Group representation

Instudymathematical groups,group representation is"description" ofgroup asconcrete grouptransformations (or automorphism group)some mathematical object. More formally, "description" means that there ishomomorphism fromgroupsome automorphism group. A faithful representationonewhich this homomorphisminjective.

Some people use realizationthis notionreserveterm representationwhat belowcalled linear representations.

Representation theory divides into subtheories depending onkindgroup being represented. The various theoriesquite differentdetail, though some basic definitionsconceptssimilar. The most important divisions are:

Finite groups: group representations arevery important tool instudyfinite groups. They also arise inapplicationsfinite group theorycrystallography andgeometry. The special case whererepresentationoverfieldcharacteristic pp dividesorder ofgroup, called modular representation theory, has very different properties (see below).

Compact or locally compact topological groups: many ofresultsfinite group representation theoryproved by averaging overgroup. These proofs can carry overinfinite groups ifaveragereplaced by an integral, which only works if an acceptable notionintegral can be defined. This can be donelocally compact groups, using Haar measure. The resulting theory iscentral partharmonic analysis. The Pontryagin duality describestheorycommutative groups, asgeneralised Fourier transform.

Lie groups: Many important Lie groupscompact, soresultscompact representation theory applythem. Other techniques specificLie groupsused as well. Most ofgroups importantphysicschemistryLie groups, andrepresentation theorycrucial toapplicationgroup theorythose fields. See RepresentationsLie groupsalgebras.

Non-compact topological groups: The classnon-compact groupstoo broadconstruct any general representation theory, but specific special cases have been studied, sometimes using ad hoc techniques. The semisimple Lie groups havedeep theory, building oncompact case. The complementary solvable Lie groups cannot insame way be classified. The general theoryLie groups dealssemidirect products oftwo types, by meansgeneral results called Mackey theory, which isgeneralizationWigner's classification methods.

Withingiven kindrepresentation theory, results differ depending onkindautomorphism group thattargeted. One target ispermutation groups. Butmost important targetsgroupsmatrices over some field, or, more generally, groupsinvertible linear transformations ofvector space.

The most important case isfieldcomplex numbers (that is,representationshomomorphisms togroupcomplex matrices or invertible linear transformations ofcomplex vector space). Ifvector spacefinite dimensional, thenrepresentationssaidbe finite dimensional as well. (Infinite dimensional representationsquite possible;vector space could be an infinite dimensional Hilbert space,example.)

The other important cases arefieldreal numbers, finite fields,fieldsp-adic numbers. Representations infinite field casecalled modular. Herecharacteristic offieldquite significant; many theorems depend onorder ofgroup not dividingcharacteristic offield.

Tablecontents

1 Set theoretic representation 2 Linear representation 2.1 Example 2.2 Equivalencerepresentations 2.3 Group actions 2.4 Reducibility 2.5 Character theory

Set theoretic representation

A set Ssaidbeset-theoretic representation ofgroup G if there isfunction, ρ from GS^S,setfunctions from SS such that

.

Equivalently,representation isgroup homomorphism from G topermutation groupS.

See Group action.

A linear representation isspecial case ofset representationadditional structure.

Linear representation

In abstract algebra,representation offinite group G isgroup homomorphism from G togeneral linear group GL(n,C)invertible complex n-by-n matrices. The studysuch representationscalled representation theory.

Representation theoryimportant becauseenablesreductionsome group theory problemslinear algebra, which hasvery well-understood theory.

Therean analoguethis theorymany important kindsinfinite groups; see RepresentationsLie groupsalgebrasPeter-Weyl theoremcompact topological groups.

A representation by projective transformations (see projective representation) can be described aslinear representation upscalar matrices. These representations occur naturally, also.

We could also have affine representations. For example,Euclidean group acts affinely upon Euclidean space.

Example

Considercomplex number u = exp(2πi/3) which hasproperty u³ = 1. The cyclic group C₃ = {1, u, u²} hasrepresentation ρ given by:

(the three matricesρ(1), ρ(u)ρ(u²) respectively).

This representationsaidbe faithful, because ρ isone-to-one map.

Equivalencerepresentations

Two representations ρ₁ρ₂saidbe equivalent if the matrices only differ bychangebasis, i.e. if there exists AGL(n,C) such thatall xG: ρ₁(x) = Aρ₂(x)A^-1. For example,representationC₃ given bymatrices:

is an equivalent representation toone shown above.

Group actions

Every square n-by-n matrix describeslinear map from an n-dimensional vector space Vitself (oncebasisV has been chosen). Therefore, every representation ρ: G -> GL_n definesgroup action on V given by g.v = (ρ(g))(v) (for gG, vV). One mayfact definerepresentation ofgroup as an actionthat group on some vector space, thereby avoidingneedchoosebasis andrestrictionfinite-dimensional vector spaces.

Reducibility

If V hasnon-trivial proper subspace W such that WcontainedV, then the representationsaidbe reducible. A reducible representation can be expressed as a direct sumsubrepresentations (Maschke's theorem) (onlyfinite groupsreducible representations necessarily decomposable!).

If V has no such subspaces, itsaidbe an irreducible representation.

Inexample above,representation givenreducible into two 1-dimensional subrepresentations (given by span{(1,0)}span{(0,1)}).

Character theory

The character ofrepresentation ρ : G -> GL_n isfunction χ : G -> C which sends gG totrace (the sum ofdiagonal elements)the matrix ρ(g). For example,character ofrepresentation given abovegiven by: χ(1) = 2, χ(u) = 1 + u, χ(u²) = 1 + u².

If ghmembersG insame conjugacy class, then χ(g) = χ(h)any character;values ofcharacter therefore havebe specified only fordifferent conjugacy classesG. Moreover, equivalent representations havesame characters. Ifrepresentation isdirect sumsubrepresentations, thencorresponding character issum ofsubrepresentations' characters.

The charactersallirreducible representations offinite group formcharacter table,conjugacy classeselements ascolumns,characters asrows. Here is the character tableC₃:

     (1)  (u)  (u2)
  1   1    1    1
  χ1  1    u    u2
  χ2  1    u2   u

The character tablealways square, androwscolumnsorthogonalrespect tostandard inner product on C^m, which allows onecompute character tables more easily. The first row ofcharacter table always consists1s,corresponds totrivial representation (the 1-dimensional representation consisting1-by-1 matrices containingentry 1).

Certain properties ofgroup G can be deduced from its character table:

• The orderGgiven bysum(χ(1))² overcharacters intable.
• Gabelian ifonly if χ(1) = 1all characters intable.
• G hasnon-trivial normal subgroup (i.e. Gnotsimple group) ifonly if χ(1) = χ(g)some non-trivial character χ intablesome non-identity element gG.

The character table does notgeneral determinegroup up to isomorphism:example,quaternion group Qthe dihedral group8 elements (D₈) havesame character table.