Representation theory is the branch of mathematics that studies abstract algebraic structures, chiefly groups, by representing their elements as linear transformations of vector spaces, turning abstract symmetry into the concrete language of matrices. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
Central QuestionGiven an abstract group or algebra, which of its representations are irreducible, the basic building blocks every other representation decomposes into, and can the irreducible representations be listed completely. 1 Key DebateWhether every irreducible representation of interest can be classified explicitly. The classification is essentially complete for finite and compact groups, but an effective description of the unitary dual, the full set of irreducible unitary representations, remains an open problem for the noncompact real reductive Lie groups central to mathematical physics. 1 Classification
Pure or Applied Connections
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Source Wikipedia: Bertram Kostant
Source Wikipedia: David Kazhdan
Source Frobenius reciprocity, Wikipedia
Source Issai Schur (Wikipedia)
Source Levi decomposition, Wikipedia
Source Lie-Kolchin theorem, Wikipedia
Source Schur Polynomial (Wikipedia)
Source Wikipedia: Walter Feit
Source Weyl's theorem on complete reducibility, Wikipedia
Source Wigner-Eckart theorem, Wikipedia
Source Wilhelm Specht, Biography (MacTutor History of Mathematics)
Source Wikipedia: Young tableau
Sources
1. Wikipedia: Representation Theory
Wikimedia FoundationLead section
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces.
Subrepresentations, quotients and irreducible representations section
Irreducible representations are the building blocks of representation theory for many groups: if a representation V is not irreducible then it is built from a subrepresentation and a quotient that are both simpler in some sense.
Unitary representations
An effective description of the unitary dual, even for relatively well-behaved groups such as real reductive Lie groups (discussed below), remains an important open problem in representation theory.
View the Source Weyl's theorem on complete reducibility, Wikipedia
Includes: Weyl's Theorem on Complete Reducibility, Lead sentenceQuote, Includes: Weyl's Theorem on Complete Reducibility, Lead sentence
Lie algebra representations (specifically in the representation theory of semisimple Lie algebras).
Wigner-Eckart theorem, Wikipedia
Wikipedia: Young tableau
Includes: Young Tableau, Lead sentenceQuote, Includes: Young Tableau, Lead sentence
al: tableaux) is a combinatorial object useful in representation theory and Schubert calculus.
Frobenius reciprocity, Wikipedia
Includes: Frobenius Reciprocity Theorem, Lead sentenceQuote, Includes: Frobenius Reciprocity Theorem, Lead sentence
In mathematics, and in particular representation theory, Frobenius reciprocity is a theorem expressing a duality between the proce
Lie-Kolchin theorem, Wikipedia
Levi decomposition, Wikipedia
Issai Schur (Wikipedia)
Includes: Issai Schur, Lead paragraph [in-branch]Quote, Includes: Issai Schur, Lead paragraph [in-branch]
group representations
Wikipedia: David Kazhdan
Includes: David Kazhdan, LeadQuote, Includes: David Kazhdan, Lead
known for work in representation theory
Wikipedia: Bertram Kostant
Includes: Bertram Kostant, LeadQuote, Includes: Bertram Kostant, Lead
who worked in representation theory
Wikipedia: Walter Feit
Includes: Walter Feit, LeadQuote, Includes: Walter Feit, Lead
Walter Feit (October 26, 1930, July 29, 2004) was an Austrian-born American mathematician who worked in finite group theory and representation theory.
Schur Polynomial (Wikipedia)
Includes: Schur Polynomial, Lead paragraphQuote, Includes: Schur Polynomial, Lead paragraph
In representation theory they are the characters of polynomial irreducible representations of the general linear groups.
Wilhelm Specht, Biography (MacTutor History of Mathematics)
Includes: Wilhelm Specht, SummaryQuote, Includes: Wilhelm Specht, Summary
Wilhelm Specht was a German mathematician who in worked group theory and representation theory.
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