicative group studied by Niels Henrik Abel and Évariste Galois; its kernel is SL(n,C). GL(n,C) is connected as a complex manifold, a fact used in topology by Henri Poincaré and in homotopy theory refined by J. H. C. Whitehead and Raoul Bott. Fundamental group and higher homotopy groups connect to classical results by Raoul Bott and computations influenced by the work at Institute for Advanced Study and Princeton University.

Applications and examples

GL(n,C) appears in differential equations studied by Sofya Kovalevskaya and Émile Picard, in quantum mechanics via Paul Dirac and John von Neumann, and in gauge theory and particle physics influenced by Chen-Ning Yang and Robert Mills. Examples include change-of-basis groups in linear algebra courses at University of Oxford and symmetry groups of vector bundles in the work of Michael Atiyah and Isadore Singer. In number theory and automorphic forms GL(n,C) interacts with the Langlands program advanced by Robert Langlands and explored at research centers such as IAS and Princeton University's mathematics department.

Category:Matrix groups