User:Jheald/Irreducible representation

(work in progress) In mathematics, specifically in the representation theory of groups, an irreducible representation (or irrep for short) is ...  no invariant subspaces  ... PIC.

In quantum physics and quantum chemistry, each set of degenerate eigenstates of the Hamiltonian operator makes up a representation of the symmetry group of the Hamiltonian, that barring accidental degeneracies will correspond to an irreducible representation. Identifying the irreducible representations therefore allows one to label the states, predict how they will split under perturbations; and predict non-zero transition elements.

Group representation theory was generalised by Richard Brauer from the 1940s to give modular representation theory, in which the matrix operators act over a field K of arbitrary characteristic, rather than a vector of real or complex numbers. The structure analogous to an irreducible representation in the resulting theory is a simple module.

misc. definitions

 * "(mathematics) A representation of a group as a family of linear operators of a vector space V where there is no proper closed subspace of V invariant under these operators. " (answers.com): McGraw-Hill dictionary of scientific and technical terms


 * "A representation of a symmetry operation of a group, which cannot be expressed in terms of a representation of lower dimension. When the representation of the group is in matrix form (i.e. a set of matrices that multiply in the same way as the elements of the group), the matrix representation cannot be put into block-diagonal form by constructing a linear combination of the basis functions. The importance of irreducible representations in quantum mechanics is that the energy levels of the system are labelled by the irreducible representations of the symmetry group of the system, thus enabling selection rules to be deduced. In contrast to an irreducible representation, a reducible representation can be expressed in terms of a representation of lower dimension, with a reducible matrix representation that can be put into block diagonal form by constructing a linear combination of the basis functions." (answers.com) A Dictionary of Chemistry, 6e.  OUP.


 * "a group representation (qv) that has no nontrivial invariant subspaces." Wolfram Mathworld

Resources

 * fr:Représentation irréductible


 * Character theory / fr:Caractère d'une représentation d'un groupe fini / de:Charaktertafel
 * Irreducibility (mathematics) /
 * Representation theory /
 * Group representation / fr:Représentation de groupe / de:Darstellungstheorie
 * Representation theory of finite groups / fr:Théorie des représentations d'un groupe fini


 * Module (mathematics) / fr:Module sur un anneau / de:Modul (Mathematik)
 * Modular representation theory /
 * Simple module / fr:Module simple / de:Einfacher Modul
 * Semisimple module


 * Regular representation /


 * List of representation theory topics