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170 relations: Admissible representation, Amenable group, Antiparticle, Artin L-function, Bell's theorem, Bilbao Crystallographic Server, Borel–de Siebenthal theory, Borel–Weil–Bott theorem, Burnside theorem, Casimir element, Change of rings, Character theory, Chemical bonding of H2O, Chirality (physics), Circle group, Clebsch–Gordan coefficients, Clebsch–Gordan coefficients for SU(3), Complexification (Lie group), Conformal family, Correlation function, Cuspidal representation, Decoherence-free subspaces, Dedekind zeta function, Degenerate energy levels, Density matrix, Depolarization ratio, E7½, E8 (mathematics), Electron configuration, Equivariant map, Ergodicity, Eutactic star, Exterior algebra, Fourier transform on finite groups, Frobenius group, Frobenius–Schur indicator, Fundamental representation, Gelfand pair, Gelfand–Naimark theorem, Gelfand–Naimark–Segal construction, Georgi–Glashow model, Glossary of representation theory, Grand Unified Theory, Group theory, Hereditary C*-subalgebra, Holstein–Primakoff transformation, Humanistic sociology, Hurwitz's theorem (composition algebras), Hyperfine structure, Infinitesimal character, ..., Invariant subspace, Irreducibility (mathematics), Isotypic component, Isotypical representation, Jahn–Teller effect, Jones polynomial, Kadison transitivity theorem, Kazhdan's property (T), Kirillov character formula, Kirillov model, Kostant partition function, Kostka number, Kronecker coefficient, L-packet, Langlands classification, Lattice gauge theory, Lie algebra representation, Lie–Kolchin theorem, Ligand field theory, Linear algebraic group, Linear combination of atomic orbitals, List of harmonic analysis topics, List of representation theory topics, Littlewood–Richardson rule, Local quantum field theory, Loop quantum gravity, Loop representation in gauge theories and quantum gravity, Magnetic monopole, Maschke's theorem, Matrix coefficient, Minuscule representation, Modular representation theory, Molecular symmetry, Molecular vibration, Monstrous moonshine, Multiplicity-one theorem, Naimark's problem, Octahedral molecular geometry, Orbit method, Parity (physics), Particle physics and representation theory, Partition (number theory), Pati–Salam model, Peter–Weyl theorem, Plancherel measure, Plancherel theorem for spherical functions, Plethysm, Projective representation, PSL(2,7), Quaternionic representation, Ramanujan–Sato series, Real representation, Reductive group, Regular representation, Representation of a Lie group, Representation rigid group, Representation theory of SU(2), Representation theory of the Galilean group, Representation theory of the Lorentz group, Representation theory of the symmetric group, Restricted representation, Ricci decomposition, Rotation matrix, Rule of mutual exclusion, Schur functor, Schur orthogonality relations, Schur polynomial, Schur's lemma, Selberg class, Semi-simplicity, Sheaf (mathematics), Simple Lie group, Simple module, Specht module, Spectrum of a C*-algebra, Spherical harmonics, Spin network, Spin representation, Spinor, Springer correspondence, Square planar molecular geometry, Stable vector bundle, Steinberg formula, Stone–von Neumann theorem, String theory, Subrepresentation, Superselection, Symmetric group, Symmetric space, Symmetry in quantum mechanics, Symmetry of diatomic molecules, T-symmetry, Tannaka–Krein duality, Tate conjecture, Td Molecular Orbitals, Telephone number (mathematics), Tempered representation, Tensor product of representations, Theta representation, Timeline of quantum mechanics, Topological group, Torsion tensor, Triality, Trivial representation, Unitary representation, Verma module, Vibrational spectroscopy of linear molecules, Waldspurger formula, Weil–Brezin Map, Weyl module, Weyl tensor, Whittaker model, Wigner D-matrix, Wigner's classification, Wigner–Eckart theorem, Wilson loop, Witten zeta function, Young symmetrizer, Young tableau, Zonal spherical function. Expand index (120 more) »
Admissible representation
In mathematics, admissible representations are a well-behaved class of representations used in the representation theory of reductive Lie groups and locally compact totally disconnected groups.
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Amenable group
In mathematics, an amenable group is a locally compact topological group G carrying a kind of averaging operation on bounded functions that is invariant under translation by group elements.
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Antiparticle
In particle physics, every type of particle has an associated antiparticle with the same mass but with opposite physical charges (such as electric charge).
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Artin L-function
In mathematics, an Artin L-function is a type of Dirichlet series associated to a linear representation ρ of a Galois group G. These functions were introduced in the 1923 by Emil Artin, in connection with his research into class field theory.
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Bell's theorem
Bell's theorem is a "no-go theorem" that draws an important distinction between quantum mechanics and the world as described by classical mechanics.
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Bilbao Crystallographic Server
Bilbao Crystallographic Server is an open access website offering online crystallographic database and programs aimed at analyzing, calculating and visualizing problems of structural and mathematical crystallography, solid state physics and structural chemistry.
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Borel–de Siebenthal theory
In mathematics, Borel–de Siebenthal theory describes the closed connected subgroups of a compact Lie group that have maximal rank, i.e. contain a maximal torus.
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Borel–Weil–Bott theorem
In mathematics, the Borel–Weil–Bott theorem is a basic result in the representation theory of Lie groups, showing how a family of representations can be obtained from holomorphic sections of certain complex vector bundles, and, more generally, from higher sheaf cohomology groups associated to such bundles.
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Burnside theorem
In mathematics, Burnside theorem in group theory states that if G is a finite group of order where p and q are prime numbers, and a and b are non-negative integers, then G is solvable.
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Casimir element
In mathematics, a Casimir element (also known as a Casimir invariant or Casimir operator) is a distinguished element of the center of the universal enveloping algebra of a Lie algebra.
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Change of rings
In algebra, given a ring homomorphism f: R \to S, there are three ways to change the coefficient ring of a module; namely, for a right R-module M and a right S-module N,.
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Character 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 the trace of the corresponding matrix.
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Chemical bonding of H2O
Water is a simple triatomic bent molecule with C2v molecular symmetry and bond angle of 104.5° between the central oxygen atom and the hydrogen atoms.
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Chirality (physics)
A chiral phenomenon is one that is not identical to its mirror image (see the article on mathematical chirality).
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Circle group
In mathematics, the circle group, denoted by T, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers The circle group forms a subgroup of C×, the multiplicative group of all nonzero complex numbers.
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Clebsch–Gordan coefficients
In physics, the Clebsch–Gordan (CG) coefficients are numbers that arise in angular momentum coupling in quantum mechanics.
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Clebsch–Gordan coefficients for SU(3)
In mathematical physics, Clebsch–Gordan coefficients are the expansion coefficients of total angular momentum eigenstates in an uncoupled tensor product basis.
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Complexification (Lie group)
In mathematics, the complexification or universal complexification of a real Lie group is given by a continuous homomorphism of the group into a complex Lie group with the universal property that every continuous homomorphism of the original group into another complex Lie group extends compatibly to a complex analytic homomorphism between the complex Lie groups.
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Conformal family
In theoretical physics, a conformal family is an irreducible representation of the Virasoro algebra.
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Correlation function
A correlation function is a function that gives the statistical correlation between random variables, contingent on the spatial or temporal distance between those variables.
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Cuspidal representation
In number theory, cuspidal representations are certain representations of algebraic groups that occur discretely in L^2 spaces.
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Decoherence-free subspaces
A decoherence-free subspace (DFS) is a subspace of a system's Hilbert space that is invariant to non-unitary dynamics.
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Dedekind zeta function
In mathematics, the Dedekind zeta function of an algebraic number field K, generally denoted ζK(s), is a generalization of the Riemann zeta function (which is obtained in the case where K is the rational numbers Q).
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Degenerate energy levels
In quantum mechanics, an energy level is degenerate if it corresponds to two or more different measurable states of a quantum system.
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Density matrix
A density matrix is a matrix that describes a quantum system in a mixed state, a statistical ensemble of several quantum states.
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Depolarization ratio
In Raman spectroscopy, the depolarization ratio is the intensity ratio between the perpendicular component and the parallel component of Raman scattered light.
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E7½
In mathematics, the Lie algebra E7½ is a subalgebra of E8 containing E7 defined by Landsberg and Manivel in order to fill the "hole" in a dimension formula for the exceptional series E''n'' of simple Lie algebras.
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E8 (mathematics)
In mathematics, E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding root lattice, which has rank 8.
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Electron configuration
In atomic physics and quantum chemistry, the electron configuration is the distribution of electrons of an atom or molecule (or other physical structure) in atomic or molecular orbitals.
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Equivariant map
In mathematics, equivariance is a form of symmetry for functions from one symmetric space to another.
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Ergodicity
In probability theory, an ergodic dynamical system is one that, broadly speaking, has the same behavior averaged over time as averaged over the space of all the system's states in its phase space.
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Eutactic star
In Euclidean geometry, a eutactic star is a geometrical figure in a Euclidean space.
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Exterior algebra
In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogs.
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Fourier transform on finite groups
In mathematics, the Fourier transform on finite groups is a generalization of the discrete Fourier transform from cyclic to arbitrary finite groups.
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Frobenius group
In mathematics, a Frobenius group is a transitive permutation group on a finite set, such that no non-trivial element fixes more than one point and some non-trivial element fixes a point.
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Frobenius–Schur indicator
In mathematics, and especially the discipline of representation theory, the Schur indicator, named after Issai Schur, or Frobenius–Schur indicator describes what invariant bilinear forms a given irreducible representation of a compact group on a complex vector space has.
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Fundamental representation
In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible finite-dimensional representation of a semisimple Lie group or Lie algebra whose highest weight is a fundamental weight.
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Gelfand pair
In mathematics, the expression Gelfand pair is a pair (G,K) consisting of a group G and a subgroup K that satisfies a certain property on restricted representations.
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Gelfand–Naimark theorem
In mathematics, the Gelfand–Naimark theorem states that an arbitrary C*-algebra A is isometrically *-isomorphic to a C*-algebra of bounded operators on a Hilbert space.
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Gelfand–Naimark–Segal construction
In functional analysis, a discipline within mathematics, given a C*-algebra A, the Gelfand–Naimark–Segal construction establishes a correspondence between cyclic *-representations of A and certain linear functionals on A (called states).
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Georgi–Glashow model
In particle physics, the Georgi–Glashow model is a particular grand unification theory (GUT) proposed by Howard Georgi and Sheldon Glashow in 1974.
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Glossary of representation theory
This is a glossary of representation theory in mathematics.
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Grand Unified Theory
A Grand Unified Theory (GUT) is a model in particle physics in which, at high energy, the three gauge interactions of the Standard Model which define the electromagnetic, weak, and strong interactions, or forces, are merged into one single force.
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Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.
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Hereditary C*-subalgebra
In mathematics, a hereditary C*-subalgebra of a C*-algebra is a particular type of C*-subalgebra whose structure is closely related to that of the larger C*-algebra.
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Holstein–Primakoff transformation
The Holstein-Primakoff transformation in quantum mechanics is a mapping to the spin operators from boson creation and annihilation operators, effectively truncating their infinite-dimensional Fock space to finite-dimensional subspaces.
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Humanistic sociology
Humanistic sociology is a domain of sociology which originated mainly from the work of the University of Chicago Polish philosopher-turned-sociologist, Florian Znaniecki.
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Hurwitz's theorem (composition algebras)
In mathematics, Hurwitz's theorem is a theorem of Adolf Hurwitz (1859–1919), published posthumously in 1923, solving the Hurwitz problem for finite-dimensional unital real non-associative algebras endowed with a positive-definite quadratic form.
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Hyperfine structure
In atomic physics, hyperfine structure refers to small shifts and splittings in the energy levels of atoms, molecules and ions, due to interaction between the state of the nucleus and the state of the electron clouds.
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Infinitesimal character
In mathematics, the infinitesimal character of an irreducible representation ρ of a semisimple Lie group G on a vector space V is, roughly speaking, a mapping to scalars that encodes the process of first differentiating and then diagonalizing the representation.
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Invariant subspace
In mathematics, an invariant subspace of a linear mapping T: V → V from some vector space V to itself is a subspace W of V that is preserved by T; that is, T(W) ⊆ W.
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Irreducibility (mathematics)
In mathematics, the concept of irreducibility is used in several ways.
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Isotypic component
The Isotypic component of weight \lambda of a Lie algebra module is the sum of all submodules which are isomorphic to the highest weight module with weight \lambda.
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Isotypical representation
In group theory, an isotypical, primary or factor representation of a group G is a unitary representation \pi: G \longrightarrow \mathcal(\mathcal) such that any two subrepresentations have equivalent sub-subrepresentations.
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Jahn–Teller effect
The Jahn–Teller effect (JT effect or JTE) is an important mechanism of spontaneous symmetry breaking in molecular and solid-state systems which has far-reaching consequences for different fields, and it is related to a variety of applications in spectroscopy, stereochemistry and crystal chemistry, molecular and solid-state physics, and materials science.
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Jones polynomial
In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984.
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Kadison transitivity theorem
In mathematics, Kadison transitivity theorem is a result in the theory of C*-algebras that, in effect, asserts the equivalence of the notions of topological irreducibility and algebraic irreducibility of representations of C*-algebras.
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Kazhdan's property (T)
In mathematics, a locally compact topological group G has property (T) if the trivial representation is an isolated point in its unitary dual equipped with the Fell topology.
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Kirillov character formula
In mathematics, for a Lie group G, the Kirillov orbit method gives a heuristic method in representation theory.
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Kirillov model
In mathematics, the Kirillov model, studied by, is a realization of a representation of GL2 over a local field on a space of functions on the local field.
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Kostant partition function
In representation theory, a branch of mathematics, the Kostant partition function, introduced by, of a root system \Delta is the number of ways one can represent a vector (weight) as a non-negative integer linear combination of the positive roots \Delta^+\subset\Delta.
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Kostka number
In mathematics, the Kostka number Kλμ (depending on two integer partitions λ and &mu) is a non-negative integer that is equal to the number of semistandard Young tableaux of shape λ and weight μ.
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Kronecker coefficient
In mathematics, Kronecker coefficients gλμν describe the decomposition of the tensor product (.
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L-packet
In the field of mathematics known as representation theory, an L-packet is a collection of (isomorphism classes of) irreducible representations of a reductive group over a local field, that are L-indistinguishable, meaning they have the same Langlands parameter, and so have the same L-function and ε-factors.
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Langlands classification
In mathematics, the Langlands classification is a description of the irreducible representations of a reductive Lie group G, suggested by Robert Langlands (1973).
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Lattice gauge theory
In physics, lattice gauge theory is the study of gauge theories on a spacetime that has been discretized into a lattice.
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Lie algebra representation
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 as a set of matrices (or endomorphisms of a vector space) in such a way that the Lie bracket is given by the commutator.
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Lie–Kolchin theorem
In mathematics, the Lie–Kolchin theorem is a theorem in the representation theory of linear algebraic groups; Lie's theorem is the analog for linear Lie algebras.
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Ligand field theory
Ligand field theory (LFT) describes the bonding, orbital arrangement, and other characteristics of coordination complexes.
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Linear algebraic group
In mathematics, a linear algebraic group is a subgroup of the group of invertible n×n matrices (under matrix multiplication) that is defined by polynomial equations.
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Linear combination of atomic orbitals
A linear combination of atomic orbitals or LCAO is a quantum superposition of atomic orbitals and a technique for calculating molecular orbitals in quantum chemistry.
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List of harmonic analysis topics
This is a list of harmonic analysis topics.
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List of representation theory topics
This is a list of representation theory topics, by Wikipedia page.
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Littlewood–Richardson rule
In mathematics, the Littlewood–Richardson rule is a combinatorial description of the coefficients that arise when decomposing a product of two Schur functions as a linear combination of other Schur functions.
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Local quantum field theory
The Haag–Kastler axiomatic framework for quantum field theory, introduced by, is an application to local quantum physics of C*-algebra theory.
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Loop quantum gravity
Loop quantum gravity (LQG) is a theory of quantum gravity, merging quantum mechanics and general relativity.
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Loop representation in gauge theories and quantum gravity
Attempts have been made to describe gauge theories in terms of extended objects such as Wilson loops and holonomies.
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Magnetic monopole
A magnetic monopole is a hypothetical elementary particle in particle physics that is an isolated magnet with only one magnetic pole (a north pole without a south pole or vice versa).
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Maschke's theorem
In mathematics, Maschke's theorem, named after Heinrich Maschke, is a theorem in group representation theory that concerns the decomposition of representations of a finite group into irreducible pieces.
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Matrix coefficient
In mathematics, a matrix coefficient (or matrix element) is a function on a group of a special form, which depends on a linear representation of the group and additional data.
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Minuscule representation
In mathematical representation theory, a minuscule representation of a semisimple Lie algebra or group is an irreducible representation such that the Weyl group acts transitively on the weights.
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Modular representation theory
Modular representation theory is a branch of mathematics, and that part of representation theory that studies linear representations of finite groups over a field K of positive characteristic.
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Molecular symmetry
Molecular symmetry in chemistry describes the symmetry present in molecules and the classification of molecules according to their symmetry.
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Molecular vibration
A molecular vibration occurs when atoms in a molecule are in periodic motion while the molecule as a whole has constant translational and rotational motion.
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Monstrous moonshine
In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group M and modular functions, in particular, the ''j'' function.
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Multiplicity-one theorem
In the mathematical theory of automorphic representations, a multiplicity-one theorem is a result about the representation theory of an adelic reductive algebraic group.
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Naimark's problem
Naimark's problem is a question in functional analysis asked by.
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Octahedral molecular geometry
In chemistry, octahedral molecular geometry describes the shape of compounds with six atoms or groups of atoms or ligands symmetrically arranged around a central atom, defining the vertices of an octahedron.
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Orbit method
In mathematics, the orbit method (also known as the Kirillov theory, the method of coadjoint orbits and by a few similar names) establishes a correspondence between irreducible unitary representations of a Lie group and its coadjoint orbits: orbits of the action of the group on the dual space of its Lie algebra.
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Parity (physics)
In quantum mechanics, a parity transformation (also called parity inversion) is the flip in the sign of one spatial coordinate.
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Particle physics and representation theory
There is a natural connection between particle physics and representation theory, as first noted in the 1930s by Eugene Wigner.
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Partition (number theory)
In number theory and combinatorics, a partition of a positive integer n, also called an integer partition, is a way of writing n as a sum of positive integers.
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Pati–Salam model
In physics, the Pati–Salam model is a Partial Unification Theory proposed in 1974 by nobel laureate Abdus Salam and Jogesh Pati.
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Peter–Weyl theorem
In mathematics, the Peter–Weyl theorem is a basic result in the theory of harmonic analysis, applying to topological groups that are compact, but are not necessarily abelian.
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Plancherel measure
In mathematics, Plancherel measure is a measure defined on the set of irreducible unitary representations of a locally compact group G, that describes how the regular representation breaks up into irreducible unitary representations.
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Plancherel theorem for spherical functions
In mathematics, the Plancherel theorem for spherical functions is an important result in the representation theory of semisimple Lie groups, due in its final form to Harish-Chandra.
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Plethysm
In algebra, plethysm is an operation on symmetric functions introduced by Dudley E. Littlewood, who denoted it by ⊗.
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Projective representation
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 from G to the projective linear group where GL(V) is the general linear group of invertible linear transformations of V over F, and F∗ is the normal subgroup consisting of nonzero scalar multiples of the identity; scalar transformations). In more concrete terms, a projective representation is a collection of operators \rho(g),\, g\in G, where it is understood that each \rho(g) is only defined up to multiplication by a constant. These should satisfy the homomorphism property up to a constant: for some constants c(g,h). Since each \rho(g) is only defined up to a constant anyway, it does not strictly speaking make sense to ask whether the constants c(g,h) are equal to 1. Nevertheless, one can ask whether it is possible to choose a particular representative of each family \rho(g) of operators in such a way that the \rho(g)'s satisfy the homomorphism property on the nose, not just up to a constant. If such a choice is possible, we say that \rho can be "de-projectivized," or that \rho can be "lifted to an ordinary representation." This possibility is discussed further below.
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PSL(2,7)
In mathematics, the projective special linear group PSL(2, 7) (isomorphic to GL(3, 2)) is a finite simple group that has important applications in algebra, geometry, and number theory.
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Quaternionic representation
In mathematical field of representation theory, a quaternionic representation is a representation on a complex vector space V with an invariant quaternionic structure, i.e., an antilinear equivariant map which satisfies Together with the imaginary unit i and the antilinear map k.
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Ramanujan–Sato series
In mathematics, a Ramanujan–Sato seriesHeng Huat Chan, Song Heng Chan, and Zhiguo Liu, "Domb's numbers and Ramanujan–Sato type series for 1/Pi" (2004)Gert Almkvist and Jesus Guillera, Ramanujan–Sato Like Series (2012) generalizes Ramanujan’s pi formulas such as, to the form by using other well-defined sequences of integers s(k) obeying a certain recurrence relation, sequences which may be expressed in terms of binomial coefficients \tbinom, and A,B,C employing modular forms of higher levels.
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Real representation
In the mathematical field of representation theory a real representation is usually a representation on a real vector space U, but it can also mean a representation on a complex vector space V with an invariant real structure, i.e., an antilinear equivariant map which satisfies The two viewpoints are equivalent because if U is a real vector space acted on by a group G (say), then V.
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Reductive group
In mathematics, a reductive group is a type of linear algebraic group over a field.
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Regular representation
In mathematics, and in particular the theory of group representations, the regular representation of a group G is the linear representation afforded by the group action of G on itself by translation.
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Representation of a Lie group
In mathematics and theoretical physics, the idea of a representation of a Lie group plays an important role in the study of continuous symmetry.
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Representation rigid group
In mathematics, in the representation theory of groups, a finitely generated group is said to be representation rigid if for every n, it has only finitely many isomorphism classes of irreducible representations of dimension n.
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Representation theory of SU(2)
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 Lie groups.
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Representation theory of the Galilean group
In nonrelativistic quantum mechanics, an account can be given of the existence of mass and spin (normally explained in Wigner's classification of relativistic mechanics) in terms of the representation theory of the Galilean group, which is the spacetime symmetry group of nonrelativistic quantum mechanics.
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Representation theory of the Lorentz group
The Lorentz group is a Lie group of symmetries of the spacetime of special relativity.
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Representation theory of the symmetric group
In mathematics, the representation theory of the symmetric group is a particular case of the representation theory of finite groups, for which a concrete and detailed theory can be obtained.
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Restricted representation
In mathematics, restriction is a fundamental construction in representation theory of groups.
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Ricci decomposition
In semi-Riemannian geometry, the Ricci decomposition is a way of breaking up the Riemann curvature tensor of a pseudo-Riemannian manifold into pieces with useful individual algebraic properties.
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Rotation matrix
In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space.
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Rule of mutual exclusion
In molecular spectroscopy, the rule of mutual exclusion states that no normal modes can be both Infrared and Raman active in a molecule that possesses a centre of symmetry.
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Schur functor
In mathematics, especially in the field of representation theory, Schur functors are certain functors from the category of modules over a fixed commutative ring to itself.
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Schur orthogonality relations
In mathematics, the Schur orthogonality relations, which is proven by Issai Schur through Schur's lemma, express a central fact about representations of finite groups.
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Schur polynomial
In mathematics, Schur polynomials, named after Issai Schur, are certain symmetric polynomials in n variables, indexed by partitions, that generalize the elementary symmetric polynomials and the complete homogeneous symmetric polynomials.
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Schur's lemma
In mathematics, Schur's lemma is an elementary but extremely useful statement in representation theory of groups and algebras.
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Selberg class
In mathematics, the Selberg class is an axiomatic definition of a class of ''L''-functions.
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Semi-simplicity
In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, category theory, and algebraic geometry.
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Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space.
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Simple Lie group
In group theory, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected normal subgroups.
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Simple module
In mathematics, specifically in ring theory, the simple modules over a ring R are the (left or right) modules over R that have no non-zero proper submodules.
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Specht module
In mathematics, a Specht module is one of the representations of symmetric groups studied by.
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Spectrum of a C*-algebra
In mathematics, the spectrum of a C*-algebra or dual of a C*-algebra A, denoted Â, is the set of unitary equivalence classes of irreducible *-representations of A. A *-representation π of A on a Hilbert space H is irreducible if, and only if, there is no closed subspace K different from H and which is invariant under all operators π(x) with x ∈ A. We implicitly assume that irreducible representation means non-null irreducible representation, thus excluding trivial (i.e. identically 0) representations on one-dimensional spaces.
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Spherical harmonics
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere.
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Spin network
In physics, a spin network is a type of diagram which can be used to represent states and interactions between particles and fields in quantum mechanics.
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Spin representation
In mathematics, the spin representations are particular projective representations of the orthogonal or special orthogonal groups in arbitrary dimension and signature (i.e., including indefinite orthogonal groups).
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Spinor
In geometry and physics, spinors are elements of a (complex) vector space that can be associated with Euclidean space.
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Springer correspondence
In mathematics, the Springer representations are certain representations of the Weyl group W associated to unipotent conjugacy classes of a semisimple algebraic group G. There is another parameter involved, a representation of a certain finite group A(u) canonically determined by the unipotent conjugacy class.
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Square planar molecular geometry
The square planar molecular geometry in chemistry describes the stereochemistry (spatial arrangement of atoms) that is adopted by certain chemical compounds.
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Stable vector bundle
In mathematics, a stable vector bundle is a (holomorphic or algebraic) vector bundle that is stable in the sense of geometric invariant theory.
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Steinberg formula
In mathematical representation theory, Steinberg's formula, introduced by, describes the multiplicity of an irreducible representation of a semisimple complex Lie algebra in a tensor product of two irreducible representations.
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Stone–von Neumann theorem
In mathematics and in theoretical physics, the Stone–von Neumann theorem is any one of a number of different formulations of the uniqueness of the canonical commutation relations between position and momentum operators.
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String theory
In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings.
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Subrepresentation
In representation theory in mathematics, a subrepresentation of a representation (\pi, V) of a group G is a representation (\pi|_W, W) such that W is a vector subspace of V and \pi|_W(g).
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Superselection
In quantum mechanics, superselection extends the concept of selection rules.
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Symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions.
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Symmetric space
In differential geometry, representation theory and harmonic analysis, a symmetric space is a pseudo-Riemannian manifold whose group of symmetries contains an inversion symmetry about every point.
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Symmetry in quantum mechanics
Symmetries in quantum mechanics describe features of spacetime and particles which are unchanged under some transformation, in the context of quantum mechanics, relativistic quantum mechanics and quantum field theory, and with applications in the mathematical formulation of the standard model and condensed matter physics.
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Symmetry of diatomic molecules
Molecular symmetry in physics and chemistry describes the symmetry present in molecules and the classification of molecules according to their symmetry.
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T-symmetry
T-symmetry or time reversal symmetry is the theoretical symmetry of physical laws under the transformation of time reversal: T-symmetry can be shown to be equivalent to the conservation of entropy, by Noether's Theorem.
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Tannaka–Krein duality
In mathematics, Tannaka–Krein duality theory concerns the interaction of a compact topological group and its category of linear representations.
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Tate conjecture
In number theory and algebraic geometry, the Tate conjecture is a 1963 conjecture of John Tate that would describe the algebraic cycles on a variety in terms of a more computable invariant, the Galois representation on étale cohomology.
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Td Molecular Orbitals
Tetrahedral (abbreviated Td) molecules are highly symmetric molecules that belong the Td symmetry group.
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Telephone number (mathematics)
In mathematics, the telephone numbers or the involution numbers are a sequence of integers that count the ways telephone lines can be connected to each other, where each line can be connected to at most one other line.
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Tempered representation
In mathematics, a tempered representation of a linear semisimple Lie group is a representation that has a basis whose matrix coefficients lie in the L''p'' space for any ε > 0.
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Tensor product of representations
In mathematics, the tensor product of representations is a tensor product of vector spaces underlying representations together with the factor-wise group action on the product.
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Theta representation
In mathematics, the theta representation is a particular representation of the Heisenberg group of quantum mechanics.
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Timeline of quantum mechanics
This timeline of quantum mechanics shows the key steps, precursors and contributors to the development of quantum mechanics, quantum field theories and quantum chemistry.
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Topological group
In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology.
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Torsion tensor
In differential geometry, the notion of torsion is a manner of characterizing a twist or screw of a moving frame around a curve.
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Triality
In mathematics, triality is a relationship among three vector spaces, analogous to the duality relation between dual vector spaces.
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Trivial representation
In the mathematical field of representation theory, a trivial representation is a representation of a group G on which all elements of G act as the identity mapping of V. A trivial representation of an associative or Lie algebra is a (Lie) algebra representation for which all elements of the algebra act as the zero linear map (endomorphism) which sends every element of V to the zero vector.
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Unitary representation
In mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π(g) is a unitary operator for every g ∈ G. The general theory is well-developed in case G is a locally compact (Hausdorff) topological group and the representations are strongly continuous.
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Verma module
Verma modules, named after Daya-Nand Verma, are objects in the representation theory of Lie algebras, a branch of mathematics.
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Vibrational spectroscopy of linear molecules
To determine the vibrational spectroscopy of linear molecules, the rotation and vibration of linear molecules are taken into account to predict which vibrational (normal) modes are active in the infrared spectrum and the Raman spectrum.
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Waldspurger formula
The Waldspurger formula is a formula from representation theory that relates the special values of two L-functions of two related admissible irreducible representations.
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Weil–Brezin Map
In mathematics, the Weil–Brezin map, named after André Weil and Jonathan Brezin, is a unitary transformation that maps a Schwartz function on the real line to a smooth function on the Heisenberg manifold.
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Weyl module
In algebra, a Weyl module is a representation of a reductive algebraic group, introduced by and named after Hermann Weyl.
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Weyl tensor
In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold.
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Whittaker model
In representation theory, a branch of mathematics, the Whittaker model is a realization of a representation of a reductive algebraic group such as GL2 over a finite or local or global field on a space of functions on the group.
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Wigner D-matrix
The Wigner D-matrix is a unitary matrix in an irreducible representation of the groups SU(2) and SO(3).
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Wigner's classification
In mathematics and theoretical physics, Wigner's classification is a classification of the nonnegative (E ≥ 0) energy irreducible unitary representations of the Poincaré group which have sharp mass eigenvalues.
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Wigner–Eckart theorem
The Wigner–Eckart theorem is a theorem of representation theory and quantum mechanics.
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Wilson loop
In gauge theory, a Wilson loop (named after Kenneth G. Wilson) is a gauge-invariant observable obtained from the holonomy of the gauge connection around a given loop.
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Witten zeta function
In mathematics, the Witten zeta function, is a function associated to a root system that encodes the degrees of the irreducible representations of the corresponding Lie group.
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Young symmetrizer
In mathematics, a Young symmetrizer is an element of the group algebra of the symmetric group, constructed in such a way that, for the homomorphism from the group algebra to the endomorphisms of a vector space V^ obtained from the action of S_n on V^ by permutation of indices, the image of the endomorphism determined by that element corresponds to an irreducible representation of the symmetric group over the complex numbers.
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Young tableau
In mathematics, a Young tableau (plural: tableaux) is a combinatorial object useful in representation theory and Schubert calculus.
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Zonal spherical function
In mathematics, a zonal spherical function or often just spherical function is a function on a locally compact group G with compact subgroup K (often a maximal compact subgroup) that arises as the matrix coefficient of a K-invariant vector in an irreducible representation of G. The key examples are the matrix coefficients of the spherical principal series, the irreducible representations appearing in the decomposition of the unitary representation of G on L2(G/K).
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Redirects here:
Irreducible representations, Irrep, Reducible representation.
References
[1] https://en.wikipedia.org/wiki/Irreducible_representation
