Meta tags:

Headings (most frequently used words):

irreducible, representation, representations, and, of, lie, groups, group, indecomposable, contents, history, overview, examples, applications, in, theoretical, physics, chemistry, see, also, references, further, reading, external, links, notation, terminology, reducible, decomposable, connection, between, trivial, one, dimensional, complex, example, an, over, fp, lorentz, associative, algebras, books, articles,

Text of the page (most frequently used words):
the (122), displaystyle (97), group (73), #representation (67), and (44), irreducible (34), #representations (34), theory (31), edit (26), groups (20), isbn (19), pdf (18), quantum (18), from (15), 978 (15), all (14), for (14), dimensional (14), matrix (14), sum (13), mathbb (13), symmetry (12), this (11), lie (11), mechanics (11), indecomposable (11), rho (11), with (10), physics (10), lorentz (10), can (10), matrices (10), over (10), that (9), finite (9), space (9), wikipedia (8), one (8), complex (8), basis (8), direct (8), dim (8), cdots (8), toggle (7), its (7), linear (7), vector (7), then (7), block (7), form (7), reducible (7), articles (6), theoretical (6), chemistry (6), irreps (6), press (6), theorem (6), also (6), other (6), let (6), every (6), has (6), trivial (6), invariant (6), have (6), decomposable (6), pmatrix (6), vdots (6), into (6), subrepresentation (6), product (6), may (5), page (5), retrieved (5), unitary (5), poincaré (5), archived (5), original (5), see (5), some (5), are (5), main (5), only (5), there (5), subspace (5), nontrivial (5), elements (5), field (5), not (5), which (5), subsection (5), contents (4), search (4), under (4), 2013 (4), dimension (4), algebra (4), links (4), further (4), vol (4), university (4), algebras (4), simple (4), where (4), article (4), label (4), size (4), free (4), identity (4), same (4), upper (4), triangular (4), general (4), history (4), algebraic (4), special (4), hide (4), move (4), sidebar (4), view (3), text (3), terms (3), use (3), was (3), statements (3), english (3), scientific (3), any (3), spacetime (3), michael (3), 2007 (3), 2008 (3), wigner (3), doi (3), bibcode (3), relativistic (3), molecular (3), introduction (3), books (3), weinberg (3), cambridge (3), 521 (3), fields (3), dover (3), applications (3), each (3), set (3), called (3), type (3), references (3), associative (3), module (3), will (3), orbit (3), element (3), since (3), example (3), action (3), orthogonal (3), standard (3), proper (3), transformation (3), examples (3), diagonal (3), blocks (3), decomposed (3), without (3), begin (3), end (3), oplus (3), said (3), say (3), invertible (3), but (3), homomorphism (3), notation (3), subset (3), overview (3), structure (3), numbers (3), abelian (3), cyclic (3), tools (3), languages (2), table (2), contact (2), about (2), privacy (2), policy (2), using (2), last (2), categories (2), date (2), pages (2), unsourced (2), written (2), american (2), short (2), description (2), wikidata (2), mcgraw (2), hill (2), 2012 (2), chapter (2), 2015 (2), 2014 (2), notes (2), young (2), 2011 (2), mathematical (2), crystallography (2), external (2), artin (2), reading (2), 1968551 (2), annals (2), mathematics (2), 211 (2), wave (2), equations (2), parts (2), 2004 (2), heine (2), courier (2), publications (2), present (2), usage (2), 084011 (2), help (2), cite (2), 216 (2), hall (2), possible (2), term (2), tung (2), 1985 (2), world (2), 997 (2), 1966 (2), 560 (2), diffeomorphism (2), generator (2), spin (2), they (2), related (2), allows (2), hamiltonian (2), states (2), system (2), being (2), orbits (2), must (2), such (2), given (2), irrep (2), known (2), std (2), left (2), right (2), triv (2), maps (2), gamma (2), particular (2), number (2), subspaces (2), mapping (2), converse (2), however (2), connection (2), between (2), even (2), choose (2), suitable (2), obtained (2), applying (2), above (2), notice (2), although (2), ddots (2), pattern (2), equiv (2), put (2), words (2), similarity (2), two (2), represented (2), context (2), terminology (2), information (2), real (2), modular (2), infinite (2), alternating (2), dihedral (2), subgroup (2), appearance (2), upload (2), file (2), changes (2), read (2), log (2), create (2), account (2), donate (2), menu (2), add, topic, mobile, cookie, statement, statistics, developers, code, conduct, legal, safety, contacts, disclaimers, available, additional, apply, site, you, agree, registered, trademark, non, profit, organization, wikimedia, foundation, inc, creative, commons, attribution, sharealike, license, rendered, parsoid, edited, april, 2026, utc, hidden, cs1, errors, deprecated, format, math, tags, july, january, 2019, different, https, org, index, php, title, irreducible_representation, oldid, 1351331568, answers, com, dictionary, technical, bekaert, xavier, boulanger, niclas, 2006, hep, 0611263, arxiv, finley, drake, kyle, feinberg, guild, david, turetsky, emma, 2009, woit, peter, mathematicians, maciejko, joseph, dermisek, radovan, 2018, hunt, labels, tableaux, useful, teleman, constantin, 2005, van, beveren, eef, 2010, commission, summer, schools, 1999, noncommutative, rings, 1937, 204, 121773411, s2cid, 1503456, jstor, 2307, 1939anmat, 149w, 149, inhomogeneous, bargmann, 1948, 16578292, pmid, 1079095, pmc, 1073, pnas, 1948pnas, 211b, proc, natl, acad, sci, discussion, atkins, 1970, oxford, 126, 855129, 125, penrose, vintage, 679, 77631, road, reality, 2000, 66000, 1996, 55002, 1995, 55001, 230, 231, martin, shaw, december, 3rd, manchester, series, john, wiley, sons, 470, 03294, particle, abers, addison, wesley, 425, 146100, 1993, 048, 6675, 855, boardman, conner, 1973, science, bunker, per, jensen, crc, 7503, 0941, fundamentals, incompatibility, book, 1950, magnetic, moments, 486, 60269, 203, weyl, jaroszewicz, kurzepa, 1992, geometry, propagation, spinning, particles, 267, 1016, 0003, 4916, 90176, 1992anphy, 226j, 226, levine, ira, 1991, 4th, prentice, 457, eigenvalues, species, 205, 12770, 1977, springer, verlag, 387, 90190, serre, jean, pierre, 2nd, pearson, 295, 0132413770, 1959, pure, applied, academic, application, atomic, spectra, highest, weight, galilean, sl2, rotations, boosts, used, build, because, them, derive, comprises, multiplet, best, studied, through, reduction, identifying, therefore, predict, how, perturbations, transition, thus, partially, completely, energy, levels, allowing, determined, selection, rules, split, operator, degenerate, eigenstates, jahn, teller, effect, stabilizer, acted, power, sizes, these, containing, itself, match, exists, forces, denoted, appear, decomposition, cdot, regular, decomposes, former, isomorphic, latter, exactly, root, unity, mapsto, characterized, results, decompose, equal, conjugacy, classes, character, when, true, char, nmid, nature, fail, conditions, maschke, dimensions, customary, superscript, brackets, authors, just, write, numerical, parentheses, independent, others, equivalent, diagonalizes, still, able, find, makes, well, ordered, sequence, minor, contains, similarly, correspond, requirement, etc, identically, definition, translated, multiplication, specific, precise, meaning, denote, signified, symbol, indicated, restriction, subrepresentations, whole, pick, thought, function, simplifies, things, greatly, think, acts, generalized, 1940s, give, operators, act, arbitrary, rather, than, analogous, resulting, citation, needed, characteristic, richard, brauer, always, cannot, hold, acting, unipotent, hilbert, specifically, nonzero, closed, elliptic, curve, variety, reductive, loop, conformal, symplectic, euclidean, circle, solenoid, topological, hyperbolic, lattice, arithmetic, psl, integers, lattices, discrete, sporadic, classification, schur, multiplier, frobenius, elementary, sylow, theorems, lagrange, cauchy, quaternion, symmetric, list, topics, glossary, solvable, nilpotent, additive, multiplicative, continuous, image, kernel, homomorphisms, wreath, semi, quotient, normal, basic, notions, encyclopedia, item, projects, printable, version, download, print, export, switch, legacy, parser, get, shortened, url, permanent, link, what, here, actions, talk, српски, srpski, русский, монгол, 한국어, 日本語, italiano, français, español, čeština, català, العربية, top, personal, recent, community, portal, learn, contribute, random, current, events, navigation, jump, content,

Text of the page (random words):
overview subsection 2 1 notation and terminology of group representations 2 2 reducible and irreducible representations 2 3 decomposable and indecomposable representations 2 4 connection between irreducible representation and indecomposable representation 3 examples of irreducible representations toggle examples of irreducible representations subsection 3 1 trivial representation 3 2 one dimensional representation 3 3 irreducible complex representations 3 4 example of an irreducible representation over f p 4 applications in theoretical physics and chemistry 5 lie groups toggle lie groups subsection 5 1 lorentz group 6 see also toggle see also subsection 6 1 associative algebras 6 2 lie groups 7 references toggle references subsection 7 1 books 7 2 articles 8 further reading 9 external links toggle the table of contents irreducible representation 12 languages العربية català čeština español français italiano 日本語 한국어 монгол русский српски srpski 中文 edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia type of group and algebra representation algebraic structure group theory group theory basic notions subgroup normal subgroup group action quotient group semi direct product direct sum free product wreath product group homomorphisms kernel image simple finite infinite continuous multiplicative additive cyclic abelian dihedral nilpotent solvable glossary of group theory list of group theory topics finite groups cyclic group z n symmetric group s n alternating group a n dihedral group d n quaternion group q cauchy s theorem lagrange s theorem sylow theorems hall s theorem p group elementary abelian group frobenius group schur multiplier classification of finite simple groups cyclic alternating lie type sporadic discrete groups lattices integers z displaystyle mathbb z free group modular groups psl 2 z displaystyle mathbb z sl 2 z displaystyle mathbb z arithmetic group lattice hyperbolic group topological and lie groups solenoid circle general linear gl n special linear sl n orthogonal o n euclidean e n special orthogonal so n unitary u n special unitary su n symplectic sp n g 2 f 4 e 6 e 7 e 8 lorentz poincaré conformal diffeomorphism loop infinite dimensional lie group o su sp algebraic groups linear algebraic group reductive group abelian variety elliptic curve v t e in mathematics specifically in the representation theory of groups and algebras an irreducible representation ρ v displaystyle rho v or irrep of an algebraic structure a displaystyle a is a nonzero representation that has no proper nontrivial subrepresentation ρ w w displaystyle rho _ w w with w v displaystyle w subset v closed under the action of ρ a a a displaystyle rho a a in a every finite dimensional unitary representation on a hilbert space v displaystyle v is the direct sum of irreducible representations irreducible representations are always indecomposable i e cannot be decomposed further into a direct sum of representations but the converse may not hold e g the two dimensional representation of the real numbers acting by upper triangular unipotent matrices is indecomposable but reducible history edit group representation theory was generalized by richard brauer from the 1940s to give modular representation theory in which the matrix operators act on a vector space over a field k displaystyle k of arbitrary characteristic rather than a vector space over the field of real numbers or over the field of complex numbers the structure analogous to an irreducible representation in the resulting theory is a simple module citation needed overview edit further information group representation let ρ displaystyle rho be a representation i e a homomorphism ρ g g l v displaystyle rho g to gl v of a group g displaystyle g where v displaystyle v is a vector space over a field f displaystyle f if we pick a basis b displaystyle b for v displaystyle v ρ displaystyle rho can be thought of as a function a homomorphism from a group into a set of invertible matrices and in this context is called a matrix representation however it simplifies things greatly if we think of the space v displaystyle v without a basis ρ displaystyle rho is d dimensional if the vector space v displaystyle v it acts over has dimension d displaystyle d a linear subspace w v displaystyle w subset v is called g displaystyle g invariant if ρ g w w displaystyle rho g w in w for all g g displaystyle g in g and all w w displaystyle w in w the co restriction of ρ displaystyle rho to the general linear group of a g displaystyle g invariant subspace w v displaystyle w subset v is known as a subrepresentation a representation ρ g g l v displaystyle rho g to gl v is said to be irreducible if it has only trivial subrepresentations all representations can form a subrepresentation with the trivial g displaystyle g invariant subspaces e g the whole vector space v displaystyle v and 0 if there is a proper nontrivial invariant subspace ρ displaystyle rho is said to be reducible notation and terminology of group representations edit group elements can be represented by matrices although the term represented has a specific and precise meaning in this context a representation of a group is a mapping from the group elements to the general linear group of matrices as notation let a b c denote elements of a group g with group product signified without any symbol so ab is the group product of a and b and is also an element of g and let representations be indicated by d the representation of a is written as d a d a 11 d a 12 d a 1 n d a 21 d a 22 d a 2 n d a n 1 d a n 2 d a n n displaystyle d a begin pmatrix d a _ 11 d a _ 12 cdots d a _ 1n d a _ 21 d a _ 22 cdots d a _ 2n vdots vdots ddots vdots d a _ n1 d a _ n2 cdots d a _ nn end pmatrix by definition of group representations the representation of a group product is translated into matrix multiplication of the representations d a b d a d b displaystyle d ab d a d b if e is the identity element of the group so that ae ea a etc then d e is an identity matrix or identically a block matrix of identity matrices since we must have d e a d a e d a d e d e d a d a displaystyle d ea d ae d a d e d e d a d a and similarly for all other group elements the last two statements correspond to the requirement that d is a group homomorphism reducible and irreducible representations edit a representation is reducible if it contains a nontrivial g invariant subspace that is to say all the matrices d a displaystyle d a can be put in upper triangular block form by the same invertible matrix p displaystyle p in other words if there is a similarity transformation d a p 1 d a p displaystyle d a equiv p 1 d a p which maps every matrix in the representation into the same pattern upper triangular blocks every ordered sequence minor block is a group subrepresentation that is to say if the representation is for example of dimension 2 then we have d a p 1 d a p d 11 a d 12 a 0 d 22 a displaystyle d a p 1 d a p begin pmatrix d 11 a d 12 a 0 d 22 a end pmatrix where d 11 a displaystyle d 11 a is a nontrivial subrepresentation if we are able to find a matrix p displaystyle p that makes d 12 a 0 displaystyle d 12 a 0 as well then d a displaystyle d a is not only reducible but also decomposable notice even if a representation is reducible its matrix representation may still not be the upper triangular block form it will only have this form if we choose a suitable basis which can be obtained by applying the matrix p 1 displaystyle p 1 above to the standard basis decomposable and indecomposable representations edit a representation is decomposable if all the matrices d a displaystyle d a can be put in block diagonal form by the same invertible matrix p displaystyle p in other words if there is a similarity transformation 1 d a p 1 d a p displaystyle d a equiv p 1 d a p which diagonalizes every matrix in the representation into the same pattern of diagonal blocks each such block is then a group subrepresentation independent from the others the representations d a and d a are said to be equivalent representations 2 the k dimensional say representation can be decomposed into a direct sum of k 1 matrices d a p 1 d a p d 1 a 0 0 0 d 2 a 0 0 0 d k a d 1 a d 2 a d k a displaystyle d a p 1 d a p begin pmatrix d 1 a 0 cdots 0 0 d 2 a cdots 0 vdots vdots ddots vdots 0 0 cdots d k a end pmatrix d 1 a oplus d 2 a oplus cdots oplus d k a so d a is decomposable and it is customary to label the decomposed matrices by a superscript in brackets as in d n a for n 1 2 k although some authors just write the numerical label without parentheses the dimension of d a is the sum of the dimensions of the blocks dim d a dim d 1 a dim d 2 a dim d k a displaystyle dim d a dim d 1 a dim d 2 a cdots dim d k a if this is not possible i e k 1 then the representation is indecomposable 1 3 notice even if a representation is decomposable its matrix representation may not be the diagonal block form it will only have this form if we choose a suitable basis which can be obtained by applying the matrix p 1 displaystyle p 1 above to the standard basis connection between irreducible representation and indecomposable representation edit an irreducible representation is by nature an indecomposable one the converse may fail however for finite groups under some conditions we do have an indecomposable representation being an irreducible representation maschke s theorem when the group g displaystyle g is finite and k displaystyle k is a field with c h a r k g displaystyle char k nmid g then an indecomposable representation of g displaystyle g over k displaystyle k is an irreducible representation in particular this is true for k c displaystyle k mathbb c 4 examples of irreducible representations edit trivial representation edit all groups g displaystyle g have a one dimensional irreducible trivial representation by mapping all group elements to the identity transformation one dimensional representation edit any one dimensional representation is irreducible since it has no proper nontrivial invariant subspaces irreducible complex representations edit the irreducible complex representations of a finite group g can be characterized using results from character theory in particular all complex representations decompose as a direct sum of irreps and the number of irreps of g displaystyle g is equal to the number of conjugacy classes of g displaystyle g 5 the irreducible complex representations of z n z displaystyle mathbb z n mathbb z are exactly given by the maps 1 γ displaystyle 1 mapsto gamma where γ displaystyle gamma is an n displaystyle n th root of unity let v displaystyle v be an n displaystyle n dimensional complex representation of s n displaystyle s_ n with basis v i i 1 n displaystyle v_ i _ i 1 n then v displaystyle v decomposes as a direct sum of the irreps v triv c i 1 n v i displaystyle v_ text triv mathbb c left sum _ i 1 n v_ i right and the orthogonal subspace given by v std i 1 n a i v i a i c i 1 n a i 0 displaystyle v_ text std left sum _ i 1 n a_ i v_ i a_ i in mathbb c sum _ i 1 n a_ i 0 right the former irrep is one dimensional and isomorphic to the trivial representation of s n displaystyle s_ n the latter is n 1 displaystyle n 1 dimensional and is known as the standard representation of s n displaystyle s_ n 5 let g displaystyle g be a group the regular representation of g displaystyle g is the free complex vector space on the basis e g g g displaystyle e_ g _ g in g with the group action g e g e g g displaystyle g cdot e_ g e_ gg denoted c g displaystyle mathbb c g all irreducible representations of g displaystyle g appear in the decomposition of c g displaystyle mathbb c g as a direct sum of irreps example of an irreducible representation over f p edit let g displaystyle g be a p displaystyle p group and v f p n displaystyle v mathbb f _ p n be a finite dimensional irreducible representation of g over f p displaystyle mathbb f _ p by orbit stabilizer theorem the orbit of every v displaystyle v element acted by the p displaystyle p group g displaystyle g has size being power of p displaystyle p since the sizes of all these orbits sum up to the size of g displaystyle g and 0 v displaystyle 0 in v is in a size 1 orbit only containing itself there must be other orbits of size 1 for the sum to match that is there exists some v v displaystyle v in v such that g v v displaystyle gv v for all g g displaystyle g in g this forces every irreducible representation of a p displaystyle p group over f p displaystyle mathbb f _ p to be one dimensional applications in theoretical physics and chemistry edit see also symmetry in quantum mechanics molecular symmetry and jahn teller effect in quantum physics and quantum chemistry each set of degenerate eigenstates of the hamiltonian operator comprises a vector space v for a representation of the symmetry group of the hamiltonian a multiplet best studied through reduction to its irreducible parts identifying the irreducible representations therefore allows one to label the states predict how they will split under perturbations or transition to other states in v thus in quantum mechanics irreducible representations of the symmetry group of the system partially or completely label the energy levels of the system allowing the selection rules to be determined 6 lie groups edit main article representation theory of lie groups lorentz group edit main article representation theory of the lorentz group the irreps of d k and d j where j is the generator of rotations and k the generator of boosts can be used to build to spin representations of the lorentz group because they are related to the spin matrices of quantum mechanics this allows them to derive relativistic wave equations 7 see also edit associative algebras edit simple module indecomposable module representation of an associative algebra lie groups edit representation theory of lie algebras representation theory of su 2 representation theory of sl2 r representation theory of the galilean group representation theory of diffeomorphism groups representation theory of the poincaré group theorem of the highest weight references edit 1 2 e p wigner 1959 group theory and its application to the quantum mechanics of atomic spectra pure and applied physics academic press p 73 w k tung 1985 group theory in physics world scientific p 32 isbn 978 997 1966 560 w k tung 1985 group theory in physics world scientific p 33 isbn 978 997 1966 560 artin michael 2011 algebra 2nd ed pearson p 295 isbn 978 0132413770 1 2 serre jean pierre 1977 linear representations of finite groups springer verlag isbn 978 0 387 90190 9 l...