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| SL(n, F) | |
|---|---|
| Name | Special linear group |
| Type | Linear algebraic group |
| Rank | n−1 |
| Center | scalar matrices with determinant 1 |
SL(n, F) is the group of n×n matrices with entries in the field F and determinant equal to 1, forming a central example of a linear algebraic group, a Lie group when F is R or C, and an arithmetic group when F is Z or a number field. It serves as a standard test case in the study of algebraic groups, representation theory, algebraic K-theory, and automorphic forms, and appears across the work of figures such as Évariste Galois, Henri Poincaré, Emmy Noether, Claude Chevalley, and Armand Borel.
By definition the group consists of invertible matrices A ∈ GL(n, F) with determinant det(A)=1; as such it is a normal subgroup of GL(n, F). Its center consists of scalar matrices ζ I with ζ^n=1 when F is algebraically closed, relating to cyclotomic phenomena studied by Carl Friedrich Gauss and Kummer. Over a general field F the group is an affine variety cut out by the polynomial det−1, linking to the constructions of Alexander Grothendieck in scheme theory and to classical results of David Hilbert. SL(n, F) is generated by elementary matrices (transvections), a fact used in the proofs of the Suslin stability theorem of Andrei Suslin and in Bass–Serre theory developed by Hyman Bass and Jean-Pierre Serre.
For n=1 the group is trivial modulo the condition det=1 and connects to the work of Joseph-Louis Lagrange on units; for n=2 the group SL(2, F) gives the classical two-dimensional examples intimately tied to Ferdinand von Lindemann’s transcendence context and to modular curves studied by Bernhard Riemann and Hecke. SL(2, R) and SL(2, C) relate to the theory of Fuchsian groups investigated by Henri Poincaré and Kleinian groups examined by Felix Klein, while SL(2, Z) is the source of modular group phenomena central to Srinivasa Ramanujan and Andrew Wiles’s work on elliptic curves. For n=3, SL(3, F) appears in classical invariant theory pursued by David Hilbert and in the theory of exceptional isomorphisms encountered in studies by Élie Cartan and John Milnor.
The algebraic structure features Borel subgroups and maximal tori studied by Armand Borel and Robert Langlands; parabolic subgroups correspond to block upper-triangular matrices and parabolic induction appears in the work of I. M. Gelfand and Harish-Chandra. Weyl groups for SL(n, F) are isomorphic to symmetric groups studied by Augustin-Louis Cauchy and Évariste Galois, reflecting root systems of type A_{n−1} analyzed by Wilhelm Killing and Élie Cartan. Important discrete subgroups include congruence subgroups such as Γ(N) ⊂ SL(2, Z) which underlie the modularity results of Yuri Manin and Ken Ribet. The classification of finite subgroups ties into the McKay correspondence addressed by John McKay and into the finite simple groups program related to work by Daniel Gorenstein and Robert Griess.
Representation theory of SL(n, F) connects highest-weight theory pioneered by Hector Carmichael and Élie Cartan to the modern formulations of Georges Lusztig and James Arthur. Over algebraically closed fields of characteristic zero, irreducible polynomial representations are indexed by dominant weights and studied in the context of Schur–Weyl duality linking to Issai Schur and Hermann Weyl. Over finite fields, representations of SL(n, q) are central to Deligne–Lusztig theory developed by Pierre Deligne and George Lusztig and to the classification of finite simple groups that engaged Bertram Kostant and Roger Howe. Modular representations over fields of positive characteristic involve work of J. A. Green and Daniel J. Benson, and connections to algebraic K-theory and stability phenomena relate to the Quillen–Suslin theorem and the efforts of Daniel Quillen.
When F=R or C, SL(n, F) is a real or complex Lie group whose Lie algebra sl_n is a simple Lie algebra appearing in the classification by Élie Cartan and in the formulation of the Killing form by Wilhelm Killing. Maximal compact subgroups such as SU(n) connect to the unitary groups studied by John von Neumann and Hermann Weyl; symmetric space structures relate to the work of Sigurdur Helgason and appear in harmonic analysis developed by Harish-Chandra. Fundamental groups and covering groups (e.g., the universal cover of SL(2, R)) play roles in the theory of discrete series representations investigated by Atle Selberg and Harish-Chandra, and in topological K-theory as studied by Michael Atiyah and Friedrich Hirzebruch.
SL(n, F) occurs in arithmetic geometry via arithmetic subgroups like SL(n, Z) and their action on locally symmetric spaces central to the Langlands program formulated by Robert Langlands and explored by James Arthur and Pierre Deligne. Modular forms for SL(2, Z) underpin the proofs of the modularity theorem by Andrew Wiles and Richard Taylor, while higher-rank cases enter the study of automorphic representations in the work of Friedrich Hirzebruch and Gerard Laumon. In algebraic geometry, vector bundle moduli and stability notions involve structure groups reducible to SL(n) as in geometric invariant theory of David Mumford and in gauge theory related to Simon Donaldson and Karen Uhlenbeck. In topology and combinatorics, cohomology of arithmetic quotients links to the study of L-functions and special values investigated by John Tate and Pierre Deligne, and to expander graphs constructed from Cayley graphs of SL(n, F) as in the work of Alexander Lubotzky and Michel L. Lapidus.
Category:Linear algebraic groups