i I I

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I Ii

Le G---roUf Rep,esef'\+ a L·ct1s

, 11h-e si""f \est f\/\1>1J~[Js or~ fi oil e ~ dil'f\ef\Sia,J '1

I vec\:or spal.es J wJ {~e si ""p\esl $Mooi~ 1YlbpJ

l !-are \11'1,?1,. '('(\ops. So:

r I

. 0 et tt\~{ iO(\ - \f G- Is a Ge :rovp J G/\ -riJi°" A:&><x~'I

,- ~

I

11

gf G-- Or ~ ;f

L

vec-h.r spate a11d e-,y:~ f'/\'llf

1- 1

We saj r:1... is ~] or comp \ei< depe"di°J ,-

on whk( case we'r(' COI\Sid~/\Aj . 'Ne of.f (f\ {rj

-L:, h'l>/\J le ba-l:~ ca.se.s SiMuf l Gfleo .. d:J·

D(t:";{;CY'\ - If \J i~ c' \Jee for space I

IJ

J G-LC\J J bt t~~ 3'ovp of i'l\vc,t b 1e l;f\ear

r r ---M'df s [ : V -7 \J

r r

t I

1 We h11ve (AiV) ~ G-U'I\> R) ;f V~ 1R~

i~ r1nl oJ G-L(Vj ~ C-LfY)J () If \[ ~ C\ J

.1so i11 etlk, cO'.><? G-LC'{) is 4 lie Jovj>. +

, !~ef\ce+orl~ \e{ G- be ?l lie Joup mJ V

6e ,; fo, ;f ~ -di rl\ef\S I QI\ ,J w,\ or coMp 1 « Ve J.,, $fn' < l.

.. L~rnl'V\o- R: Cr x V ~ V is a rcpre.s cJpfi()()

1f ~"d ol\l~ if p , G- ~ G-LL'v) Is o L1j/ . , , 3rovp ~""o""c::,rphisYY) 1 whw! ' I

I I

I

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I

I·

. 1~<r~ l.s 4 uJ~~o5 Ad:. ( Cr) INkre' +

. • obsects 11re i.<l,·~<, of G-: A, G-xX "7 'X +

. • a l'l'c)r~~ i SW\ f rolY\ A : G-')( )<'. ~ X .Jo +

I ..

+

. j

' +

' .

~

I

l I

f. I

r I · I

8,: G-x '(--? ~ is o SV'l\oo\:~ Mop f x~ ~ sue~ -l\J

A G-xX ~ ~

(xt t tf G-)( '( B>'(

CoMMulc:s .) j .e.

or

ex L 5 J G< J -=- A ( 3 /1-)

~(3) (~) -,:_ 8{J;j)

'I 1,

I I

·; Acl CG-J 1-% o sobcate3oj Rep (G) wkre: + I ' • objc.,t Oft' rert~e"loli'ot15 of c- :

+

+

+ I

T

+ I

I t t

t +

t t

• a N\Orp~iSIV\ .f rol'f\ R= G->< V ~ V io

S : G-)( W-? W is ~ \i"'e<V fV\op

f: v~ ~ $U<h t~~l G-xV rz~ V

\xf l l+ G-xW ~ W s

or

V'Jhtr<? ~ ruuJ pcy Cv) = R{31v)

6 l::) (w J ::. Sl31wJ

j '

i . A" gkie J of Act ( 6-) 1 s Sol'l\etl'l\e.5 cJ I eJ I

j I

·Ia G--s~ote J co,.J o rl\OrpksVYI is of{e/\ called I

» a G- -eivi'Jor ivJ """:f or et,ui.P.riani l'.Y'~.

J A rnirp~\SV'I\ ii\ ~p ( G-) is 0fitf\ ca\leJ ?J/\

!

i" +er twi I\ i "J ~er ~t,r or i" teA wi<W.

t- t

1-

. E)(-aMp\e: 1\c 5rovf' G-L~J !K) ht1s a

t- +

, i -a J o loji ttl I f!V reseiJ ~ Li" Ot'\ It" J where i'lnj

I I 1 1

3 f G-L{", W.J acts QI\ ""j )(f R" bj ""Jnx

klY\ofV'O rf ~lS ff)

A"j closed suby'Otf ot e-u.,) w...) d Isa ~OS

.I -zi ,b.Jab3ic~I r~,e setJ~t,0/\ J Q~.

;'. ; , SLC{)J~) - G-LCli2.")

i' = OCY\) ~ G-LCrK"J

i 11 : So({\) ~ G-LClV)J

I' Simi\z,r\j G-Un) <C) or d/\j close j <;;ubrovp OF

G-LCI'\, () ~~ cl bvlolo31a,I rep 01) (",

+ r t'

,[ G11r(\p\Q: G-i~e/\ f\/,,,lo rep~ of G-, S21j

p , G-- ~ G-L(\/) ,ind 6, G-~ (i--L(W) ; {~ei,

~j d~,t S\J'I{\ is " rep p~ 6: G---""=' G-L(\/ © W)

bj

+ (p©6) c3J Cv J vJ) = C p(3) v) 6{_3) w) +

H lhEue °dre Morp~\.S.MS o-f rep~

. ~ I j I 1,

I l

+

j I (\J) : ( VJ O) i2.. CwJ::: CoJv-J)

p, {vJ w J .::. V

P1. CV J w) :: 'vV

. I 6, G--:) G-LC W \ f~eir {e11Sor e._roduct i$ ~

j ~ I j ~

I j

I i

p@o, G- ~ G-L(V@\rJ) }}/e" bj

CpQ96)CsJ Cv@w) = pCfvJ ~d3J{w)

;. E)(<IM'f)\e: ~( Cl"J >"er p: G ~ G-UV)

P®P ~ ~'LP ® 1\-i..p + l

~~r (C,-) J w~w.ce SL p g Np ire repJ .

i I{\

l ~'\ \ de{;"e.

52. V = { 'Z. I/;~ v( + v( ~v; , 1/;;V(t V? £ V®V I

of SjV'MY\e.(.(i( {e.l\SorS g <l Spxe

. '. ~ 1.. \J = f l v; 19 v/ - v;' & V; , V;J v( t V? ~ V@ V I

~ I

J I

Proof- Def ii\~ a l;C\~¥" l'V\?if bj 6: V69 \J ~ V~\J

V ® \JI ~ 'J' 6," t

ti Note 6'L== 1 al\J I

... s~v [tEVr1,V 6 Ct) .,, )<] --._

• A 1.. V {"'f v~v 6 ('t<):-x] -~

+

H sutf ic~ s to s~ow i{ T: L9? L ' Y\OW I .S

a"j l;Mar M?.P w;.\:~ T'\,. I {he'l

I L "::: {x~L:Tx=xJ@ {x1'L,T)u=-xi

if 1o see -l~is J t;r.sl l\o\.e if Tx"' X .£ Tx:::-x i~efl

X -=- o s i I) c. e )( =- TX == -x. 1h e ri Y\O{ e a i i<f L i.5 , I

-a suf() o( ,Je,-lors if\ t~~ two subsp~(e..S ;

·l x+T)( x-Tx X - +

- --i_ ~

\rlh,re T ( x:Tx) :: ± ( >:~Tx). ~

Ne~l, 11

1-l:~zit I 1

+ +

if p: G-~ G-LL\JJ LS a rep , note

V 3 t G- -l:"'e IJl\e2r MOP

cf ri9f) {::) , V 6P V ~ V® V · M~s {~ subspi(e S.,_Y i,Jo ifst ~ J ofld

L

SiNA,r\j .for t,_7- V' (''l... \/ _...._ s- , I I ~ .

1 xt .:) =;, x = L._ v, @vi + vi V, l 1 ;

~ C p(ii) p) C ~) Cx) = Z:. fl5J(v; J ® f('j{v;') + pt3){vn6?i ,t3Xv;J I . I

I I

I

I~ I I I

I

j ~ Y -~ pl3) :j ~ Y V 3 f G-

p reslncls lo o rep 'jf G---:> G-L('() wi+h

p'{~) ~ p(jl'<, Y~Y V3tG-

11

. So J j~e() a rep p = G--:) G-L{V) , we 3ef t :a rep P® p : G- ~ G-L CV® V) ) 2.J v-Je ell/\

1

l 2\f\d -

I J\'L P l:) :~ (p~ p) lf-v , /\2-V ~ f\'lV •

I a11J Sl",ce V © V 6 s'J..-v $ /\'l. V we 3el

p~ p ~ S'Lp ~ l\'Lp

I 11

Cp@p)C3 ') I

,r

" ,,

j

t '

\ s1.v , j } I

I I

I \ VflJV +

\ ) • ' J

t ,. .....

..- --......J

I I

[xomp\e: G-LC2.}[ J ~s a hvbo lo_fq 1

~P 011 <C'1._ l="'ollowil\3 Re phjsicish, wn le

f -1:~t ::J11nJo-l basis Qt ((1.. as 1'; l

l

+ I

lhef\

([-i-® ('l. = < 't'wf J 1\i-1., J .J., ®1'1

,!, ®Jt > s,,_ Ci:: 1- = < f ®t J i ~ i

1 t®t+ J,@ 1' >

-/\~('2- =. <t~i-t~t> d/'\J {~e taufolo3ic:11\ rep

I ' f

1 i tse \f is {~ clir-Q,c.~

+

I

.. s~r{)e .fcy- SU 2-J () i UC2-) J;'. sv(2 )_ +.

I

t t

I

- t

+

I

( {lr.n we call Y 1111 il\vori?I\{ subspoce of )(

1,f we {~" d(.(11\e -a rep 7r G- ~ GL( Y) OS

j i" -t\e I ef'I\Mo:

'pC3) = pl3) /y : Y-"'<

p' d su~re~resc:/'ILfiQA or sub-rep of p - - - , X

r . '

:~I /

) \( /

t -- _,

1 Noh {he i/\clvsiOI\ i: Y ~ '/. Jives a l'V\0rphisYV1 I

of reps i: f ->t p :;inc e

p { ~) C U1~)') ==- i ( pc5:x~J) V3t G--> jt Y JI /I

pC~X!j) plj) ('j)

11

ExoMp\e: I • P~P l I c)S s0b,.ep.s -ard

~

~ P®P ~ I

/ WQ Cof\ 3rzn<rokz.e .fh1s {o p@f ~ p; I 4\1 I 'l p © p © p © p J e'lc. usij \ o~~ clia3"7lv'1'\S '

11

p69p ~ [I] ffi B [t Sf A2 p ~1

I

p9p:p ~ c I r 1 @ EP ~ I

etc.~ ~

1 8,ul wi bo-ll,e, ? Ir\ l,UMlvM mechllflic.S J

if p d(scribes ?i. pa,Lc \e of some kit-, J J

I p@p dc.sc-ribe.s 7._ pc\rl5,\e.s of ~~;~ kil\J;

•

1

I'. k So we need -lh\s rf\ai~ ~ p~jSics !