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Branch of mathematics that studies abstract algebraic structures
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of
Representation_theory
Representation of the symmetry group of spacetime in special relativity
theories. The development of the representation theory has historically followed the development of the more general theory of representation theory of
Representation theory of the Lorentz group
Representation_theory_of_the_Lorentz_group
Group homomorphism into the general linear group over a vector space
In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector
Group_representation
Representation theory (RT) is a theoretical linguistic framework in the generative tradition, created and developed by Edwin S. Williams – chiefly in
Representation theory (linguistics)
Representation_theory_(linguistics)
Framework for exploring meaning
In formal linguistics, discourse representation theory (DRT) is a framework for exploring meaning under a formal semantics approach. One of the main differences
Discourse representation theory
Discourse_representation_theory
Concept in Lie algebra representation theory
In the mathematical field of representation theory, the concept of weights of an algebra A over a field F is a generalisation of that of eigenvalues.
Weight (representation theory)
Weight_(representation_theory)
Studies linear representations of finite groups over fields of positive characteristic
Modular representation theory is a branch of mathematics, and is the part of representation theory that studies linear representations of finite groups
Modular_representation_theory
Area of discrete mathematics
In mathematics and computer science, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects
Graph_theory
Type of group and algebra representation
In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation ( ρ , V ) {\displaystyle (\rho ,V)} or
Irreducible_representation
Area of mathematics
In mathematics, the representation theory of the symmetric group is a particular case of the representation theory of finite groups, for which a concrete
Representation theory of the symmetric group
Representation_theory_of_the_symmetric_group
Group of unitary complex matrices with determinant of 1
looking at the induced long exact sequence on homotopy groups. The representation theory of SU(3) is well-understood. Descriptions of these representations
Special_unitary_group
Mathematical representation
theory of covering spaces, Bn acts on H1(Cn), and this representation is called the reduced Burau representation. The unreduced Burau representation has
Burau_representation
Algebraic structure used in analysis
is an important result, the primary goal of representation theory is not to find a faithful representation of a given Lie algebra g {\displaystyle {\mathfrak
Lie_algebra
Representations of finite groups, particularly on vector spaces
The representation theory of groups is a part of mathematics which examines how groups act on given structures. Here the focus is in particular on operations
Representation theory of finite groups
Representation_theory_of_finite_groups
Quantum field theory enjoying conformal symmetry
space of states of a theory is a representation of the product of the two Virasoro algebras. This space is a Hilbert space if the theory is unitary. This
Conformal_field_theory
This is a glossary of representation theory in mathematics. The term "module" is often used synonymously for a representation; for the module-theoretic
Glossary of representation theory
Glossary_of_representation_theory
First case of a Lie group that is both compact and non-abelian
In the study of the representation theory of Lie groups, the study of representations of SU(2) is fundamental to the study of representations of semisimple
Representation theory of SU(2)
Representation_theory_of_SU(2)
Concept in mathematics
V} . Serre 2001, Ch. VII, § 6. Etingof, Pavel. "Lecture Notes on Representation Theory". Kac, Victor (1990). "Integrable Representations of Kac–Moody Algebras
Special_linear_Lie_algebra
Branch of mathematics that studies the properties of groups
group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is
Group_theory
Concept in mathematical group theory
In mathematics, more specifically in group theory, the character of a group representation is a function on the group that associates to each group element
Character_theory
Area of mathematical analysis
partial differential equations, potential theory, ergodic theory, representation theory, and number theory. Harmonic analysis shares many methods with
Harmonic_analysis
Supposition or system of ideas intended to explain something
theory — Perturbation theory — Potential theory — Probability theory — Ramsey theory — Rational choice theory — Representation theory — Ring theory —
Theory
German mathematician (1882–1935)
noncommutative algebras and hypercomplex numbers and united the representation theory of groups with the theory of modules and ideals. In addition to her own publications
Emmy_Noether
Psychological theory
Dual representation theory (DRT) is a psychological theory of post-traumatic stress disorder (PTSD) first developed by Chris Brewin, Tim Dalgleish, and
Dual_representation_theory
Study of abstract machines and automata
Automata theory is the study of abstract machines and automata, as well as the computational problems that can be solved using them. It is a theory in theoretical
Automata_theory
Writing Lie algebra sets as matrices
In the mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra
Lie_algebra_representation
Group representation
Representation theory of connected compact groups Lie algebra representation Projective representation Representation theory of SU(2) Representation theory
Representation_of_a_Lie_group
Group that is also a differentiable manifold with group operations that are smooth
In M-theory, for example, a 10-dimensional SU(N) gauge theory becomes an 11-dimensional theory when N becomes infinite. Adjoint representation of a Lie
Lie_group
discrete and Euclidean geometries, graph theory, group theory, mathematical logic, number theory, set theory, Ramsey theory, dynamical systems, and partial differential
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Russian mathematician (born 1969)
mathematician who works on representation theory and its applications to algebraic geometry, mathematical physics, probability theory and special functions
Andrei_Okounkov
Chinese-American mathematician (born 1982)
Professor of Mathematics at MIT specializing in number theory, algebraic geometry and representation theory, with a particular focus on the Langlands program
Zhiwei_Yun
Representation theory of an important group in physics
In mathematics, the representation theory of the Poincaré group is an example of the representation theory of a Lie group that is neither a compact group
Representation theory of the Poincaré group
Representation_theory_of_the_Poincaré_group
Topological group with compact topology
fashion. Compact groups have a well-understood theory, in relation to group actions and representation theory. In the following we will assume all groups
Compact_group
Theory of subatomic structure
symplectic geometry and representation theory. Prior to 1995, theorists believed that there were five consistent versions of superstring theory (type I, type IIA
String_theory
Mathematical terminology
Galois representation is frequently used when the G-module is a vector space over a field or a free module over a ring in representation theory, but can
Galois_representation
Direct sum of simple Lie algebras
Lie algebras, which were classified by Élie Cartan. Further, the representation theory of semisimple Lie algebras is much cleaner than that for general
Semisimple_Lie_algebra
An object whose endomorphisms are isomorphic to another structure
the term representation theory is well established in the algebraic sense discussed above, there are many other uses of the term representation throughout
Representation_(mathematics)
Unitary representations of a Lie group
produces only one unitary representation of SL(2, R), the trivial representation. The finite-dimensional representation theory of the noncompact group SL(2
Representation theory of SL2(R)
Representation_theory_of_SL2(R)
Physics-mathematics connection
There is a natural connection between particle physics and representation theory, as first noted in the 1930s by Eugene Wigner. It links the properties
Particle physics and representation theory
Particle_physics_and_representation_theory
Linear representation in mathematics
mathematics, the Steinberg representation, or Steinberg module or Steinberg character, denoted by St, is a particular linear representation of a reductive algebraic
Steinberg_representation
Australian mathematician (born 1981)
in automorphic forms and number theory, in particular representation theory, locally symmetric spaces, ergodic theory, and algebraic topology. He was
Akshay_Venkatesh
Soviet mathematician (1913–2009)
contributions to many branches of mathematics, including group theory, representation theory and functional analysis. The recipient of many awards, including
Israel_Gelfand
Branch of mathematics concerning probability
Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations
Probability_theory
Representation theory of groups
and in particular the theory of group representations, the regular representation of a group G is the linear representation afforded by the group action
Regular_representation
In group theory, restriction forms a representation of a subgroup using a known representation of the whole group. Restriction is a fundamental construction
Restricted_representation
Group of flat spacetime symmetries
group Galilean group Particle physics and representation theory Pauli–Lubanski pseudovector Representation theory of the Poincaré group Super-Poincaré algebra
Poincaré_group
Mathematical group
central role in symplectic geometry, Hamiltonian mechanics, and representation theory. A related but different family is the compact symplectic group
Symplectic_group
algebra. Representation theory of groups Representation theory of the Galilean group Representation theory of the Lorentz group Representation theory of the
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Study of the properties of codes and their fitness
Coding theory is the study of the properties of codes and their respective fitness for specific applications. Codes are used for data compression, cryptography
Coding_theory
Hypothetical internal cognitive symbol that represents external reality
A mental representation (or cognitive representation), in philosophy of mind, cognitive psychology, neuroscience, and cognitive science, is a hypothetical
Mental_representation
American mathematician
mathematician who works in the areas of analytic number theory, automorphic forms and representation theory, L-functions, harmonic analysis, and homogeneous
Alex_Kontorovich
Set with associative invertible operation
point of view of representation theory (that is, through the representations of the group) and of computational group theory. A theory has been developed
Group_(mathematics)
Mathematical term
In mathematics, the adjoint representation (or adjoint action) of a Lie group G is a way of representing the elements of the group as linear transformations
Adjoint_representation
Geometric arrangements of points, foundational to Lie theory
concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representation theory of semisimple Lie algebras.
Root_system
Representation theory of the symmetries of non-relativistic quantum space
Wigner's classification of relativistic mechanics) in terms of the representation theory of the Galilean group, which is the spacetime symmetry group of
Representation theory of the Galilean group
Representation_theory_of_the_Galilean_group
Map from algebra to geometric transforms
In the field of representation theory in mathematics, a projective representation of a group G on a vector space V over a field F is a group homomorphism
Projective_representation
Decomposition of an integer as a sum of positive integers
of symmetric polynomials and of the symmetric group and in group representation theory in general. The seven partitions of 5 are 5 4 + 1 3 + 2 3 + 1 +
Integer_partition
direct sum of irreducible representations of the group. By the representation theory of the general linear group we know that each summand is isomorphic
Plethysm
(combinatorics) Birkhoff's representation theorem (lattice theory) Boolean prime ideal theorem (mathematical logic) Bourbaki–Witt theorem (order theory) Cantor's isomorphism
List_of_theorems
Aspect of mathematical representation theory
In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible finite-dimensional representation of a semisimple
Fundamental_representation
System of ideas establishing social order
communication among the members of groups and communities. Social representation theory is a body of theory within social psychology and sociological social psychology
Social_representation
Classification scheme for hadrons
towards the quark model, which proved to be the solution. Group representation theory is the mathematical underpinning of the eightfold way, but technical
Eightfold_way_(physics)
Concept in mathematics
the case of an abelian group G, a fairly complete picture of the representation theory of G is given by Pontryagin duality. In general, the unitary equivalence
Unitary_representation
Group representation
(g)} . In the representation theory of SU(2), the dual of each irreducible representation does turn out to be isomorphic to the representation. But for the
Dual_representation
Physical theory with fields invariant under the action of local "gauge" Lie groups
In physics, a gauge theory is a type of field theory in which the Lagrangian, and hence the dynamics of the system itself, does not change under local
Gauge_theory
Matrices named after Élie Cartan
lattice and root lattice, respectively. In modular representation theory, and more generally in the theory of representations of finite-dimensional associative
Cartan_matrix
Representation theory of the symplectic group
Neumann theorem, was later interpreted within group representation theory, in particular the theory of induced representations initiated by George Mackey
Oscillator_representation
Linear representation in abstract algebra
abstract algebra known as representation theory, a faithful representation ρ of a group G on a vector space V is a linear representation in which different elements
Faithful_representation
Methods of mathematical approximation
In mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related
Perturbation_theory
Representation of a group or algebra that is a direct sum of simple representations
specifically in representation theory, a semisimple representation (also called a completely reducible representation) is a linear representation of a group
Semisimple_representation
Branch of mathematics that studies algebraic structures
reciprocity Induced representation Restricted representation Affine representation Projective representation Modular representation theory Quiver (mathematics)
List of abstract algebra topics
List_of_abstract_algebra_topics
Mathematical theorem
Hn. One representation of the Heisenberg group which is important in number theory and the theory of modular forms is the theta representation, so named
Stone–von_Neumann_theorem
Graphical representation of supersymmetric algebras
In supergravity and supersymmetric representation theory, Adinkra symbols are a graphical representation of supersymmetric algebras. Mathematically they
Adinkra_symbols_(physics)
Branch of applied mathematics
mathematical formulation of quantum field theory has also brought about some progress in fields such as representation theory. There is a tradition of mathematical
Mathematical_physics
Branch of discrete mathematics
group and in group representation theory in general. Graphs are fundamental objects in combinatorics. Considerations of graph theory range from enumeration
Combinatorics
In mathematical representation theory, a minuscule representation of a semisimple Lie algebra or group is an irreducible representation such that the Weyl
Minuscule_representation
Generalization of the discrete Fourier transform
relationship between the Fourier transform on finite groups and the representation theory of finite groups. The set of complex-valued functions on a finite
Fourier transform on finite groups
Fourier_transform_on_finite_groups
Process of extending a representation of a subgroup to the parent group
In group theory, the induced representation is a representation of a group, G, which is constructed using a known representation of a subgroup H. Given
Induced_representation
Fundamental unit of cognition
exemplar theory, similarity derives from individual memories of concrete instances rather than a single prototypical summary representation. Theory theory maintains
Concept
Type of group in abstract algebra
important to diverse areas of mathematics such as Galois theory, invariant theory, the representation theory of Lie groups, and combinatorics. Cayley's theorem
Symmetric_group
Non-commutative group with 6 elements
a representation on C 3 {\displaystyle \mathbb {C} ^{3}} by permuting the entries of the vector, the fundamental representation. This representation is
Dihedral_group_of_order_6
Mathematical study of invariants under symmetries
{\displaystyle k} (which in classical invariant theory was usually assumed to be the complex numbers). A representation of G {\displaystyle G} in V {\displaystyle
Invariant_theory
Physical quantities taking values at each point in space and time
characterized by numbers or quantum operators respectively. In this theory an equivalent representation of field is a field particle, for instance a boson. To Isaac
Field_(physics)
Study of discrete mathematical structures
part of number theory and analysis, partition theory is now considered a part of combinatorics or an independent field. Order theory is the study of
Discrete_mathematics
Directed graph which is also a multigraph
In mathematics, especially representation theory, a quiver is another name for a multidigraph; that is, a directed graph where loops and multiple arrows
Quiver_(mathematics)
Topics referred to by the same term
well as other theories Representative democracy, type of democracy in which elected officials represent a group of people Representation in contract law
Representation
simple expression in group representation theory. Let G be a group and ρ a representation of G with character χ. The representation ψk(ρ) has character χ ψ
Adams_operation
Combinatorial object in representation theory
ˈtæbloʊ/; plural: tableaux) is a combinatorial object useful in representation theory and Schubert calculus. It provides a convenient way to describe
Young_tableau
248-dimensional exceptional simple Lie group
compact Lie groups in that its non-trivial representation of smallest dimension is the adjoint representation (of dimension 248) acting on the Lie algebra
E8_(mathematics)
The history of representation theory concerns the mathematical development of the study of objects in abstract algebra, notably groups, by describing
History of representation theory
History_of_representation_theory
Mathematical formula for the number of Young tableaux
given Young diagram. It has applications in diverse areas such as representation theory, probability, and algorithm analysis; for example, the problem of
Hook_length_formula
the representation theory of semisimple Lie algebras is one of the crowning achievements of the theory of Lie groups and Lie algebras. The theory was
Representation theory of semisimple Lie algebras
Representation_theory_of_semisimple_Lie_algebras
Branch of mathematics that studies dynamical systems
applications in probability theory. Ergodic theory has fruitful connections with harmonic analysis, Lie theory (representation theory, lattices in algebraic
Ergodic_theory
Mathematical method to analyse Lie groups
In Lie theory and representation theory, the Levi decomposition, conjectured by Wilhelm Killing and Élie Cartan and proved by Eugenio Elia Levi (1905)
Levi_decomposition
Concepts from linear algebra
algebra representation – an associative algebra acting on a module. The study of such actions is the field of representation theory. The representation-theoretical
Eigenvalues_and_eigenvectors
Sentence that resists simple formalization
formalisms including systems of dynamic semantics such as Discourse representation theory. Their name comes from the example sentence "Every farmer who owns
Donkey_sentence
Branch of applied probability theory
Decision theory or the theory of rational choice is a branch of probability, economics, and analytic philosophy that uses expected utility and probability
Decision_theory
American Jewish mathematician
– February 2, 2017) was an American mathematician who worked in representation theory, differential geometry, and mathematical physics. Kostant grew up
Bertram_Kostant
Representation theory
In mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms
Weyl_character_formula
Chinese mathematician (born 1982)
work deals primarily with geometric representation theory and in particular the Langlands program, tying number theory to algebraic geometry and quantum
Xinwen_Zhu
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