This article was accepted into the corpus but its outbound wikilinks were never NER-processed — typical at the deepest BFS hop or when the run's entity cap was reached. No expansion funnel to show.
| PGL(n) | |
|---|---|
| Name | PGL(n) |
| Type | Algebraic group |
PGL(n)
PGL(n) is the projective general linear group associated to an n-dimensional vector space, a central object in algebraic group theory and algebraic geometry. It arises as the quotient of the general linear group by its center and acts naturally on projective space, linking classical projective geometry with modern Galois theory, representation theory, and arithmetic aspects of moduli spaces. PGL(n) appears across topics including the study of quadratic forms, Brauer group, Tate cohomology, and the classification of algebraic fiber bundles.
PGL(n) is defined as GL(n)/Z where GL(n) denotes the general linear group of invertible n×n matrices and Z its center consisting of scalar matrices; this construction parallels quotients in group cohomology and Galois cohomology. As an algebraic group over a field, PGL(n) is an example of a linear algebraic group, related to notions in Chevalley group theory and reductive group structure. For fields such as the complex numbers, the real numbers, finite fields like F_q, or local fields such as Q_p, PGL(n) has distinct topological and arithmetic properties studied via Lie groups and p-adic analysis. The group is often nonabelian for n≥2 and its center-triviality influences classification results in the theory of simple groups and inner forms.
Concretely, PGL(n) can be realized as equivalence classes of matrices in GL(n) under scalar multiplication, a perspective used in computations in matrix theory and determinant-based invariants. Over algebraically closed fields like C or algebraic closures of F_p, PGL(n) coincides with an adjoint form of the simple algebraic group of type A_{n−1} appearing in Dynkin diagram classification and in constructions by Cartan and Weyl. In arithmetic contexts, central simple algebras and Azumaya algebras over schemes yield twisted forms of PGL(n) via the Skolem–Noether theorem and connections to the Brauer group, with descent data encoded by Galois descent and Tsen's theorem in function field settings.
PGL(n) acts faithfully on projective (n−1)-space, providing the full group of projective linear automorphisms of classical spaces studied by Pascal, Desargues, and later formalized by Hermann Grassmann and Felix Klein. This action underlies the study of projective transformations, collineations, and automorphism groups of projective varieties such as Grassmannians, Veronese embeddings, and classical plane curves like Fermat curves and Cremona transformation-related surfaces. In moduli problems, PGL(n) quotients describe isomorphism classes of framed vector bundles and appear in geometric invariant theory constructions by David Mumford for constructing moduli of stable vector bundles and projective hypersurfaces.
The short exact sequence 1 → Z → GL(n) → PGL(n) → 1 links PGL(n) to GL(n) and to special linear group SL(n) through determinants and central quotients; SL(n)/{±I} often yields PSL(n) which for many n coincides with the derivation of PGL(n) in adjoint form. Relationships between these groups are crucial in classification theorems by Jordan, structural results in Lie algebra theory by Elie Cartan, and arithmetic dualities in the work of Langlands and Tate. For finite fields, the comparison of PGL(n,q), PSL(n,q), and GL(n,q) features heavily in the classification of finite simple groups and in the analysis of permutation representations studied by Frobenius and Burnside.
Subgroups of PGL(n) include images of parabolic subgroups corresponding to stabilizers of flags, Levi subgroups tied to block-diagonal matrices, and maximal tori given by classes of diagonal matrices, concepts central to the work of Borel and Tits. Finite subgroups in low dimensions relate to classical lists such as the icosahedral group, dihedral group, and polyhedral groups studied by Klein and Schwarz. The classification of conjugacy classes, centralizers, and Weyl groups connects PGL(n) to Bruhat decomposition, the Borel–Tits theorem, and buildings introduced by Jacques Tits for describing subgroup geometry.
Representation theory for PGL(n) over local and global fields interacts with the representation theories of GL(n) and SL(n) via lifting and trace formulas by Harish-Chandra and Selberg. Automorphic representations for PGL(n) are central in the Langlands program, with contributions from Godement, Jacquet, and Shahidi on L-functions and functoriality. For finite fields, character tables of PGL(n,q) are computed using methods pioneered by Green, Lusztig, and Deligne through étale cohomology and perverse sheaves, while harmonic analysis on PGL(n,R) leverages the Plancherel theorem developed by Gelfand and Naimark.
PGL(n) appears in moduli of projective bundles, geometric invariant theory quotients by Mumford and in the study of rational points on projective varieties via descent and obstruction theories by Manin and Colliot-Thélène. In arithmetic theory, PGL(n) features in the analysis of algebraic cycles, the Brauer–Manin obstruction, and the study of central simple algebra forms in the work of Albert and Brauer. Connections to modular forms, Shimura varieties, and reciprocity laws arise through PGL(n)-type groups in the Langlands correspondence and in the arithmetic of Galois representations studied by Serre and Wiles.
Category:Linear algebraic groups