History of representation 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.^[1]^[2] 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.^[3] At first this method was not widely appreciated, but with the development of character theory and the proof of Burnside's $p^{\alpha }q^{\beta }$ solvability criterion using such methods,^[4] its power was soon appreciated.^[5] Later Richard Brauer and others developed modular representation theory.^[6]

Notes

^ Lam 1998.
^ Cayley 1854.
^ Frobenius 1896, Frobenius 1897.
^ Burnside 1904.
^ In the first edition of his famous treatise, Burnside 1897 writes "It may then be asked why... modes of representation, such as groups of linear transformations, are not even referred to. My answer to this question is that while, in the present state of our knowledge, many results in the pure theory are arrived at most readily by dealing with properties of substitution groups, it would be difficult to find a result that could be most directly obtained by the consideration of groups of linear transformations." In the second edition (Burnside 1911) he writes instead, "The theory of groups of linear substitutions has been the subject of numerous and important investigations by several writers; and the reason given in the original preface for omitting any account of it no longer holds good. In fact it is now more true to say that for further advances in the abstract theory one must look largely to the representation of group as a group of linear substitutions."
^ Curtis (2003).

References

Borel, Armand (2001), Essays in the History of Lie Groups and Algebraic Groups, American Mathematical Society, ISBN 978-0-8218-0288-5.
Burnside, William (1897). Theory of groups of finite order (1st ed.). Cambridge University Press; xxiv+388 pp.{{cite book}}: CS1 maint: postscript (link)
Burnside, William (1904), "On Groups of Order p^αq^β", Proceedings of the London Mathematical Society (s2-1 (1)): 388–392, doi:10.1112/plms/s2-1.1.388
Burnside, William (1911). Theory of groups of finite order (2nd ed.). Cambridge University Press; xxiv+512 pp.{{cite book}}: CS1 maint: postscript (link)
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.
Conrad, Keith. "The origin of representation theory" (PDF).
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
Frobenius, Ferdinand G. (1896), "Uber Gruppencharaktere", Sitzungsberichte der Koniglich Preussischen Akademie der Wissenschaften zu Berlin: 985–1021
Frobenius, Ferdinand G. (1897), "Uber die Darstellung der endlichen Gruppen durch lineare Substitutionen", Sitzungsberichte der Koniglich Preussischen Akademie der Wissenschaften zu Berlin: 944–1015
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).

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[1] Lam 1998.

[2] Cayley 1854.

[3] Frobenius 1896, Frobenius 1897.

[4] Burnside 1904.

[5] In the first edition of his famous treatise, Burnside 1897 writes "It may then be asked why... modes of representation, such as groups of linear transformations, are not even referred to. My answer to this question is that while, in the present state of our knowledge, many results in the pure theory are arrived at most readily by dealing with properties of substitution groups, it would be difficult to find a result that could be most directly obtained by the consideration of groups of linear transformations." In the second edition (Burnside 1911) he writes instead, "The theory of groups of linear substitutions has been the subject of numerous and important investigations by several writers; and the reason given in the original preface for omitting any account of it no longer holds good. In fact it is now more true to say that for further advances in the abstract theory one must look largely to the representation of group as a group of linear substitutions."

[6] Curtis (2003).

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