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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:
V
½
(
g
)
/
V
Á
Á
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.
1
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).
2
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
3
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
=
X
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,
X
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
C[
G
]=(dim
½
1
)
½
1
©¢¢¢©
(dim
½
k
)
½
k
;
andinparticular
X
jGj
=
(dim
½
i
)
2
:
Columnorthogonality
Fix
g
and
h2G
.Then
X
(
jC
G
(
h
)
j
if
g
isconjugateto
h
0 if
g
isnotconjugateto
h
.
Â
(
g
)
Â
(
h
)=
Â
irreducible
Thisisaformalconsequenceoftheorthogonalityofcharacters.
4
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:
5
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