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 Connections
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 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.
View the Source Stone-Weierstrass Theorem (Wikipedia)
Wikipedialead paragraph
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces.
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.
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