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# Irreducible representation

In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper subrepresentation (\rho|_W,W), W \subset V closed under the action of \. Every finite-dimensional unitary representation on a Hermitian vector space V is the direct sum of irreducible representations. [1]

51 relations: Algebra over a field, Algebra representation, Block matrix, Character theory, Characteristic (algebra), Complex number, Degenerate energy levels, Diagonal matrix, Diagonalizable matrix, Direct sum, Energy level splitting, Field (mathematics), General linear group, Group (mathematics), Hamiltonian (quantum mechanics), Hermann Weyl, Homomorphism, Identity element, Identity matrix, Indecomposable module, Lie algebra representation, Linear subspace, List of things named after Charles Hermite, Mathematics, Matrix (mathematics), Matrix addition, Matrix multiplication, Matrix similarity, Modular representation theory, Quantum chemistry, Quantum mechanics, Real number, Regular representation, Relativistic wave equations, Representation theory, Representation theory of diffeomorphism groups, Representation theory of SL2(R), Representation theory of SU(2), Representation theory of the Galilean group, Representation theory of the Poincaré group, Restriction (mathematics), Richard Brauer, Root of unity, Selection rule, Sesquilinear form, Simple module, The Road to Reality, Triviality (mathematics), Unitary representation, Vector space, ... Expand index (1 more) »

## Algebra over a field

In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product.

## Algebra representation

In abstract algebra, a representation of an associative algebra is a module for that algebra.

## Block matrix

In mathematics, a block matrix or a partitioned matrix is a matrix that is interpreted as having been broken into sections called blocks or submatrices.

## 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.

## Characteristic (algebra)

In mathematics, the characteristic of a ring R, often denoted char(R), is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0) if the sum does indeed eventually attain 0.

## Complex number

A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.

## 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.

## Diagonal matrix

In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero.

## Diagonalizable matrix

In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P such that P−1AP is a diagonal matrix.

## Direct sum

The direct sum is an operation from abstract algebra, a branch of mathematics.

## Energy level splitting

In quantum physics, energy level splitting of a quantum system occurs when a degenerate energy level of two or more states is split because corresponding Hamiltonian's eigenvalues become different.

## Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.

## General linear group

In mathematics, the general linear group of degree n is the set of invertible matrices, together with the operation of ordinary matrix multiplication.

## Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.

## Hamiltonian (quantum mechanics)

In quantum mechanics, a Hamiltonian is an operator corresponding to the total energy of the system in most of the cases.

## Hermann Weyl

Hermann Klaus Hugo Weyl, (9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher.

## Homomorphism

In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces).

## Identity element

In mathematics, an identity element or neutral element is a special type of element of a set with respect to a binary operation on that set, which leaves other elements unchanged when combined with them.

## Identity matrix

In linear algebra, the identity matrix, or sometimes ambiguously called a unit matrix, of size n is the n × n square matrix with ones on the main diagonal and zeros elsewhere.

## Indecomposable module

In abstract algebra, a module is indecomposable if it is non-zero and cannot be written as a direct sum of two non-zero submodules.

## 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.

## Linear subspace

In linear algebra and related fields of mathematics, a linear subspace, also known as a vector subspace, or, in the older literature, a linear manifold, is a vector space that is a subset of some other (higher-dimension) vector space.

## List of things named after Charles Hermite

Numerous things are named after the French mathematician Charles Hermite (1822–1901).

## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

## Matrix (mathematics)

In mathematics, a matrix (plural: matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.

In mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together.

## Matrix multiplication

In mathematics, matrix multiplication or matrix product is a binary operation that produces a matrix from two matrices with entries in a field, or, more generally, in a ring or even a semiring.

## Matrix similarity

In linear algebra, two n-by-n matrices and are called similar if for some invertible n-by-n matrix.

## 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.

## Quantum chemistry

Quantum chemistry is a branch of chemistry whose primary focus is the application of quantum mechanics in physical models and experiments of chemical systems.

## Quantum mechanics

Quantum mechanics (QM; also known as quantum physics, quantum theory, the wave mechanical model, or matrix mechanics), including quantum field theory, is a fundamental theory in physics which describes nature at the smallest scales of energy levels of atoms and subatomic particles.

## Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

## 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.

## Relativistic wave equations

In physics, specifically relativistic quantum mechanics (RQM) and its applications to particle physics, relativistic wave equations predict the behavior of particles at high energies and velocities comparable to the speed of light.

## Representation theory

Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures.

## Representation theory of diffeomorphism groups

In mathematics, a source for the representation theory of the group of diffeomorphisms of a smooth manifold M is the initial observation that (for M connected) that group acts transitively on M.

## Representation theory of SL2(R)

In mathematics, the main results concerning irreducible unitary representations of the Lie group SL(2,'''R''') are due to Gelfand and Naimark (1946), V. Bargmann (1947), and Harish-Chandra (1952).

## 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.

## 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.

## Representation theory of the Poincaré group

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 nor a semisimple group.

## Restriction (mathematics)

In mathematics, the restriction of a function f is a new function f\vert_A obtained by choosing a smaller domain A for the original function f. The notation f is also used.

## Richard Brauer

Richard Dagobert Brauer (February 10, 1901 &ndash; April 17, 1977) was a leading German and American mathematician.

## Root of unity

In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that gives 1 when raised to some positive integer power.

## Selection rule

In physics and chemistry, a selection rule, or transition rule, formally constrains the possible transitions of a system from one quantum state to another.

## Sesquilinear form

In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space.

## 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.

The Road to Reality: A Complete Guide to the Laws of the Universe is a book on modern physics by the British mathematical physicist Roger Penrose, published in 2004.

## Triviality (mathematics)

In mathematics, the adjective trivial is frequently used for objects (for example, groups or topological spaces) that have a very simple structure.

## 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.

## Vector space

A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.

## Zero object (algebra)

In algebra, the zero object of a given algebraic structure is, in the sense explained below, the simplest object of such structure.

## References

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