GL_n

GL_n
Name	GL_n
Caption	General linear group of degree n
Type	Algebraic group
Base field	Varies (e.g., R, C, F_q)
Dimension	n^2 (as a Lie group)

GL_n

The general linear group of degree n is the group of invertible n-by-n matrices over a specified field or ring, central to linear algebra, algebraic geometry, and number theory. It underlies constructions in representation theory, algebraic groups, and topology, and interacts with objects such as Bernoulli numbers, Modular forms, Galois groups and classical groups like SL_n, O(n), and Sp(2n). Historically it appears in work of Arthur Cayley, Camille Jordan, Évariste Galois, and in modern contexts such as the Langlands program, Algebraic K-theory, and the theory of Automorphic forms.

Definition and basic properties

GL_n is defined as the set of n×n matrices with entries in a field or ring R that have a two-sided inverse under matrix multiplication, forming a group under multiplication. Over a field such as R or C it is an affine algebraic group; over a finite field F_q it is a finite group whose order is (q^n−1)(q^n−q)…(q^n−q^{n−1}). The group contains classical subgroups related to S_n actions via permutation matrices and to torus subgroups; it admits matrix factorizations such as LU, QR, and polar decompositions connected historically to work by G. H. Golub and John von Neumann.

Determinant and connected components

The determinant map det: GL_n(R) → R^× is a surjective group homomorphism for fields R, and its kernel is the special linear group SL_n, a normal subgroup studied by Élie Cartan and Hermann Weyl. Over R the sign of the determinant splits GL_n(R) into two connected components, one containing the identity and one containing reflections; over C GL_n(C) is connected. For local fields like Q_p determinants control volume forms, and for finite fields F_q the determinant map targets the cyclic group F_q^×, relevant in counting conjugacy classes and in the study of Weil conjectures via zeta functions.

Group structure and subgroups

GL_n contains many important subgroups: the Borel of invertible upper-triangular matrices, maximal tori of diagonal matrices, unipotent radicals, and parabolic subgroups classified by flag varieties studied by Alexander Grothendieck and Maurice Auslander. Weyl groups appearing as normalizers of tori are isomorphic to S_n, linking to Schubert calculus on Grassmannians investigated by Hermann Schubert. Central elements form the scalar matrices (isomorphic to the multiplicative group of the base field), and Levi decompositions relate GL_n to products of general linear factors as in block-diagonal embeddings used in the study of reductive groups.

Representations and actions

Linear representations of GL_n on vector spaces are fundamental: the defining representation on R^n or C^n, exterior powers ∧^k giving determinant bundles, symmetric powers Sym^k tied to Schur constructions, and polynomial representations classified by highest weights corresponding to partitions and Young diagrams developed by Alfred Young and Issai Schur. Actions on flag varieties and Grassmannians yield geometric representations linked to cohomology theories pioneered by Jean-Pierre Serre and Alexander Grothendieck. Over finite fields, representations of GL_n(F_q) enter the construction of modular forms and Deligne–Lusztig theory associated with Pierre Deligne and George Lusztig.

Topological and Lie group aspects

As a real Lie group GL_n(R) has dimension n^2 with Lie algebra gl_n consisting of all n×n matrices, studied in the classification of semisimple Lie algebras by Élie Cartan and Claude Chevalley. Compact forms relate to U(n) and O(n), and exponential and logarithm maps connect Lie algebra elements to group elements in neighborhoods of the identity, as in work by Sophus Lie and Wilhelm Killing. For complex groups GL_n(C) the topology is that of a complex manifold, and over p-adic fields GL_n(Q_p) is a totally disconnected locally compact group central to Bruhat–Tits theory and the study of smooth representations used in the Langlands correspondence.

Applications in geometry and number theory

GL_n acts on vector bundles and local systems appearing in the study of algebraic varieties by Alexander Grothendieck and in moduli problems like moduli of vector bundles on curves studied by David Mumford and Michael Atiyah. In number theory, automorphic representations of GL_n(A) for the adele ring A underpin the Langlands program and reciprocity laws related to Artin reciprocity and Hecke characters. Determinant and monodromy representations connect to Galois representations studied by Jean-Pierre Serre and Andrew Wiles, and counting points on GL_n over finite fields features in explicit formulas in Algebraic geometry and in the computation of zeta functions as in the work of André Weil.

Category:Algebraic groups