xl yl m n m n r m r m r r! The inner sum in the last term simplifies because it is a binomial expansion of ( x + y) r : e +.

Transcription

1 Ler Trsfortos d Group Represettos Hoework #3 (06-07, Aswers Q-Q re further exerses oer dots, self-dot trsfortos, d utry trsfortos Q3-6 volve roup represettos Of these, Q3 d Q4 should e quk Q5 s espelly useful for the upo terl Q: (y e skpped, d result ssued for Q: Let L e ler trsforto, d x d y xl xl xl yl ( x y L slrsdefe e y the power seres e = å x L Show tht e e = e +! = 0 ö ö xl yl + e e = x L y L = x L y L = x y L çå 0! çå = = 0! åå åå è øè ø = 0 = 0!! = 0 = 0!! Now ollet ters wth the se power of L, ely, r= + For y r, there s otruto wth ll the re 0,, r, tk = r- So, r ö r xl yl r r r r! ö + - r- e e = x y L = L x y = L x y 0 0!! ç = = r= 0 è= 0!( r-! ø r= 0 r! çè= 0!( r-! ø The er su the lst ter splfes euse t s ol expso of ( x + y r : r xl yl r r! ö r- r r r r e e = å L x y = L ( x+ y = ( x+ y L r 0 r! ç å = = 0!( r! å å è - ø r= 0 r! r= 0 r! ( The fl expresso s the defto of x y L e + åå å å å å * QA For eerl ler trsforto A wth dot A d (possly oplex slr z, show tht the dot of za s za * * zavw, = z Avw, = z v, Aw = v, zaw, where the steps follow fro ( lerty of the er produt the frst ruet, ( defto of the dot of A, ( oute lerty of the dot the seod ruet B Show tht f A s self-dot d x s rel (e, x = x, the xa defed s Q Ht: do ths y oput the dot of e xa e s utry, where xa e s xa e v, w = å ( x A v, w = å ( x A v, w = å ( x v, A w, = 0! = 0! = 0! xa where the equltes follow fro ( the defto of e s Q, ( lerty of the er produt the frst ruet, d ( the ssupto tht A s self-dot Now reverse the steps: -xa å ( x v, A w = å v, (- x A w = v, å (- x A w = v, e w (frst d = 0! = 0! = 0! seod equlty fro oute lerty of the er produt ts seod ruet, thrd fro the

2 xa defto of e - xa xa So the dot of e s e - Flly, Q shows tht these re verses, se xa -xa ( x-x A 0A xa e e = e = e = I So the dot of e s ts verse, d t s therefore utry C A terest spel se Rell tht prevous hoework, we showed tht the ler df opertor L, defed y Lf ( x = s self-dot the Hlert spe of dfferetle futos o the le for whh f ( x ò s fte, d the er produt f, ( ( - - = ò f x x Now we further ssue tht ll dervtves of f exst Apply the results of prt B to - sl (for s rel slr d develop ore flr expresso for the result opertor sl Aord to prt B, e - s utry, where - df df slf ( x = ( - s s = Apply the defto of expoetl of opertor Q: d s -sl d e f( x = e f( x =å s f( x = 0! Ths s the flr Tylor expso for f roud x So f f s lyt (e, equl to ts Tylor expso, s typl futos re, the -sl s d e f( x = å f( x = f( x+ s! Tht s, e -sl = 0 d s = e s shft y out s e utry represetto of roup G Show tht det( Q3: Let represetto, vetor spe of deso det s lso utry Frst, we eed to show tht the opertor (, e, ultplto y the slr det(, s utry Ths follows euse ( s utry so ts eevlues ll hve tude ; det( the produt of the eevlues of Seod, we eed to show tht the pp det( det( h = det( det( h Ths follows euse, so t s ultplto y slr of tude preserves struture, e, tht s preserves struture, d the propertes of the detert Se h = h, det( h = det( h = det( det( h Q4: Let e utry represetto of roup G Is ( - utry represetto? No, euse eerl, t fls to preserve the roup struture: the verse of produt s the =, ut for ( - to e produt of the verses, ut reverse order ( h h

3 represetto, we d eed ( - - hve worked - = If G s outtve, the ths ostruto would h h Q5 Chrter tles Cosder the roup of rottos d rror-flps of equlterl trle Speflly, deste the three vertes s,, d ( lokwse order, wth t the top, d the roup opertos s I for the detty, R d L for rotto rht d left y /3 of yle, d,, d for rror flps lo the les throuh eh of the vertes Copute the hrters t eh of these eleets for the represettos desred the tle elow Wth rerd to S, rell (fro erler weeks tht perutto s odd f t e eerted y odd uer of pr-swps, d eve f t requres eve uer of pr swps Group eleet: I R L Represetto: E : the trvl represetto (ll roup eleets p to P : Represetto s perutto tres o the letters {,,} S : Represetto tht ps eve peruttos o {,,} to +, odd peruttos to - C : Represetto s he-of-oordte tres the ple The opleted tle follows ths lyss: E : I the trvl represetto ll roup eleets p to, so the hrter (the tre s P : Eh roup eleet s pped to 3 3 perutto trx Its tre s the uer of 's o the dol, whh s the uer of letters {,, } tht re preserved All three re preserved for the detty For rottos, oe re preserved, so the tre s 0 For rror-flps, the vertex tht s o the rror le s preserved; the others re swpped, so the tre s S : A rotto s yl perutto (,, (,,, d e ult y o two prswps, e, (, (, d the (, (,, so t s eve A rror flp preserves oe vertex d swps the other two, so t s odd

4 C : The detty ps to the detty trx, whose tre s A rotto y le q ps osq s qö to the trx ç ç- è s q osq ø, whose tre s osq For p q =, ths s - The rror-flp 3 s flp lo the vertl, so ts trx s 0 ö çè0 - ø, whh hs tre of 0 d dffer fro y he of oordtes, so they ust hve the se tre Surz: Group eleet: I R L Represetto: E (trvl P (the letters {,, } S (eve d odd pers C ( tres Note tht P = E + C, suest (ut ot prov tht the represetto P e redued to dret su of E d C Q6 Chrter of syetr d tsyetr prts of tesor produt of roup represettos We strt wth the stdrd setup ( the lss otes for tesor produt of roup represettos: represetto of G V, represetto of G V, led to roup represetto Ä V V, Ä, ( v Ä v =, ( v Ä, ( v Here we dd to ths the further supposto tht = =, d V= V = V der these rustes, rell (see otes oer the dervto of the detert tht V Ä V e deoposed to two prts: syetr prt sy( V Ä whh hs ss osst of eleets v v v Ä v + v Ä v (for <, d v ss of V, d tsyetr prt, Ä whose to s defed y ( Ä d ( t( V Ä, whh hs ss osst of eleets ( v Äv -v Ä v (for < A Show tht Ä ps Ä = Ä e redued to two opoets, use oordte-free pproh, where Pv ( Ä v = ( vä v + v Ä v, d sy( V Ä to tself d lso ps t( V Ä sy( V Ä sy( Ä to tself So d t( Ä Here t s helpful to s the re of the proeto P defed y t( V Ä s the re of the opleetry proeto I - P

5 We eed to show tht Ä ppled to eleet the re of P res the re of P Ths wll follow f we show tht ( Ä P= P( Ä To show the ltter: ö ( Ä P( vä v = ( Ä ç ( vä v + v Ä v = (( Ä ( vä v + ( Ä ( v Äv çè ø = ( ( v Ä ( v + ( v Ä ( v = P( ( v Ä ( v = P( Ä ( väv Ä P= P Ä Se ths holds for ll v d v, the ( ( A slr ruet e used to show tht Ä d I - P oute But t s eser ot to rry out other lulto,ut to oserve tht f y trsfortos Y d Z oute, the so do Y d I - Z, se I outes wth everyth: Y( I- Z = YI- YZ = IY- ZY = ( I- Z Y B Detere the hrters of these two opoet represettos ( sy( Ä d t( Ä, ters of the hrter of Here, to lulte ( ( ( Ä Ä = tr sy sy( t s helpful use the ses ( v Ä v + v Ä v (for for sy( V Ä, d to lulte ( ( ( Ä Ä = tr t t(, where the v k re the eevetors of (for < for t( V Ä For sy Ä ( Slrly, t s helpful to use the ss ( v Äv -v Ä v : The tre s the su of the eevlues, whh we opute oe eevetor t v Ä v + v Ä v, wth ( v = l, te For the eevetor ( k k k Ä ö Ä Ä sy( ç ( v Ä v + v Ä v = ( sy( ( v Ä v + sy( ( v Äv çè ø é = Ä Ä + Ä Ä + ëê Ä Ä + Ä Ä = (( Ä ( v Ä v + ( Ä ( v Äv = Ä + Ä = Ä (( ( v v ( ( v v (( ( v v ( ( v v ú (( ( v ( v ( ( v ( v (( l l + ( l Äl (( ( = ll Ä + Ä ù úû

6 Suessve equltes use: ( lerty of sy( Ä, ( defto of sy( Ä, (, o lke ters, (v defto of tesor produt of ler trsfortos, (v the ft tht the v k re eevetors of, d (v lerty of the tesor produt sy = Ä P (where P Ä We ould lso hve doe ths ore qukly y oserv tht ( ( s defed ove, d tht ( v v v v s the re of the proeto So ( v v v v Ä + Ä s eevetor of P wth eevlue, se t Ä + Ä hs eevlue ll We therefore hve to su up these produts for ll Note tht the se = tkes re of the eevetors v Ä v Ä sy( - Ä = = = å åå ( = tr( sy( = l + ll ö - Ths e splfed euse of the detty l = l + ll ç è ø ö ö Ä ( = l sy( + l ç å ç å çè = è = ø ø Flly, ote tht f l s eevlue of, the ( å å åå Therefore, = = = = l s eevlue of ( = se s represetto So l = tr( = (, so å = ö l l çè ø ö Ä ( = ( ( sy( + = + ç ç = è = ø ( ( å å, d tht The tsyetr prt follows loously, reoz tht the se = s exluded Frst, the eevlues: ç ( Ä - Ä = (( Ä -( Ä Ä t( ç v v v v ö ll çè ø The, su these (for strtly less th yelds the hrter: - Ä Ä ( tr( t( t( ll = = = =åå, whh slrly splfes: ö ö Ä ( = l ( ( t( l - + = - + ç çèç = è = ø ø ( ( å å