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.{{Harvnb|Lam|1998}}.{{Harvnb|Cayley|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.{{Harvnb|Frobenius|1896}}, {{Harvnb|Frobenius|1897}}. 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,{{Harvnb|Burnside|1904}}. its power was soon appreciated.In the first edition of his famous treatise, {{Harvnb|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 ({{Harvnb|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." Later Richard Brauer and others developed modular representation theory.{{Harvb|Curtis|2003}}.

Notes

References

{{Citation|title=Essays in the History of Lie Groups and Algebraic Groups|first=Armand|last= Borel|author-link=Armand Borel|publisher=American Mathematical Society|year= 2001|isbn=978-0-8218-0288-5}}
{{Citation | last1=Brauer | first1=R. | author1-link=Richard Brauer | title=Über die Darstellung von Gruppen in Galoisschen Feldern | url=https://books.google.com/books?id=hkexAAAAIAAJ | publisher=Hermann et cie |location=Paris | series=Actualités Scientifiques et Industrielles | year=1935 | volume=195 | pages=1–15}}
{{cite book|last1=Burnside|first1=William|authorlink=William Burnside|title=Theory of groups of finite order|year=1897|publisher=Cambridge University Press|edition=1st|url=https://catalog.hathitrust.org/Record/006199565|pages=xxiv+388 pp|no-pp=yes}}
{{Citation|first=William|last= Burnside|title=On Groups of Order p^αq^β

|journal=Proceedings of the London Mathematical Society |year=1904|issue= s2-1 (1)|pages= 388–392 |doi=10.1112/plms/s2-1.1.388|url=https://zenodo.org/record/1433459}}

{{cite book|last1=Burnside|first1=William|title=Theory of groups of finite order|year=1911|publisher=Cambridge University Press|edition=2nd|url=https://catalog.hathitrust.org/Record/000419368|pages=xxiv+512 pp|no-pp=yes}}
{{cite journal|last1=Cayley |first1=A.|authorlink=Arthur Cayley|title=On the theory of groups, as depending on the symbolic equation θⁿ = 1 |journal=Philosophical Magazine |date=1854 |volume=7 |issue=42 |pages=40–47 |url=https://babel.hathitrust.org/cgi/pt?id=pst.000068485757;view=1up;seq=54 |series=4th series |doi=10.1080/14786445408647421|url-access=subscription }}
{{cite web|last1=Conrad|first1=Keith|title=The origin of representation theory|url=https://kconrad.math.uconn.edu/articles/groupdet.pdf}}
{{Citation|last1=Curtis | first1=Charles W. | authorlink = Charles W. Curtis | title=Pioneers of Representation Theory: Frobenius, Burnside, Schur, and Brauer | publisher=American Mathematical Society | location=Providence, R.I. | series=History of Mathematics | isbn=978-0-8218-2677-5 | year=2003}}
{{Citation|first1=Walter|last1=Feit|authorlink1=Walter Feit|first2=John G.|last2=Thompson|authorlink2=John G. Thompson|title=Solvability of groups of odd order|journal= Pacific J. Math.|volume=13|year=1963|pages=775–1029|doi=10.2140/pjm.1963.13.775 }}
{{Citation|last1=Frobenius|first1=Ferdinand G.|authorlink=Ferdinand Georg Frobenius|title=Uber Gruppencharaktere|journal=Sitzungsberichte der Koniglich Preussischen Akademie der Wissenschaften zu Berlin|year=1896|pages=985–1021}}
{{Citation|last1=Frobenius|first1=Ferdinand G.|title=Uber die Darstellung der endlichen Gruppen durch lineare Substitutionen|journal=Sitzungsberichte der Koniglich Preussischen Akademie der Wissenschaften zu Berlin|year=1897|pages=944–1015}}
{{Citation | last1=Gabriel | first1=Peter | authorlink = Pierre Gabriel | title=Unzerlegbare Darstellungen. I | doi=10.1007/BF01298413 | mr=0332887 | year=1972 | journal=Manuscripta Mathematica | issn=0025-2611 | volume=6 | pages=71–103| s2cid=119425731 }}
{{Citation |last=King |first=Alastair | journal=Quart. J. Math. |year=1994 |title= Moduli of representations of finite-dimensional algebras | volume=45 |issue= 180 | pages=515–530 |doi=10.1093/qmath/45.4.515|doi-access=free }}
{{Citation|first=Felix|last=Klein|authorlink=Felix Klein|year=1872|title=Vergleichende Betrachtungen über neuere geometrische Forschungen|trans-title=A comparative review of recent researches in geometry|journal=Mathematische Annalen|volume= 43|pages=63–100|doi=10.1007/BF01446615 |arxiv=0807.3161 }}
{{cite book |last1=Kleiner |first1=Israel |editor-first1=Israel |editor-last1=Kleiner |title=A history of abstract algebra |date=2007 |publisher=Birkhäuser |location=Boston, Mass. |doi=10.1007/978-0-8176-4685-1 |isbn=978-0-8176-4685-1 |url=https://link.springer.com/book/10.1007/978-0-8176-4685-1}}
{{Citation|title=Representations of finite groups: a hundred years|first=T. Y.|last=Lam|journal=Notices of the AMS|volume = 45| issue= 3,4|year=1998|pages =[https://www.ams.org/notices/199803/lam.pdf 361–372 (Part I)], [https://www.ams.org/notices/199804/lam2.pdf 465–474 (Part II)]}}
{{Citation|last=Langlands|first=Robert|authorlink=Robert Langlands|title=Letter to Prof. Weil|year=1967|url=http://publications.ias.edu/rpl/section/21}}
{{cite journal|first=George W.|last=Mackey|authorlink=George W. Mackey|doi=10.1090/S0273-0979-1980-14783-7|title= Harmonic analysis as the exploitation of symmetry – a historical survey|journal=Bull. Amer. Math. Soc.|volume=3|year=1980|pages=543–698|doi-access=free|hdl=1911/63317|hdl-access=free}}
{{cite book |first=Shlomo|last=Sternberg|authorlink=Shlomo Sternberg|year=1994|title=Group Theory and Physics| publisher=Cambridge University Press|isbn=0-521-24870-1}}
{{cite journal|first=Eugene P. |last=Wigner |authorlink=Eugene Wigner |year=1939 |title=On unitary representations of the inhomogeneous Lorentz group |journal=Annals of Mathematics |issue=1 |volume=40 | pages=149–204 |doi=10.2307/1968551 |mr=1503456 |jstor=1968551|bibcode=1939AnMat..40..149W }}

Category:Representation theory

Representation theory