Branches of Mathematics
Representation Theory
Citation Formats
General Reference
APA Style
BibTeX
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.
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 Cross-Tradition Connections
Sources
1. Wikipedia: Representation Theory
Wikimedia FoundationLead sectionQuote, Lead section
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces.
1. Wikipedia: Representation Theory
Wikimedia FoundationSubrepresentations, quotients and irreducible representations sectionQuote, 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.
1. Wikipedia: Representation Theory
Wikimedia FoundationUnitary representations sectionQuote, Unitary representations section
An effective description of the unitary dual, even for relatively well-behaved groups such as real reductive Lie groups, remains an important open problem in representation theory.
Stone-Weierstrass Theorem (Wikipedia)
Wikipedialead paragraphQuote, lead paragraph
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces.
Stone-Weierstrass Theorem (Wikipedia)
WikipediaUnitary representations sectionQuote, Unitary representations section
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.
Reader Challenges (0 open reader challenges)
No disputes yet. Spotted an error or a better source? Open the first one.
Sign in to dispute this or suggest a correction.
View At A Past Year
The atlas records no dated fact of its own for this entry, so there is no other year to choose.