representation theory

representation theory
Name	Representation theory
Field	Mathematics
Subfieldof	Algebra
Notable	Évariste Galois, Camille Jordan, Ferdinand Frobenius, Issai Schur, Hermann Weyl, Emmy Noether, Harish-Chandra, George Mackey, John von Neumann, André Weil, Bernhard Kostant, Bertram Kostant, David Kazhdan, Robert Langlands

representation theory

Representation theory studies how algebraic objects act by linear transformations on vector spaces. It connects abstract structures such as Évariste Galois's permutation groups, Camille Jordan's matrix methods, and Emmy Noether's ring-theoretic ideas to concrete linear algebra, enabling classification, decomposition, and computation of invariants. The subject interacts deeply with areas associated with Hermann Weyl, André Weil, and Robert Langlands, yielding tools for number theory, geometry, and mathematical physics.

Introduction

Representation theory examines homomorphisms from algebraic objects into endomorphism algebras of vector spaces; classical instances include mappings from groups to matrix groups studied by Ferdinand Frobenius, Issai Schur, and Hermann Weyl. Historically motivated by problems in permutation groups treated by Camille Jordan and character theory advanced by Ferdinand Frobenius and Issai Schur, the field expanded through contributions by Emmy Noether, John von Neumann, and Harish-Chandra. Modern developments link representation-theoretic methods with conjectures of Robert Langlands, categorical frameworks promoted by Alexander Grothendieck, and homological techniques influenced by Jean-Pierre Serre and Daniel Quillen.

Basic concepts and definitions

Key notions include modules over rings as representations of associative algebras, group representations realized as homomorphisms into general linear groups, and Lie algebra representations given by linear actions respecting brackets; foundational contributors include Emmy Noether and Nathan Jacobson. Irreducible and indecomposable representations, Maschke-type decompositions traced to Ferdinand Frobenius, and characters introduced by Issai Schur serve as primary invariants. Intertwining operators, Schur's Lemma, and complete reducibility criteria echo results attributed to Issai Schur, Hermann Weyl, and Harish-Chandra. Construction methods involve induced representations from Frobenius reciprocity-style principles and restriction functors analyzed in the work of George Mackey.

Representation theory of groups

Finite group representation theory, advanced by Ferdinand Frobenius and Issai Schur, uses characters, block theory influenced by Richard Brauer, and modular methods with inputs from Graham Higman and Alperin. Compact Lie group representations, systematized by Hermann Weyl and Élie Cartan, employ highest-weight theory and Weyl character formulae. Reductive groups over local and global fields, central to the Langlands Program articulated by Robert Langlands and expanded by Harish-Chandra and André Weil, connect automorphic representations studied by Jacquet and Langlands collaborators. Methods for unitary representations trace to George Mackey and David Kazhdan; harmonic analysis on groups links with work by Israel Gelfand, I. M. Gelfand, and Mark Naimark.

Representation theory of associative algebras and Lie algebras

Representation theory of associative algebras uses module categories, Auslander–Reiten theory developed by Maurice Auslander and Idun Reiten, and Gabriel's classification of quivers credited to Pierre Gabriel. Lie algebra representation theory, rooted in structure theory by Élie Cartan and classification by Cartan and Weyl, utilizes highest-weight modules, Verma modules introduced by Daya-Nand Verma, and the Kazhdan–Lusztig theory initiated by David Kazhdan and George Lusztig. Universal enveloping algebras, Poincaré–Birkhoff–Witt theorems linked to Henri Poincaré and George David Birkhoff, and category O formulated by James Lepowsky and Joseph Bernstein underpin many classification results.

Categories, modules, and homological methods

Categorical language, championed by Alexander Grothendieck, recasts representations as objects in module categories with morphisms as intertwiners; this perspective enabled derived and triangulated techniques by Jean-Louis Verdier and Alexandre Grothendieck. Homological algebra tools from Henri Cartan and Samuel Eilenberg produce Ext and Tor computations critical for extension and resolution problems; Hochschild cohomology and cyclic cohomology studied by Gerald Hochschild and Alain Connes refine deformation theory. Tensor categories and braided monoidal categories influenced by Vladimir Drinfeld, Michael Atiyah, and Graeme Segal inform quantum group representations and modular tensor categories tied to Edward Witten's work in mathematical physics.

Applications and connections

Representation-theoretic methods apply across number theory through the Langlands Program and modular forms explored by Kurt Gödel-era contemporaries and Pierre Deligne, algebraic geometry via perverse sheaves and geometric representation theory advanced by Masaki Kashiwara and Lusztig, and mathematical physics via conformal field theory and quantum groups developed by Vladimir Drinfeld and Michio Jimbo. Coding theory and combinatorics connect through symmetric group representations studied by Frank Harary-era combinatorialists and Richard Stanley; chemistry and spectroscopy exploit symmetry group methods dating to Hermann Weyl and Fritz London's molecular orbital theory. Representation theory underpins index theorems associated with Atiyah–Singer and trace formulae originating in Atle Selberg's analytic work.

Major theorems and classification results

Principal results include Maschke's theorem for finite groups, Frobenius reciprocity, Schur's Lemma, Weyl character formula, the Peter–Weyl theorem, the Borel–Weil–Bott theorem credited to Armand Borel and Raoul Bott, the classification of finite-dimensional semisimple Lie algebras by Élie Cartan, and the Kazhdan–Lusztig conjectures proved by Soergel and others. Gabriel's theorem classifies quivers of finite type; the Tannaka–Krein duality framework developed by Tadao Tannaka and Mark Krein reconstructs groups from representation categories. The Local and Global Langlands correspondences, formulated by Robert Langlands and furthered by Michael Harris and Richard Taylor, constitute deep classification paradigms linking Galois representations and automorphic representations.

Category:Mathematics