Representation theory

Representation theory is a branch of <a href="/facts/Mathematics/pxTouaz4">mathematics</a> that studies <a href="/facts/Abstract_algebra/YNNE1CNz">abstract</a> <a href="/facts/Algebraic_structure/TkuXca24">algebraic structures</a> by representing their <a href="/facts/Element_(set_theory)/Q2mx1vHL">elements</a> as <a href="/facts/Linear_transformation/Q1PBKNVV">linear transformations</a> of <a href="/facts/Vector_space/M1pxMLx2">vector spaces</a>, and studies <a href="/facts/Module_(mathematics)/jP0LIhFL">modules</a> over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrices</a> and their <a href="/facts/Algebraic_operation/wl69LmeN">algebraic operations</a> (for example, <a href="/facts/Matrix_addition/6zzLWyL0">matrix addition</a>, <a href="/facts/Matrix_multiplication/lYDB6Ro2">matrix multiplication</a>).
The <a href="/facts/Algebra/nMdqczjo">algebraic</a> objects amenable to such a description include <a href="/facts/Group_(mathematics)/IQTYqINt">groups</a>, <a href="/facts/Associative_algebra/DRfoF6KV">associative algebras</a> and <a href="/facts/Lie_algebra/n2gAUkNq">Lie algebras</a>. The most prominent of these (and historically the first) is the <a href="/facts/Group_representation/FsuuDagQ">representation theory of groups</a>, in which elements of a group are represented by invertible matrices such that the group operation is matrix multiplication.
Representation theory is a useful method because it reduces problems in <a href="/facts/Abstract_algebra/YNNE1CNz">abstract algebra</a> to problems in <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a>, a subject that is well understood. Representations of more abstract objects in terms of familiar linear algebra can elucidate properties and simplify calculations within more abstract theories. For instance, representing a group by an infinite-dimensional <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a> allows methods of <a href="/facts/Mathematical_analysis/XXeR26Ih">analysis</a> to be applied to the theory of groups. Furthermore, representation theory is important in <a href="/facts/Physics/a7aH2y08">physics</a> because it can describe how the <a href="/facts/Symmetry_group/CNqeseMp">symmetry group</a> of a physical system affects the solutions of equations describing that system.
Representation theory is pervasive across fields of mathematics. The applications of representation theory are diverse. In addition to its impact on algebra, representation theory

<ul><li>generalizes <a href="/facts/Fourier_analysis/sUA8JP6q">Fourier analysis</a> via <a href="/facts/Harmonic_analysis/CyG5bIPb">harmonic analysis</a>,</li>
<li>is connected to <a href="/facts/Geometry/wTQf5ffQ">geometry</a> via <a href="/facts/Invariant_theory/5fIUh9h3">invariant theory</a> and the <a href="/facts/Erlangen_program/HAVoCLHG">Erlangen program</a>,</li>
<li>has an impact in number theory via <a href="/facts/Automorphic_form/H0Pn7MyP">automorphic forms</a> and the <a href="/facts/Langlands_program/koK9PhID">Langlands program</a>.</li></ul>
There are many approaches to representation theory: the same objects can be studied using methods from <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a>, <a href="/facts/Module_theory/jP0LIhFL">module theory</a>, <a href="/facts/Analytic_number_theory/mrHbWwrx">analytic number theory</a>, <a href="/facts/Differential_geometry/c04XmyLu">differential geometry</a>, <a href="/facts/Operator_theory/Ays1ofel">operator theory</a>, <a href="/facts/Algebraic_combinatorics/9b47n3Fh">algebraic combinatorics</a> and <a href="/facts/Topology/2EqW47cU">topology</a>.
The success of representation theory has led to numerous generalizations. One of the most general is in <a href="/facts/Category_theory/6zS3DXPa">category theory</a>. The algebraic objects to which representation theory applies can be viewed as particular kinds of categories, and the representations as <a href="/facts/Functor/l2u3rIBE">functors</a> from the object category to the <a href="/facts/Category_of_vector_spaces/AnnF9JKf">category of vector spaces</a>. This description points to two natural generalizations: first, the algebraic objects can be replaced by more general categories; second, the target category of vector spaces can be replaced by other well-understood categories.

Representation theory open-in-new

Representation theory