History of representation theory

See also: History of group theory

The history of representation theory concerns the mathematical development of the study of objects in abstract algebra, notably groups, by describing these objects more concretely, particularly using matrices and linear algebra. In some ways, representation theory predates the mathematical objects it studies: for example, permutation groups (in algebra) and transformation groups (in geometry) were studied long before the notion of an abstract group was formalized by Arthur Cayley in 1854. Thus, in the history of algebra, there was a process in which, first, mathematical objects were abstracted, and then the more abstract algebraic objects were realized or represented in terms of the more concrete ones, using homomorphisms, actions and modules.

An early pioneer of the representation theory of finite groups was Ferdinand Georg Frobenius. At first this method was not widely appreciated, but with the development of character theory and the proof of Burnside's solvability criterion, its power was was soon appreciated. Later Richard Brauer and others developed modular representation theory.

References

• Borel, Armand (2001), Essays in the History of Lie Groups and Algebraic Groups, American Mathematical Society, ISBN 978-0-8218-0288-5.
• Cayley, A. (1854). "On the theory of groups, as depending on the symbolic equation θⁿ = 1". Philosophical Magazine. 4th series. 7 (42): 40–47. doi:10.1080/14786445408647421.
• Curtis, Charles W. (2003), Pioneers of Representation Theory: Frobenius, Burnside, Schur, and Brauer, History of Mathematics, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2677-5
• Kleiner, Israel (2007). Kleiner, Israel (ed.). A history of abstract algebra. Boston, Mass.: Birkhäuser. doi:10.1007/978-0-8176-4685-1. ISBN 978-0-8176-4685-1.
• Lam, T. Y. (1998), "Representations of finite groups: a hundred years", Notices of the AMS, 45 (3, 4): 361–372 (Part I), 465–474 (Part II).