Representation.Theory.[sharethefiles.com].pdf

RepresentationTheory

Representations

Let G beagroupand V avectorspaceoveraﬁeld k .A representation of G on V isagroup

homomorphism ½ : G! Aut( V ).The degree (or dimension )of ½ isjustdim V .

Equivalentrepresentations

Let ½ : G! Aut( V )and ½ 0 : G! Aut( V 0 )betworepresentationsof G .Thena G-linearmap

from ½ to ½ 0 isalinearmap Á : V!V 0 suchthat

Á± ( ½ ( g ))=( ½ 0 ( g )) ±Á

forall g2G ,orequivalentlysuchthatthefollowingdiagramcommutes:

½ ( g ) /

² ²

V 0 ½ 0 ( g ) / / V 0

Ifadditionally Á isanisomorphismofvectorspacesthenwesaythat Á isan isomorphism from

½ to ½ 0 ,andthat ½ is isomorphic to ½ 0 .Noticethat Á : V!V 0 isanisomorphismfrom ½ to ½ 0

i® Á ¡ 1 isanisomorphismfrom ½ 0 to ½ ,soisomorphismisanequivalencerelation.

Notation

If g2G and v2V ,weoftenwrite gv insteadof ½ ( g ) v .Inthisnotation, Á : V!V 0 isan

isomorphismi®

gÁ ( v )= Á ( gv )

forall g2G andall v2V .

Subrepresentations

If G actson V ,and W isasubspaceof V suchthat g ( W ) µW forall g2G ,thenwesaythat

W isa subrepresentation of V .

Asubrepresentationof V is trivial ifitis0or V ,or non-trivial otherwise.

Irreducibleandindecomposiblerepresentations

Arepresentationiscalled irreducible ifithasnonon-trivialsubrepresentations.

V isa directsum of W and W 0 ,written V = W©W 0 ,if W and W 0 aresubrepresentationsof

V and V = W©W 0 asvectorspaces.Givenarepresentation V ,wewanttobreakitupinto

smallerpieces,thatis,writeisas

V = W 1 ©W 2 ©¢¢¢©W k

whereeach W i doesnotbreakupintosmallerpieces.

Wesaythatarepresentationis indecomposible ifitisnotadirectsumofsmallerrepresentations.

If V isindecomposiblethenitisirreducible,buttheconversedoesnotfollowingeneral.

Permutationrepresentations

Let G actonaset X .Thenthe permutationrepresentation of G withrespecttothisaction,

k [ X ],isa jXj -dimensionalvectorspaceover k withbasis fe x jx2Xg .Theactionof G onthis

vectorspaceisdeﬁnedby

ge x = e gx

forall g2G andall x2X . k [ X ]isneverirreducible,forthe1-dimensionalsubspacespanned

x2X e x isinvariantunder G .

Faithfulrepresentations

If ½ : G! Aut( V )isarepresentationthenthe kernel of ½ isker ½ = fg2Gj½ ( g )=id g .

Arepresentationof G is faithful ifker ½ = f 1 g ;inthiscasewesaythat G acts faithfully on V ,

and G isisomorphictoasubgroupofAut( V ).Notethatker ½ C G ,soif G issimplethenevery

non-trivialrepresentationisfaithful.

If G isaﬁnitegroupthenitposessesafaithfulﬁnite-dimensionalrepresentation.For G actson

itselffaithfullybyleft-multiplication;thusthepermuationrepresentation k [ G ]forthisactionis

faithful.

Completereducibility

Let G beaﬁnitegroupand V arepresentationof G overaﬁeldofcharacteristiczero.Then

1.If WµV isa G -invariantsubspacethenthereexistsa G -invariantcomplementto W .

2. V isirreducible ()V isindecomposible.

Proof

1.Let W 0 beanyvectorspacecomplementto W .Let ¼ : V!W betheprojectionof V

onto W deﬁnedby ¼ ( w + w 0 )= w forall w2W and w 0 2W 0 ,anddeﬁne

¯ ¼ ( v )= jGj ¡ 1 X

g2G

g¼ ¡ g ¡ 1 v ¢ :

Then

(a)If v2V then¯ ¼ ( v ) 2W ,andif w2W then¯ ¼ ( w )= w ,so¯ ¼ isaprojectiononto W .

(b)im¯ ¼ = W and¯ ¼j W =id,so

ker¯ ¼© im¯ ¼ = V:

(c)Forall v2V

h ¯ ¼ ( v )= jGj ¡ 1 X

g2G

hg¼ ¡ g ¡ 1 v ¢ = jGj ¡ 1 X

g2G

( hg ) ¼ ¡ ( hg ) ¡ 1 hv )=¯ ¼ ( hv ¢

so¯ ¼ is G -linear.

(d)If¯ ¼ ( v )=0then h ¯ ¼ ( v )=¯ ¼ ( hv )=0,soker¯ ¼ is G -invariant.

Henceker¯ ¼ isa G -invariantcomplementto W .

2.Thisfollowseasilyfrom(1).

by P

Characters

Forthewholeofthissections,allgroupswillbeﬁniteandallrepresentationswillbeonﬁnite-

dimensionalvectorspacesoverC.

Deﬁnition

If ½ : G! Aut( V )isarepresention,the character of ½ isthefunction

Â : G¡! C

g7¡! tr( ½ ( g )) :

Propertiesofthecharacter

1. Â doesnotdependonachoiceofbasisfor V .

2.If ½ and ½ 0 areisomorphicrepresentationsthen Â ½ ( g )= Â ½ 0 ( g )forall g2G .

3. Â (1)=dim V .

4. Â ( g )= Â ( hgh ¡ 1 )andso Â isconstantonconjugacyclassesof G .

5. Â ( g )= Â ( g ¡ 1 ).

6. Â ½©½ 0 ( g )= Â ½ ( g )+ Â ½ 0 ( g ).

Thespaceofclassfunctions

A classfunction on G isafunction f : G! Cwhichisconstantonconjugacyclassesof G .So

if V isarepresentationof G overCthen Â isaclassfunctionon G .Wewrite C G forthesetof

allclassfunctionson G .Thisisacomplexvectorspace,withabasis

± O : G¡! C

(

1 if g2O

0 if g=2O

g7¡!

where O rangesovertheconjugacyclassesof G .

WecandeﬁneaHermitianinnerproducton C G by

hf;f 0 i = jGj ¡ 1 X

g2G

f ( g ) f 0 ( g ) :

If ½ and ½ 0 areirreduciblerepresentations,then

(

1 if ½ isisomorphicto ½ 0

0 if ½ isnotisomorphicto ½ 0 .

hÂ;Â 0 i =

Thustheirreduciblecharactersformpartofanorthonormalbasisfor C G ,andsothenumberof

distinctirreduciblereresentationsisatmostthenumberofconjugacyclassesof G .Infact,the

irreduciblecharactersformanorthonormalbasisforC G ,andhencetherearepreciselyasmany

distinctirreduciblerepresentationsasthereareconjugacyclassesof G .

Consequencesoforthogonality

If ½ isanarbitraryrepresentationof G withcharacter Â ,and Â 1 ;:::;Â k arethedistinctirre-

duciblecharacters,thenbycompletereducibility

Â = n 1 Â 1 + ¢¢¢ + n k Â k :

forsome n i 2 N.Therefore hÂ;Â i i = n i byorthogonality,andso

Â =

n i Â i

where n i = hÂ;Â i i .Thus

1.Inanydecompositionof ½ intoasumofirreduciblerepresentations,eachirreduciblerep-

resentationoccursthesamenumberoftimes.

2.If ½ and ½ 0 arerepresentationsof G withthesamecharacter,then ½ » = ½ 0 .

3.With n i deﬁnedasabove,

hÂ;Âi =

n 2 i

andso ½ isirreduciblei® hÂ;Âi =1.

Theregularrepresentation

Anygroup G actsonitselfbyleft-multiplication.Thepermutationrepresentationofthisaction

iscalledthe regularrepresentation of G .If Â isthecharacteroftheregularrepresentationof G

then

(

jGj if g =1

0 otherwise.

Â ( g )=

Hence

andinparticular

jGj =

(dim ½ i ) 2 :

Columnorthogonality

Fix g and h2G .Then

(

jC G ( h ) j if g isconjugateto h

0 if g isnotconjugateto h .

Â ( g ) Â ( h )=

Â irreducible

Thisisaformalconsequenceoftheorthogonalityofcharacters.

Proofoforthogonality

Firstweneedthefollowinglemma.

Theorem(Schur’sLemma)

Supposethat( ½;V )and( ½ 0 ;V 0 )areirreduciblerepresentationsof G overC,withcharacters Â

and Â 0 respectively.Supposethat Á : V!V 0 isa G -linearmap.Then

1.Either Á isanisomorphismor Á =0.

2.If Á : V!V isanisomorphismthen Á ismultiplicationbyascalar ¸2 C,andso

(

C if ½ isisomorphicto ½ 0

0 if ½ isnotisomorphicto ½ 0 .

Hom G ( V;V 0 )=

Proof

1.Observethatker Á isasubrepresentationof V andim Á isasubrepresentationof V 0 .Since

V and V 0 arebothirreducible,ker Á =0or V andim Á =0or V 0 .Theresultfollows.

2.SinceCisalgebraicallyclosed, Á hasaneigenvalue ¸ andaneigenvector v for ¸ .Then

˜ Á = Á¡¸I isalsoa G -linearmap V!V .But ˜ Á ( v )=0andsoker Á6 =0.Butthensin ce

V isirreducible,ker Á = V andso Á = ¸I .

Nowlet( ½;V )and( ½ 0 ;V 0 )beasaboveandlet Á : V!V 0 be any linearmap.Deﬁne

Av Á = jGj ¡ 1 X

g2G

g ¡ 1 Ág:

ThenAv Á isa G -linearmap.Furthermore,tr(Av Á )=tr Á ,soinparticulariftr Á6 =0then

Av Á6 =0.

Nowontothemainpartoftheproof.Choosebasesfor V and V 0 andwrite ½ ( g )and ½ 0 ( g )as

matriceswithrespecttothesebases.Then

hÂ;Âi = jGj ¡ 1 X

g2G

Â ( g ) Â 0 ( g )

= jGj ¡ 1 X

g2G

tr( ½ ( g ))tr( ½ ( g ))

= jGj ¡ 1 X

g2G

i;j

½ ( g ) ii ½ 0 ( g ¡ 1 ) jj :

Tobecontinued...

ObserveinpassingthataconsequenceofSchur’sLemmaandtheorthogonalityofcharactersis

thatif( ½;V )and( ½ 0 ;V 0 )aretworepresentationsof G withcharacters Â and Â 0 ,then

dimHom G ( V;V 0 )= hÂ;Â 0 i:

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