LECTURES ON REPRESENTATION THEORY AND INVARIANT THEORY

Transcription

1 LECTUES ON EPESENTATION THEOY AND INVAIANT THEOY These are the otes for a lecture course o the syetrc group, the geeral lear group ad varat theory. The a of the course was to cover as uch of the beautful classcal theory as te allowed, so, for exaple, I have always restrcted to workg over the coplex ubers. The result s a course whch requres o prevous kowledge beyod a satterg of rgs ad odules, character theory, ad affe varetes. These are certaly ot the frst otes o ths topc, but I hope they ay stll be useful, for [Deudoé ad Carrell] has a uber of flaws, ad [Weyl], although beautfully wrtte, requres a lot of hard work to read. The oly ew part of these otes s our treatet of orda s Theore 13: the oly referece I foud for ths was [race ad Youg] where t was proved usg the sybolc ethod. The lectures were gve at Belefeld Uversty the wter seester , ad ths s a ore or less fathful copy of the otes I prepared for that course. I have, however, reordered soe of the parts, ad rewrtte the secto o sesple algebras. The refereces I foud ost useful were: [H. Boerer] "Darstelluge vo ruppe" (1955). Eglsh traslato "epresetatos of groups" (North-Hollad, 196,1969). [J. A. Deudoé ad J. B. Carrell] "Ivarat Theory, Old ad New," Advaces Matheatcs 4 (1970) Also publshed as a book (1971). [J. H. race ad A. Youg] by Chelsea). "The algebra of varats" (1903, eprted [H. Weyl] "The classcal groups" (Prceto Uversty Press, 1946). ()

2 My thaks go to A. J. Wassera who suggested that I gve such a course ad explaed soe of the cetral deas to e, ad also to y studets, both for ther patece ad for potg out ay accuraces the orgal verso of these otes. Wlla Crawley-Boevey, Aprl 1990 Fakultät für Matheatk Matheatcal Isttute Uverstät Belefeld Oxford Uversty Postfach St. les 4800 Belefeld 1 Oxford OX1 3LB West eray Eglad ()

3 CONTENTS The Syetrc roup 1. Sesple algebras Youg syetrzers Stadard Tableaux A Character Forula The Hook Legth Forula The eeral Lear roup 6. Multlear ad Polyoal Algebra Schur-Weyl Dualty Decoposto of Tesors atoal epresetatos of L(V) Weyl s Character Forula Ivarat Theory 11. Soe Exaples of Ivarats The Frst Fudaetal Theore of Ivarat Theory Covarats of Blear Fors ()

4 1. SEMISIMPLE ALEBAS The facts about sesple algebras whch we shall eed for the syetrc group should be well-kow, ad eed ot be repeated here. For the geeral lear group, however, we shall eed soe ore delcate results, so soe presetato s ecessary. Not kowg what to clude ad what to exclude, we gve a very quck developet of the whole theory here. All rgs are assocatve ad have a detty whch s deoted by 1 or 1. By "odule" we always ea left odule. Defto. A -algebra s rg whch s also a -vector space wth the sae addto, satsfyg (rr ) = ( r)r = r( r ) r,r ad. Oe has the obvous otos of subalgebras ad algebra hooorphss. We shall be partcularly terested the case whe s fte desoal. earks ad Exaples. (1) s a -algebra. M ( ) s a -algebra. If s a group, the the group algebra s the -algebra wth bass the eleets of ad ultplcato lfted fro. () If ad S are -algebras, the so s S. Here the vector space structure coes fro the detfcato of S wth S. (3) If 1 = 0 the s the zero rg. Otherwse, for, we have 1 = 1 = so we ca detfy wth 1. Ths akes a subalgebra of. (4) If M s a -odule, the t becoes a -vector space va = ( 1 ), M. If N s aother -odule, the Ho (M,N) s a -subspace of Ho (M,N). I partcular t s a -vector space. The structure s gve by ( f)() = f() = f( ) for M, ad f Ho (M,N). (5) If M s a -odule, the Ed (M) s a -algebra, wth ultplcato gve by coposto 1

5 (fg)() = f(g()) for M, f,g Ed (M) I partcular, f V s a -vector space, the Ed (V) s a -algebra. Of course Ed (V) M ( ). d V (6) If s a -algebra ad X s a subset, the the cetralzer of X s a c () of. c (X) = { r rx=xr x X } -subalgebra of. I partcular ths holds for the cetre s a (7) If s a -algebra ad M s a -odule, the the ap so that : Ed (M), -algebra ap. By defto r ( r) Ed (M) = c ( ()), Ed (M) () c (Ed (M)). Ed (M) (8) If M s a -odule, the t s aturally a Ed (M)-odule wth the acto gve by evaluato, ad ths acto coutes wth that of. The cluso above says that the eleets of act as Ed (M)-odule edoorphss of M. If N s aother -odule, the Ho (N,M) s also a Ed (M)-odule, wth the acto gve by coposto. Lea 1. If s a fte desoal -algebra, the there are oly ftely ay soorphs classes of sple -odules, ad they are fte desoal. POOF. If S s a sple odule, pck 0 s S ad defe a ap S sedg r to rs. Ths s a -odule ap, ad the age s o-zero, so t s all of S. Thus d S d <. Moreover, S ust occur ay coposto seres of, so by the Jorda-Hölder Theore there are oly ftely ay soorphs classes of sple odules. Schur s Lea. Let be a fte desoal -algebra. (1) If S T are o-soorphc sple odules the Ho (S,T)=0. () If S s a sple -odule, the Ed (S). POOF. () The usual arguets show that D = Ed (S) s a dvso rg.

6 Sce S s fte desoal, so also s D, ad therefore f d D the eleets 1,d,d,... caot all be learly depedat, ad so there s soe o-zero polyoal p(x) over wth p(d)=0. Sce s algebracally closed ths polyoal s a product of lear factors p(x) = c(x-a )...(X-a ), 0 c, a,...,a 1 1 so (d-a 1 )...(d-a 1 )=0. Now D has o zero-dvsors, so oe of the ters 1 D D ust be zero. Thus d=a 1 1. Sce d was arbtrary, D= 1. D D D Defto. A -odule s sesple f t s a drect su of sple subodules. Lea. Subodules, quotets ad drect sus of sesple odules are aga sesple. Every subodule of a sesple odule s a suad. POOF. Otted. Defto. If ad S are -algebras, ad M ad N are a -odule ad a S-odule, the M N (the tesor product over ) has the structure of a -odule gve by r( ) = r for r, M, N. ad the structure of a S-odule gve by s( ) = s for s S, M, N. earks. (1) Ths s copletely dfferet to the tesor product of two -odules whch we shall cosder later. () These two actos coute, sce r(s( )) = r( s) = r s = s(r ) = s(r( )). Thus the ages of ad S Ed (M N) coute. (3) If N has bass e,...,e the the ap 1 M... M M N, (,..., ) e e s a soorphs of -odules. Slarly, f M has bass f,...,f 1 l the ap N... N M N, (,..., ) f +...+f 1 l 1 1 l l s a soorphs of S-odules. the 3

7 Lea 3. If M s a sesple -odule, the the evaluato ap S Ho (S,M) M S s a soorphs of -odules ad of Ed (M)-odules. Here S rus over a coplete set of o-soorphc sple -odules, ad we are usg the acto of o S ad of Ed (M) o Ho (S,M). POOF. The ap s deed a -odule ap ad a Ed (M)-odule ap. To see that t s a soorphs of vector spaces, oe ca reduce to the case whe M s sple, whch case t follows fro Schur s Lea. Lea 4. If M s a fte desoal sesple -odule, the Ed (M) Ed (Ho (S,M)) S where S rus over a coplete set of o-soorphc sple -odules. POOF. The product E = Ed (Ho (S,M)) acts aturally o S S Ho (S,M), ad sce ths acto coutes wth that of, there s a S hooorphs E Ed (M) whch s fact jectve. To show that t s a soorphs we cout desos. By Lea 3, M s soorphc to the drect su of d Ho(S,M) copes of each sple odule S, ad so by Schur s Lea, d Ed (M) (d Ho(S,M)), S whch s also the deso of E. Defto. A fte desoal sesple -odule. -algebra s sesple f s a earks. (1) If s a sesple algebra, the ay -odule s sesple, for ay odule s a quotet of a free odule, but these are sesple. () If s a fte group the s sesple by Maschke s Theore. Art-Wedderbur Theore. Ay fte desoal sesple -algebra s soorphc to a product h = Ed (V ) =1 where the V are fte desoal vector spaces. Coversely, f has ths for t s sesple, the o-zero V for a coplete set of 4

8 o-soorphc sple -odules, ad as a -odule, s soorphc to the drect su of d (V ) copes of each V. POOF. We prove the assertos about the product frst. Clearly we ay suppose that all V 0. The V are aturally -odules, wth the factors other tha Ed (V ) actg as zero. Now L(V ), ad hece also, acts trastvely o V \{0}, ad t follows that the V are sple -odules. If the V have bases (e,...,e ), the the ap,1, V... V... V... V, 1 1 h h (f,..,f ) (f (e ),..,f (e ),...,f (e ),..,f (e )) 1 h 1 1,1 1 1, h h,1 h h, 1 h s a -odule ap, ad s jectve, so s a soorphs by desos. Thus s sesple. The V are a coplete set of sple -odules by the Jorda-Hölder Theore, as Lea 1, ad they are o-soorphc sce f j the the eleet (0,..,0,1,0,..,0) but ot V. (wth the 1 the -th place) ahlates V j If s ay rg the the atural ap : Ed (), r (x rx) Ed () s a soorphs, for t s certaly jectve, ad f rght had sde, the t coutes wth the edoorphss Now f r the f Ed (), f (x) = xr. r r les the (r) = (1r) = (f (1)) = f ( (1)) = (1)r, r r so acts as left ultplcato by (1), ad hece = ( (1)) so s surjectve. (), If ow s a sesple -algebra the Ed () s sesple by Lea 4 ad the proof above, so s a sesple Ed ()-odule. A secod applcato of Lea 4 ad the soorphs shows that has the requred for. 5

9 Lea 5. If s sesple ad M s a fte desoal -odule the the atural ap s surjectve. : Ed (M), r ( r) Ed (M) POOF. The kerel of ths ap s the ahlator I = {r rm=0} of M. Now /I s sesple, M s a /I-odule ad Ed (M) = Ed (M), /I so we ca replace by /I ad hece we ay suppose that M s fathful ad that s jectve. By Lea 4, Ed (M) Ed (Ho (S,M)), ad sce M s fathful, all S the spaces Ho (S,M) are o-zero, so they are precsely the sple Ed (M)-odules. By Lea 3, M s soorphc as a Ed (M)-odule to S Ho (S,M), so S t s the drect su of d S copes of the sple odule Ho (S,M) for each S. As Lea 4 ths ples that d Ed (M) = (d S), Ed (M) S but ths s the deso of, so s a soorphs. Fally, we ote the followg fact Lea 6. If s a -algebra ad h s a eleet wth h = h for soe o-zero, the for ay -odule M we have a soorphs of Ed (M)-odules. Ho (h,m) hm POOF. Note frst that hm s a Ed (M)-subodule of M, sce f Ed (M) ad h hm the (h) = h () hm. eplacg h by h/ we ay suppose that h s depotet. Now we have a Ed (M)-odule ap Ho (h,m) hm, f f(h) wth verse as requred. hm Ho (h,m), (r r), 6

10 . YOUN SYMMETIZES. ecall that the represetatos : L(V) of a group correspod to -odules by settg gv = (g)(v) for g ad v V. The trval represetato s the ap sedg all g to 1; the correspodg -odule s deoted by. The syetrc group s S = bjectos {1,...} {1,...,} wth ultplcato gve by coposto. I ths secto we copute ts represetatos usg certa eleets of the group algebra S called Youg Syetrzers. Oe represetato, the sgature ( - j) : S { 1} gve by =, (-j) 1 <j s of course well-kow. For coveece, ths secto we set A = S. Ths s a fte desoal sesple -algebra by Maschke s Theore. Defto. A partto of s a sequece = (,,...) wth, 1... ad 1 =1 lexcographcally, so that =. The parttos of are ordered < such that = for j < ad < j j Ths s a total orderg o the set of parttos of. Exaple. The parttos of 5 are 5 3 (1 ) < (,1 ) < (,1) < (3,1 ) < (3,) < (4,1) < (5). Defto. If s a partto of, the the Youg frae [ ] of s the subset { (,j) 1, 1 j We draw a pcture for ths. For exaple [(5,,1)] = } 7

11 Defto. A Youg tableau s a bjecto [ ] {1,...,}. For exaple we ght have (5,,1) = If s a Youg tableau ad S we defe a Youg tableau by ( )(x) = ( (x)) for x [ ]. For each partto we eed to pck oe represetatve of all the 0 correspodg tableaux, so for defteess we deote by the Youg tableau ubered the order that oe reads a book. For exaple = (5,,1) We defe subgroups ow( ) ad Col( ) of S by ow( ) each {1,...,} s the sae row of ad. Col( ) each {1,...,} s the sae colu of ad. Defto. If eleet s a Youg tableau, the the Youg syetrzer s the h( ) = rc c r ow( ) c Col( ) 0 of A. We also set h = h( ). The rest of ths secto s devoted to showg that the left deals A of the for Ah wth rug through the parttos of, are a coplete set of o-soorphc sple A-odules. Exaples. (1) If = () the h = h( ) = s the syetrzer A. Clearly S gh = h for all g S, so Ah = h s the trval represetato of S. () If = (1 ) the h = h( ) = s the alterzer A. Now S gh = h for all g S, so Ah = h s the sgature represetato of S. g Lea 1. Let be a partto of, a Youg tableau ad S (1) ow( ) Col( ) = {1}. 8

12 () The coeffcet of 1 h( ) s (3) ow( ) = ow( ) ad Col( ) = Col( ). -1 (4) h( ) = h( ). (5) The A-odules Ah( ) ad Ah are soorphc. 0 POOF. (5) = for soe S. Postultplcato by defes a soorphs Ah( ) Ah. Lea. If are parttos of ad ad are Youg tableaux wth fraes [ ] ad [ ], the oe of the followg s true (1) there are dstct tegers,j whch occur the sae row of ad the sae colu of. () = ad = rc for soe r ow( ) ad c Col( ). POOF. Suppose (1) fals. If the >, so [ ] has fewer colus tha [ ], ad hece two of the ubers the frst row of are the sae colu of, so (1) holds, cotrary to the assupto. Thus =, ad sce (1) fals soe c Col( ) esures that c has the sae eleets ubers the frst row as. Now gore the frst rows of ad c. By the sae arguet we fd = ad ca fd c such that c c have the sae ubers each of 1 the frst two rows. 1 Evetually we fd = ad soe c Col( ) such that = c have the sae ubers each row. The r = c for soe r ow( ). Fally = rc where c = r c r r c Col( )c r = r Col(c )r = Col( ) sce c Col( ). Lea 3. If S caot be wrtte as rc for ay r ow( ) ad c Col( ) the there are traspostos u ow( ) ad v Col( ) wth u = v. 9

13 POOF. () fals for ad, so there are j the sae row -1 ad the sae colu. Let u = ( j) ad v = u. Lea 4. Let be a Youg tableau ad a A. The followg are equvalet (1) rac = a for all r ow( ) ad c Col( ). c () a = h( ) for soe. POOF. () (1) rh( )c = h( ) sce as r rus through ow( ) so does rr, ad as c rus through Col( ) so does c c, wth =. c c c c (1) () Say a = a. If s ot of the for rc the there are S traspostos u ow( ) ad v Col( ) wth u v =. By assupto uav = a, ad the coeffcet of gves a = a, so a = -a, ad v u v v hece a = 0. Now the coeffcet of rc (1) gves a = a. Thus 1 c rc a = a rc = a rc = a h( ). r,c rc r,c c 1 1 Lea 5. If a A the h( ) a h( ) = h( ) for soe. POOF. Let x = h( ) a h( ). Ths has property (1) above. Defto. Let f = d (Ah ). Lea 6. (1) h( ) = (!/f ) h( ) () f dvdes! (3) h( ) A h( ) = h( ). I partcular (f /!) h( ) s a depotet. POOF. (1) Let h = h( ). We kow that h = h for soe. ght ˆ ultplcato by h duces a lear ap h:a A. For a A we have ˆ (ah)h = (ah), so h acts as ultplcato by. Take a bass of Ah ad Ah ˆ exted t to a bass of A. Wth respect to ths h has atrx I * f

14 ˆ (f sce d Ah = d Ah ), so Trace(h) = f. Now H Wth respect to the bass S ˆ = 1 so Trace(h) =!. ˆ of A, h has atrx H wth H = coeffcet of h. () The coeffcet of 1 h = h s =. c c r,r ow( ) c,c Col( ) r c r c = (3) By Lea 5 the oly other possblty s hah = 0. But h 0. Lea 7. Ah( ) s a sple A-odule. POOF. Let h = h( ). Sce Ah s o-zero, ad A s sesple, t suffces to prove that Ah s decoposable. Say Ah = U V. The h = hah = hu + hv, so oe of hu ad hv s o-zero. Wthout loss of geeralty hu 0, but the hu = h, so Ah = AhU U, ad hece U = Ah ad V = 0. Lea 8. If > are parttos ad ad are Youg tableau, the h( ) A h( ) = 0. POOF. Sce >, by Lea there are be two tegers the sae row of ad the sae colu of. The correspodg trasposto ow( ) Col( ). The h( ) h( ) = h( ) h( ) = - h( ) h( ), so h( ) h( ) = 0. Applyg ths to ad for S gves 0 = h( ) h( ) = h( ) h( ) so h( ) h( ) = 0. Thus h( ) A h( ) = Lea 9. If are parttos ad ad are Youg tableau, the Ah( ) ad Ah( ) are ot soorphc. POOF. We ay assue that >. If there s a soorphs 11

15 of A-odules, the f : Ah( ) Ah( ) f( h( )) = f(h( )Ah( )) = h( )f(ah( )) = h( )Ah( ) = 0, a cotradcto. eark. The parttos of correspod to cojugacy classes S, wth say (5,,1) correspodg to the perutatos S of the for 10 (...)(..)(..)(.). Theore. The left deals Ah wth rug through the parttos of are a coplete set of o-soorphc sple A-odules. POOF. They are sple, o-soorphc, ad the uber of the s equal to the uber of cojugacy classes S, whch we kow fro character theory s the uber of sple A-odules. 1

16 3. STANDAD TABLEAUX Ths secto s ot really ecessary for the a developet, but s cluded because of ts cleveress, ad because the stadard tableaux gve a explct decoposto of tesor space to sple subodules. Defto. A Youg tableau s stadard f the ubers crease fro left to rght each row ad fro top to botto each colu. The stadard tableaux wth frae [ ] are ordered so that s saller tha f t s saller the frst place that they dffer whe you read [ a book. ] lke Exaple. For = (3,) the stadard tableaux are < < < < We deote by F the uber of stadard tableaux wth frae [ ]. We shall show that F = f ad as a frst step we prove that F =!. I the ext few leas we wrte / to ea that s a partto of ad s a partto of -1 for soe, ad that [ ] [ ]. Lea 1. If s a partto of the F = F. st / -1 POOF. If s a stadard tableau, the ({1,...,-1}) s the frae of a partto of -1, ad s a stadard tableau. Ad coversely. [ ] Lea. If are parttos of, the { / ad / } = { / ad / } {0,1}. POOF. If / ad / the [ ] [ ] [ ], so there ust be equalty here. Slarly f / ad / the [ ] = [ ] [ ]. Now [ ] [ ] ad [ ] [ ] are always fraes of parttos, so there s a [ ] [ ] = +1 [ ] [ ] = -1 there s a. Lea 3. If s a partto of the (+1)F = F. st / 13

17 POOF. Ths s true for = 1. We prove t by ducto, so suppose t s true for all parttos of -1. Now F = F by Lea 1 st / st / st / = { / } F + F., st /, /, By spectg the Youg fraes oe sees that ad usg Lea we get = { { / } = { / } + 1, / } +1 F + F, st /, /, = F + F, st /, / = F + F by the ducto st / = F + F = (+1)F by Lea 1. Lea 4. F =! a partto of POOF. It s true for = 1. We prove t by ducto o. Now F = F F by Lea 1. a partto of a partto of ad / = F by Lea 3. a partto of -1 =! by the ducto. Lea 5. If > are stadard tableaux the h( ) h( ) = 0. POOF. It suffces to show that there are two ubers j the sae row ad the sae colu, for the the trasposto t=( j) s ow( ) ad Col( ). Thus so so ths product s zero. h( ) = t h( ) ad h( ) = - h( ) t, h( ) h( ) = h( ) t t h( ) = - h( ) h( ). Cosder where ad frst dffer. Pctorally we have 14

18 [ ] = = = = = = = = = =z = x y where a "=" eas that the two tableaux are the sae at that box, ad x s -1 the frst place where they dffer. Let (x) = ad y = (). By the assuptos, y ust be below ad to the left of x; partcular x caot be the frst colu or the last row. Let z be the eleet of [ ] the sae row as x ad the sae colu as y, ad let j be the coo value of ad at z. Now ad j satsfy the assuptos above. Theore. S = S h( ) wth rug over all stadard tableaux for all parttos of. POOF. We show frst that the su s drect, so suppose that there s a o-trval relato a( )h( ) = 0 (*) wth a( ) S. Pck axal such that soe a( )h( ) 0, ad the pck al wth respect to a( )h( ) 0. Multplyg (*) o the rght by h( ) we obta a( ) h( ) = 0 by Lea 5 ad Lea 8, so a( ) h( ) = 0. A cotradcto. Now S cotas S h( ). By the Art-Weddurbur Theore, S s soorphc as a S -odule to the drect su of f copes of each S h, whle ths drect su there are F copes of S h, so by the Jorda Hölder Theore, F f. O the other had F =! = f so we ust have F = f for each. But ths eas that the drect su s equal to S. Corollary. f = d ( S h ) s equal to the uber F of stadard tableaux wth frae [ ]. Lea 6. If M s a fte desoal S -odule, the M = h( )M, where rus over all stadard tableaux for all parttos of. 15

19 POOF. If ( ) = 0 s a o-trval relato wth ( ) h( )M, choose al ad the axal, such that ( ) 0. Now preultply the relato by h( ) to obta a cotradcto by Lea 5 ad Lea 8. Thus the su s drect. Now h( )M Ho ( S h( ), M) by 1 Lea 6 S Ho S h( ), M Ho ( S,M) M, S S ad all we eed s that the desos are equal. Exercse. If = M ( ) ad h,g are depotets wth = h g, the arguet used the proof above shows that s soorphc to the exteral drect su of h ad g. Show, however, that t s stll possble that h+g. 16

20 4. A CHAACTE FOMULA ecall that f M s a fte desoal correspodg character s -odule, the the (g) = trace of the ap M M, g. M It s a class fucto, so f s a cojugacy class we ca wrte ( ). M If s a partto of, the character of the S -odule S h s deoted by. I ths secto we derve a very useful forula whch eables oe to copute the ( ). I the preset course we shall ot use ths forula to copute ay characters explctly; stead we use t later to derve Weyl s character forula for the geeral lear group. If s a cojugacy class S, the cossts of all the perutatos wth a fxed cycle type, whch we deote by eag that the perutatos volve -cycles,..., -cycles ad 1 1-cycles. The uber of perutatos s deoted by.! Lea 1. = !!... POOF. Ay perutato s oe of the! of the for (*)(*)..(*) (**)...(**) (***) wth the * s replaced by the ubers 1,...,. However, each such 1 perutato ca be represeted!! ways by perutg the 1 -cycles! ways, or rotatg a -cycle ways. Orthogoalty relatos. (1) If ad are parttos of the ( ) ( ) = coj class S! ( = ) 0 (else) () If ad are cojugacy classes S, the!/ ( = ) ( ) ( ) = partto of 0 (else) 17

21 POOF. Every eleet S s cojugate to ts verse, so -1 (g) = (g ) = (g) for g S. Wth ths observato these relatos becoe the stadard orthogoalty relatos for fte groups. Notato. ve x,...,x ad l,...,l defe 1 1 l1 l l j x,...,x = det(x ). 1,j Usually the l 0, whch case t s a hoogeeous polyoal of degree l +...+l the x. 1-1 Exaple. The Vaderode x,...,x,1. Lea. The Vaderode = (x -x ). <j j POOF. Subtractg the secod row fro the frst, the eleet posto (1,j) s j j j-1 j- j-1 x - x = (x - x )(x + x x x ) so the etre frst row s dvsble by x -x. Thus the deterat V s 1 dvsble by x -x [x,...,x ]. Slarly for x -x wth <j. Sce 1 1 j polyoal rgs are UFDs, V s dvsble by the product P. Now V s a polyoal of degree (-1), whch s the degree of the product, so V = ap for soe a. We show by ducto o that a=1. If =1 the ths s clear. I geeral, f x =0 the expadg the deterat ad usg the ducto V = x x..x (x -x ), so a= <j< j eark. The sae arguet shows that f l 0 the x,...,x s dvsble by the Vaderode, so that l1 l s a polyoal x,...,x. 1 l 1 l x,...,x -1 x,..., 1 Cauchy s Lea. If x,y (1 ) ad always x y 1 the j det( ) = x,..,1. y,..,1. ( ) 1-x y,j 1-x y j j POOF. By ducto o. True for =1. Now 18

22 x -x y j - =. 1-x y 1-x y 1-x y 1-x y j 1 j 1 j j so subtractg the frst row fro each other row the deterat oe ca reove the factor x -x fro each row 1 ad 1/(1-x y ) fro each colu, 1 1 j ad the deterat equals y1/(1-xy1) y/(1-xy)... (x -x ). ( ). det (*) >1 1 j 1-x y y1/(1-x3y1) y/(1-x3y)... 1 j Now subtract the frst colu fro each other, ad use y y y -y j 1 j =.. 1-x y 1-x y 1-x y 1-x y j 1 1 j so the deterat (*) becoes * 1/(1-xy) 1/(1-xy3)... (y -y ). ( ). det j>1 j 1 >1 1-x y * 1/(1-x3y) 1/(1-x3y3) ad the asserto follows. Lea 3. If x,...,x ad y,...,y have odulus < 1, the l1 l l1 l det( ) = x,...,x. y,..,y. 1-x y l1>...>l 0 j POOF. The deterat s (1 + x y + x y +...) S =1 () () l 1 l 1 l y S =1 () = l1 l y,...,y. ad the ooal x...x (wth l ) occurs wth coeffcet I partcular t zero uless the l are dstct, so the deterat s l 1 l l1 l x...x y,...,y l1,...,l dstct 1 l l l l (1) () (1) () = x... x y,...,y l1>...>l, S 1 l l (1) () l1 l l1>...>l, S 1 = x... x y,...,y l1 l l1 l = x,...,x. y,..,y. l1>...>l Notato. If = (,..., ) (for exaple f s a partto wth 1 parts), we set 19

23 so l = +-1,..., l =. 1 1 l = + -. earks. (1)... l >l >...>l. 1 1 () If =, 0, the the polyoal l 1 l x,...,x -1 x,..., 1 has degree the x. Notato. If x,...,x ad y,...,y are coplex ubers, for we set 1 1 s = x + x x ad t = y + y y. 1 1 the power sus of the x ad the y. Lea 4. l 1 l l 1 l x,...,x y,...,y = s...s t...t -1-1! 1 1 x,..., 1 y,..., 1 where the frst su s over the parttos secod su s over the cojugacy classes S. of wth parts, ad the eark. The quotets o the left are polyoals, so ths akes sese eve f the x or y are ot dstct. POOF. Sce both sdes are polyoals, we eed oly prove ths whe the x ad y have odulus < 1. Now 3 3 x y x y x y 1 j j j log ( ) = ( ),j=1 1-x y,j 1 3 j s t s t s t = so s t s t s t ( ) = exp ,j=1 1-x y 1 3 j s t s t s t = ! 1 3 0

24 By the ultoal theore ths s s t s t 1! =...!!! where the su exteds over all sequeces (,,...) of o-egatve 1 tegers wth oly ftely ay o-zero ters. 1 1 s s... t t = !!... By Lea 3 ad Cauchy s Lea, l 1 l l 1 l x,...,x y,...,y 1. = ( ) -1-1,j=1 1-x y x,..., 1 y,..., 1 j where the su s over all l >...>l 0. So 1 l 1 l l 1 l 1 1 x,...,x y,...,y s s... t t = x,..., 1 y,..., !!... We ca ow equate the ters ths whch are of degree the x, gettg the requred equalty. Defto. For = (,..., ) ad a cojugacy class S, let ( ) be the coeffcet of the ooal x...x s...s. Thus s...s = ( ) x...x. 1 1,..., 1 Equvaletly we ca thk of the as class fuctos :S. earks. (1) ( ) = 0 f ay <0 or f () ( ) s a syetrc fucto of the. Defto. Set ( ) = ( ). S (l +1-,...,l ) (1) () We are evetually gog to show that =, but frst we eed to verfy the orthogoalty relatos for the. To do ths we eed the followg lea, whch wll evetually be our character forula. Lea l1 l s...s x,...,1 = ( ) x,...,x 1 1

25 wth suato over the parttos of wth parts. 1-1 POOF. s...s x,...,1 = (1) () = ( ) x...x x... x, S +-1 (1) (), S (1) () = ( ) x... x Let l = +- stead of the usual coveto. Sce ( ) s () syetrc the, we get l 1 l = ( ) x... x l1,..,l, S (l1+1-,...,l) (1) () l1 l = ( ) x,..., x l1,..,l (l1+1-,...,l) Sce the ters wth the l ot dstct are zero, ths becoes l1 l l >...>l, S (l +1-,...,l ) 1 (1) () = ( ) x,..., x. Now settg = l +- as usual, t becoes = ( ) x,..., x, S (l +1-,...,l ) (1) () where the su s over all (,..., ) wth.... Now the ters 1 1 for whch ths s ot a partto of are zero by the rearks above, for -1 f < 0 the certaly l + ()- < 0. l1 l Lea 6. If ad are parttos of wth parts the ( ) ( ) = coj class! ( = ) 0 (else) POOF. By Lea 4, the su l1 l l1 l x,...,x y,...,y over the parttos By Lea 5, ths s of wth parts s equal to s...s t...t x,...,1 y,...,1.! l1 l l1 l!,, ( ) ( ) x,...,x y,...,y wth suato over the parttos, of wth parts ad cojugacy classes. The asserto follows sce as ad vary, the polyoals l1 l l1 l x,...,x y,...,y

26 are learly depedet [x,...,x,y,...,y ]. 1 1 We ow start to relate these deas wth the syetrc group (whch has, so far, played o role). The key result s: Lea 7. If s a partto of wth parts the s the character of the S -odule S r where r = 0. ow( ) POOF. Let be the character of S r, let S be the cojugacy 0 class, let = ow( ), ad let N S = g =1 be a coset decoposto. The S r has bass (g r ). We use ths 1 N bass to copute traces. Now g r = g r f g g. j j Thus -1 ( ) = { 1 N g g } Now g g f ad oly g s a coset g wth g g, ad =!!..., so 1-1 ( ) = 1/!!... { g S g g } Sce g g = g g g g c ( ), each value take by g g s take S by c ( ) eleets g S. Now S so! 1 c ( ) = = 1...!!... S 1 1 ( ) = 1...!!.../!! Now a perutato restrcts to a perutato of the ubers the -th row of 0. If ths restrcto volves say j j-cycles, the the j satsfy (*): = (1 ) 1j + j + 3j +... = j (1 j ) The uber of perutatos of ths type s so (!/1...!!...)(!/1...!!...)

27 !! 1 ( ) =...!..!...!! where the suato s over all satsfyg (*). j 1 By the ultoal theore s...s s equal to 1!! ( x x...x )( x x...x )...!.!... 1!.! where the su s over all (1, 1 j ) satsfyg j = (1 j ). 1j j j j 1 Thus ( ) (the coeffcet of the ooal x...x ) s equal to 1!! 1...!..!...!! where the suato s over all ( ). j satsfyg (*), ad hece s equal to Lea 8. Let be parttos of. The sple odule S h s a subodule of S r f ad oly f =. POOF. If < ad S, the by Lea there are two tegers the sae row of ad the sae colu of, so f s ther -1 0 trasposto the Col( ) ad -1 h r = h r = -h r = 0. Thus 0 = h S r Ho ( S h, S r ). Coversely S so 0 h S r Ho ( S h, S r ). S h r ( 0 ) = h 0, Col( ) Lea 9. If s a partto of wth parts, the =. POOF. () If S, let be the partto wth parts l +1-,..., l (1) () the approprate order, so wth parts -1 + ()-. Sce ( ) s syetrc the, we ca wrte 4

28 = S -1-1 If 1 the >, for + (1)-1, wth equalty oly f (1)= The + ()-, wth equalty oly f ()=, etc.. If = 1 the =. Thus s a -lear cobato of wth ad wth coeffcet of equal to 1. () By Leas 7 ad 8, s a -lear cobato of s wth ad wth o-zero coeffcet of s wth ad wth postve coeffcet of. Thus s a -lear cobato of = k partto of wth the k, k > 0 ad k = 0 f <. () We kow that ( ) ( ) =. Say! ( = ) coj class 0 (else) I the case = the orthogoalty of the gves k = 1, so k = 0 f ad k = 1, as requred., At last our character forula! ecall that ad x,...,x 1 arbtrary, l = +- ad s =x +...+x. 1 are Theore. 1-1 l1 l s...s x,...,1 = ( ) x,...,x 1 wth suato over parttos of wth parts. POOF. Follows fro Leas 5 ad 9. eark. I partcular, takg, we ca esure that the rght had sde volves all parttos of. eark. If s a partto of wth parts, the ( ) s the l 1 l coeffcet of the ooal x...x the expaso of s...s x,...,1. 1 5

29 5. THE HOOK LENTH FOMULA We already have oe forula for the deso of the sple S -odules, the uber of stadard tableaux. I ths secto we derve two ore forulae, oe of whch s easy to use. Theore. If f of s equal to s a partto of wth exactly parts the the degree! (l -l ) / l!...l! 1 <j j 1 POOF. f = l 1 l (1), whch s the coeffcet of x...x 1 the expaso of -1 (1)-1 ()-1 (x +...+x ) x,...,1 = (x +...+x ) x...x. 1 S 1 1 By the ultoal theore ths coeffcet s equal to! S (l +1- (1))!...(l +1- ())! 1 where, by coveto 1/x! = 0 f x < 0. Now ths s equal to =! 1/(l-+1)!,... 1/(l-1)!,1/l! =!/l!...l!...,l(l-1),l,1 1-1 =!/l!...l! l,...,l,l,1 1 by addg approprate colus, ad ths s what we wat sce the last deterat s the Vaderode. Defto. If s a partto of the the hook at (,j) [ ] s the set of (a,b) [ ] wth (a ad b=j) or (a= ad b j). The hook legth h j s the uber of eleets of the hook at (,j), so that f [ ] has colu legths,,... the h = + --j+1. 1 j j Exaple. If so h = 6. = (7,5,3,1) the the hook at (,) s the shaded part of [(7,5,3,1)] =! Theore (Hook legth forula). f =. h (,j) [ ] j 6

30 that POOF. Let have parts. By the prevous theore t suffces to show (l -l ). h = l! k=+1 k j=1 j for each. Now the product o the left s a product of +- = l ters, so t suffces to show that the ters are precsely 1,,...,l soe order. Now l -l > l -l > l -l > h > h > h > Sce has exactly parts, = ad h = l, so each ter s l. Thus 1 1 t suffces to show that o h s equal to ay l -l. However, f r = j k j the j ad < j so r r+1 h -l +l = +r--j r = +1-j > 0, j r r r h -l +l = +r--j r-1 = -j < 0, j r+1 r+1 r+1 ad hece l -l < h < l -l. r j r+1 Exaple. If = 11 ad so that = (6,3,) the [ ] = f = =

31 6. MULTILINEA AND POLYNOMIAL ALEBA I ths secto we recall soe rather stadard ultlear algebra for fte desoal -odules where s a group, whch ay be fte, or t ay be 1, so that we just deal wth vector spaces. Let V,W be fte desoal -odules. Tesor products. The tesor product V W (over ) s a -odule va g(v w) = (gv) (gw). Propertes. V, V W W V, (V W) Z V (W Z). If :V V ad :W W are -odule aps, the so s : V W V W. Ho Spaces. Ho (V,W) s a -1 -odule va (gf)(v) = gf(g v). I * partcular the dual of V s V -1 = Ho (V, ), so (gf)(v)=f(g v). * * * * Propertes. (V W) V W. If V s oe-desoal the V V. If * * * * :V W s a -odule ap, the so s :W V. The ap V W Ho (V,W) takg f w to the ap v f(v)w s a -odule soorphs. Tesor powers. The -th tesor power of V s 0 T V = V... V ( copes, f >0), T V =. Propertes. (1) If V has bass e,...,e the T V has bass e... e so t has 1 1 deso. () T V s a S -odule va (v... v ) = v v -1 ( S ). 1 f (1) (f) f ad the actos of S ad coute: g x = gx for g, S, x T V. Defto. The -th exteror power of V s V = T V / X where X = <x - x x T V, S >. The age of v... v V s deoted by v... v. We also defe 1 1 T V = { x T V x = x S }, at the set of atsyetrc tesors. 8

32 Propertes. (1) v... v = v v -1 for S. I partcular, 1 (1) () cosderg a trasposto, v... v 1 = 0 wheever two of the v are equal. () T V ad V are -odules. at (3) V has bass e... e wth <...<, so t has deso ( ) I partcular V s oe-desoal ad V = V =... = 0. eark. For a vector space V over a arbtrary feld k oe should defe the exteror powers by V = T V/X where X s spaed by the tesors of for v v... v wth two of the v equal. If k has characterstc 1 ths reduces to the gve defto. at S atural ap T V V s a soorphs of -odules. at Lea 1. T V = at V where a = s the alterzer. The POOF. If x s atsyetrc, the ax = (!)x, so x at V. Coversely sce a= a, ay eleet of at V s atsyetrc. The ap at V T V V s a -odule ap wth kerel X at V ax sce x = 1/! ax for x atsyetrc. However ax = 0 sce for y T V ad S we have a(y- y)=0. The ap s surjectve sce f x V s the age of y T V, the x s also the age of 1/! ay. * * Lea. (V ) ( V). POOF. The atural ap T V V gves a cluso * * * ( V) (T V) T (V ). By the uversal property of V - that ay alteratg ultlear ap * V... V factors through V - the age of ths ap s T (V ), at * whch s soorphc to (V ). Defto. The -th syetrc power of V s S V = T V / Y where Y = <x - x x T V, S >. 9

33 The age of v... v S V s deoted by v... v. We also defe 1 1 T V = { x T V x = x S }, sy the set of syetrc tesors. Propertes. (1) For S oe has v... v = v v (1) () () T V ad S V are a -odules. sy +-1 (3) S V has bass e... e wth..., so has deso ( ). 1 1 To see ths, ote that (1-X )...(1-X ) = X...X 1 1,..., 1 so the uber of ters wth total degree s the coeffcet of X (1-X), whch s (-1) ( ) = ( ). As the case of exteror powers oe has sy S atural ap T V S V s a soorphs of -odules. sy Lea 3. T V = st V where s = s the syetrzer. The * * Lea 4. S (V ) (S V). Next we cosder polyoal aps betwee vector spaces. These geeralze the usual oto of lear aps. Defto. Let V ad W be fte desoal fucto -vector spaces. A : V W s a polyoal (resp. hoogeeous -c) ap provded that V ad W have bases e,...,e ad f,...,f such that for all 1 1 h X,...,X we have 1 where the degree ). (X e +...+X e ) = (X,...,X )f (X,...,X )f h 1 h (X,...,X ) are polyoals (resp. hoogeeous polyoals of 1 Lea 5. If there are such fuctos wth respect to soe bases, the there are such fuctos wth respect to ay bases. POOF. Suppose that e = p e ad f = q f. The j j j j j j 30

34 ( X e ) = ( X p e ),j j j ad the fuctos = = ( X p,..., X p ) f r r 1 r ( X p,..., X p ) q f r,s r 1 sr s ( X p,..., X p ) q r r 1 sr are polyoals or hoogeeous polyoals of degree lke the. Notato. We deote by Poly (V,W) ad Ho aps. Clearly these are vector spaces. (V,W) the spaces of such, Lea 6. A coposto of polyoal aps X W ad W Z s a polyoal ap. The coposto of a hoogeeous -c ad a hoogeeous -c ap s a hoogeeous -c ap. POOF. (X ) = X. Exaples. (1) Ho (V,W) = W ad Ho (V,W) = Ho (V,W),,0,1 () The ap : v v... v les Ho (V,S V), sce, ( X e ) = X...X e... e,..., for sutable costats c.,..., 1 = c X...X e... e..,.., Theore. If V ad W are vector spaces, the soorphs Ho (S V,W) Ho (V,W)., duces a POOF. To show that the ap s jectve, suppose that = 0. We show by descedg ducto o that (v... v ) = 0 wheever of the ters 1 are equal. The case = s true by assupto, ad the case =1 s what we wat. Suppose true for +1, the for, 0 = ((x+ y)... (x+ y) v... v ) + +1 j +1 = ( ) (x... x y... y v... v ) j=0 j + +1-j j 31

35 Sce ths s zero for each, each ter s zero. I partcular +1 ( ) (x... x y v... v ) = 0, 1 + as requred. Now f d V =, d W = h the +-1 d Ho (S V,W) = h ( ) = d Ho (V,W), so the ap s a soorphs. Lea 7. The eleets of the for v... v wth v V spa S V. POOF. Take W =. If these eleets do ot spa S V the there s a o-zero lear ap S V whose coposto wth s zero. eark. We ca costruct a verse explctly. Let Ho (V,W), so, (X e +...+X e ) = (X,...,X )f (X,...,X )f h 1 h wth a hoogeeous polyoal of degree. We defe the total polarzato P Ho (S V,W) of by h j (P )(e... e ) = f j=1 j 1 X... X 1 Ths akes sese sce the rght had sde s syetrc,...,. Note 1 that the partal dervatve s a coplex uber sce has degree. Now j for v V we have (P ) =!. Naely, (P ) (X e +...+X e ) = X..X (P )(e... e ) 1 1 1,..., 1 1 j = X...X f. j 1,..., 1 j X... X 1 By terato of Euler s Theore, that f F s hoogeeous of degree r varables X the X F/ X = rf, we obta =! (X,...,X ) f =! (X e +...+X e ). j j 1 j 1 1 Exaple. If :V s a quadratc for, so (X e +...+X e ) = a X X 1 1,j j j wth a j = a, the j s ( (P ) ( X e ) ( Y e ) = a X Y j j j,j j j ) the correspodg syetrc blear for. 3

36 7. SCHU-WEYL DUALITY Let V be a vector space of deso ad let. We kow that T V s a S -odule, so we have a ap S Ed (T V) sedg S to (x x) Also, regardg V as a represetato of L(V) the atural way, T V becoes a L(V)-odule, ad we have a correspodg ap L(V) Ed (T V) sedg L(V) to T =.... eark. I ths secto we prove Schur-Weyl dualty, that the ages of S ad L(V) Ed (T V) are each others cetralzers. Despte ts ocuous appearace ths result s absolutely fudaetal. For exaple t s precsely ths fact whch explas why the syetrc group ad the geeral lear group are related. Defto. The algebra A (V) of bsyetrc trasforatos s the subalgebra of Ed (T V) cosstg of the edoorphss whch coute wth the age of S. Thus A (V) = Ed (T V). S Sce S s sesple ad T V s a fte desoal S -odule, A (V) s a sesple -algebra by 1 Lea 4. We set W = Ed (V), whch s a L(V)-odule by cojugato. Lea 1. There s a soorphs : T W Ed (T V) sedg f... f to the ap 1 v... v f (v )... f (v ) Ths s a soorphs of L(V)-odules, ad of S -odules. * * POOF. T W = W... W (V V )... (V V ) * * (V... V) (V... V ) * T V (T V) Ed (T V). Now T W has a atural structure of S -odule, ad Ed (T V) herts ts structure fro T V (as cojugato). Oe ca check that s a S -odule ap (exercse). 33

37 Lea. A (V) = (T W ). sy POOF. A (V) s the set of x Ed (T V) fxed uder the acto of S, ad T W s the set of y T W fxed uder the acto of S. sy Lea 3. Affe -space s rreducble, that s, f = X Y ad X ad Y are Zarsk-closed subsets, the X = or Y =. POOF. The rg of regular fuctos o s = [X,...,X ]. If X ad 1 Y are the zero sets of deals I,J, the the assupto s that ay axal deal cotas ether I or J. If I ad J are both o-zero the we ca pck 0 I ad 0 j J. Now ay axal deal cotas j, so (j)(a,...,a ) = 0 1 for all a,...,a. Thus by Hlbert s Nullstellesatz j 1 whch cotradcts the fact that s a tegral doa. {0} = {0}, d d d Lea 4. If Y s a subspace of, the detfyg =, Y s Zarsk-closed. d POOF. Choose a bass f,...,f of Ho ( /Y, ), ad regard these as 1 h d aps. The Y s the zero set of the f. Lea 5. T W s spaed by the... wth L(V). sy POOF. Let X be the subspace of T W spaed by the... wth sy L(V). Now the ap W T W,... s a regular ap betwee the affe spaces W ad T W. -1 Sce X s a subspace, t s Zarsk-closed by Lea 4, ad hece (X) s Zarsk-closed. Thus -1 W = (X) {the edoorphss wth deterat zero} -1 s a uo of Zarsk-closed subsets. But s rreducble, so (X)=W. Thus X cotas all aps of the for... wth W. But these spa 34

38 T W, sce the... spa S W by 6 Lea 7. sy estatg ths, we have Theore. A (V) s spaed by the T wth L(V). Fally, we have Schur-Weyl dualty Theore. The ages of S ad L(V) Ed (T V) are each others cetralzers. POOF. The stateet that the age of L(V) s the cetralzer of the age of S s just a reforulato of the asserto that A (V) s spaed by the T wth L(V), whch was the last theore. ecall that A (V) s a sesple -algebra. By 1 Lea 5 we kow that S aps oto Ed (T V), ad sce the age of L(V) spas A (V) t A (V) follows that Ed (T V) = Ed (T V). A (V) L(V) Thus S aps oto Ed (T V), or other words, the age of S L(V) Ed (T V) s the cetralzer of the age of L(V). 35

39 8. DECOMPOSITION OF TENSOS Stll V s a vector space of deso. Oe lears school physcs that ay rak two tesor, e ay eleet of V V, ca be wrtte a uque way as a su of a syetrc ad a atsyetrc tesor. The Youg syetrzers eable oe to geeralze ths to hgher rak tesors, aely by 3 Lea 6 we have (*) T V = h( )T V a partto of ad stadard Exaple. h T V = T V V ad h T V = T V S V, so takg (1 ) at () sy = ths decoposto becoes T V = T V T V V S V. at sy Sce the actos of S ad L(V) o T V coute, f s a partto of ad s a Youg tableau wth frae [ ], the h( )T V s a L(V)-subodule of T V. Note that h( )T V h T V as L(V)-odules, for -1 for soe h = h( ) S, so preultplcato by duces a soorphs h( )T V h T V. Lea 1. The o-zero odules h T V wth a partto of, are o-soorphc sple L(V)-odules. If M s a A (V)-odule, ad M s regarded as a L(V)-odule by restrcto va the atural ap L(V) A (V), the M s soorphc to a drect su of copes of the h T V. POOF. ecall that A (V) = Ed (T V) ad h T V Ho ( S h, T V) S S by 1 Lea 6. By the Art-Wedderbur Theore ad 1 Lea 4, the o-zero spaces h T V are a coplete set of o-soorphc sple A (V)-odules. Note also that A (V) s sesple, so the lea follows fro the ext two assertos, whch both follow edately fro the fact proved 7 that the ap L(V) A (V) s oto. (1) If M s a A (V)-odule ad N s a L(V)-subodule of M, the N s 36

40 a A (V)-subodule, ad () If M ad N are A (V)-odules ad :M N s a L(V)-odule ap, the s a A (V)-odule ap. eark. Thus (*) s a decoposto of T V to L(V)-subodules whch are ether zero or sple. Obvously t s portat to kow whch of these subodules are o-zero, ad that s what the rest of ths secto s devoted to. Frst we have a rather techcal lea. Let be a partto ad suppose that [ ] s parttoed to two o-epty parts, say of ad j = - boxes, by a vertcal bar. Let boxes j boxes be a tableau whose ubers the left had part are {1,...,} ad the rght had part are {+1,...,}. Let be the partto of correspodg to the left had part, ad let be the restrcto of to [ ]. Let be the partto of j correspodg to the rght had part, ad let be the correspodg tableau. Ths s a ap fro [ ] to {1,...,j } f we set 1 =+1, =+,..., j =. Lea. There s a L(V)-odule surjecto j h( )T V h( )T V h( )T V. POOF. S = Aut{1,...,} ad S = Aut{1,..., } are ebedded S, so j we ca regard S ad S as subsets of S whch coute. Now j Col( ) = Col( ) Col( ) ad H = ow( ) ow( ) s the subgroup of ow( ) o the perutatos whch keep each uber o the sae sde of the bar. Let ow( ) = r H be a coset decoposto. h( ) = rc r ow( ) c Col( ) c = r r r c c r ow( ) r ow( ) c Col( ) c Col( ) c c = r h( )h( ). 37

41 Thus h( )h( )h( ) = r h( ) h( ) = h( ) where =!j!/f f. j We have a L(V)-odule ap T V T V T V h( )T V gve by preultplyg by h( ). The restrcto of ths ap to j h( )T V h( )T V s oto, sce h( ) (x y) = 1/ h( ) h( )x h( )y. Lea 3. (1) If = 0 the h T V () If >0 ad = 0 the d h T V. POOF. 0 (1) Let = row whch j occurs, ad x = e... e. The for j 1 S x = x = -1 for 1 j j (j) -1 j ad (j) occur the sae row 0 ow( ). Thus the coeffcet of x the decoposto of h x wrt the stadard 0 bass of T V s ow( ) 0, so h x 0. () If y = e... e the the arguet above shows that h x ad h y are learly depedet. Lea 4. If = 0 ad > 0 the +1 - h T V (V) h T V. ( -1,..., -1) 1 POOF. Dvde [ ] to the frst colu ad the rest. Let be a tableau whose frst colu cossts of the ubers {1,...,}. By Lea there s a surjecto - (V) h( )T V h( )T V, where = ( -1, 1-1,..., -1). Usg the usual soorphss ths gves a ap (V) - h T V h T V. - Now both h T V ad h T V are o-zero, ad hece are sple 38

42 L(V)-odules by Lea 1. Sce (V) s oe-desoal, both sdes are sple odules ad the ap ust be a soorphs. Theore. If s a partto of ad = d V, the 0 ( 0) +1 d h T V = 1 ( = =... =, = 0) 1 +1 (else) POOF. If 0 the [ ] has > rows ad as the prevous lea +1 - there s a surjecto (V) h T V h T V. But (V) = 0. O the other had, f = 0, the by teratg Lea 4 we have +1 - d h T V = d h T V ( -,..., - ) 1-1 whch s oe f =... =, ad otherwse s by Lea

43 9. ATIONAL EPESENTATIONS OF L(V) Throughout, V s a vector space wth bass e,...,e. 1 Defto. A fte desoal L(V)-odule W wth bass w,...,w s 1 h sad to be ratoal (resp. polyoal, resp. hoogeeous -c) provded that there are ratoal fuctos (resp. polyoals, resp. hoogeeous polyoals of degree ) f (X ) (1,j h) the varables X j rs rs (1 r,s ), such that the ap bass e represetato bass w L ( ) L(V) Ed (W) M ( ) h seds a atrx (A ) L ( ) to the atrx (f (A )). rs rs j rs j Lea 1. These otos do ot deped o the bases e,...,e 1 w,...,w. 1 h ad eark. W s a ratoal L(V)-odule f ad oly f the ap L(V) L(W) s a regular ap of affe varetes. ecall that a ratoal ap of affe varetes s ot everywhere defed: we deftely do t wat that. Exaples. (1) T V s a hoogeeous -c L(V)-odule. () V s polyoal, but ot hoogeeous. * (3) V s ratoal, but ot polyoal. Lea. (1) Subodules, quotet odules ad drect sus of ratoal (resp. polyoal, resp. hoogeeous -c) odules are of the sae type. * () If U s ratoal, the so s U. (3) If U ad W are ratoal (resp. polyoal, resp hoogeeous -c ad -c) the U W s ratoal (resp. polyoal, resp. hoogeeous + -c). (4) If U s hoogeeous -c ad hoogeeous -c, the U=0. -1 POOF. For () ote that the etres of A the etres of A. are ratoal fuctos of 40

44 Defto. If... 0 ad =, we set 1 D (V) = h T V. 1,.., ( 1,..., ) Theore. Every hoogeeous -c L(V)-odule s a drect su of sple = are a 1,.., 1 coplete set of o-soorphc sple hoogeeous -c L(V)-odules. subodules. The odules D (V) wth... 0 ad POOF. I vew of 8 Lea 1, t suffces to show that ay hoogeeous -c L(V)-odule s obtaed fro soe A (V)-odule by restrcto. Let U be a hoogeeous -c L(V)-odule, so U correspods to a ap :L(V) Ed (U). I the followg dagra, the aps across the top are the atural aps, ad ther coposte :L(V) A (V) s fact the atural ap we use for restrctg A (V)-odules to L(V)-odules. We shall show that there are aps akg the dagra coute. L(V) Ed (V) S (Ed (V)) T (Ed (V)) A (V) sy Ed (U) Ed (U) Ed (U) Ed (U) Ed (U) Sce U s hoogeeous -c, we ca exted the doa of defto of to obta a hoogeeous -c ap. By the property of syetrc powers there 1 s a lear ap. Sce the reag aps across the top are soorphss there are certaly lear aps ad, as requred. 3 4 Now (1) = ( (1)) = (1) = 1 ad 4 4 ( (gg )) = (gg ) = (g) (g ) = ( (g)) ( (g )) for g,g L(V). Sce (L(V)) spas A (V) t follows that s a 4 -algebra ap. Ths turs U to a A (V)-odule, ad the restrcto to L(V) s the odule we started wth, as requred. Lea 3. Every polyoal odule for L( ) = decoposes as a drect su of subodules o whch g acts as ultplcato by g (soe ). POOF. Here s a slly proof. If : L(U) L ( ) s a h polyoal represetato, the each (g) s a polyoal g, ad we j ca choose N such that each (g) has degree strctly less tha N. By j restrcto, U becoes a -odule where = {exp j/n 0 j<n} 41

45 s cyclc of order N. Now U = U U... U 1 h as a -odule, wth each U oe-desoal ad g actg as ultplcato by g o U (0 < N) (sce these are the possble sple -odules). Choosg o-zero eleets of the U gves a bass of U, ad f ( (g) ) ow deotes the atrx of (g) wth respect to ths j bass the g ( = j) (g) = j 0 ( j) for g = exp{ j/n} wth 0 j < N, ad hece for all g sce the (g) are polyoals of degree < N g. j Lea 4. Every polyoal hoogeeous -c odules. L(V)-odule decoposes as a drect su of POOF. Say :L(V) L(U) s a polyoal represetato of L(V). The cluso of, so by Lea 3, L(V) eables us to regard U as a polyoal represetato U = U 0... U N wth 1 L(V) actg as ultplcato by o U. If u U ad g L(V), let gu = u u 0 N wth u U. Now ( 1)gu = g( 1)u for, so N u + u + u u = u + u + u u 0 1 N 0 1 N ad hece u = 0 for. Thus gu U ad the spaces U are L(V)-subodules of U. Sce 1 acts as ultplcato by o U t follows that U s a hoogeeous -c L(V)-odule. Theore. Every polyoal L(V)-odule s a drect su of sple subodules. The odules D (V) wth... 0 are a coplete set of 1,.., 1 o-soorphc sple polyoal L(V)-odules. Defto. If the the oe-desoal L(V)-odule correspodg to the represetato L(V), g [det(g)] 4

46 1 1 s deoted by det. Thus det (V), det T (det ) f 0, ad - * det (det ) f 0. Defto. If... but < 0, we defe 1 D (V) = D (V) det,.., -,.., -, eark. If... >0 the we have already see that 1 D (V) D (V) det,.., -,.., -, Theore. Every ratoal L(V)-odule s a drect su of sple subodules. The odules D (V) wth... are a coplete set of 1,.., 1 o-soorphc sple ratoal L(V)-odules. POOF. The ratoal fuctos f:l(v) are all of the for f = p/det wth p a polyoal fucto. Thus f U s a ratoal L(V)-odule, the N W = U det s a polyoal L(V)-odule for soe N. Sce W decoposes as a drect su of sples, so does U. If U s sple, the so s W, ad thus W D (V) for soe.... Fally U D (V), 1,.., 1 1-N,..., -N usg the reark above. Theore. The oe-desoal ratoal det wth. L(V)-odules are precsely the POOF. After passg, as above, to polyoal odules, ths follows fro the theore 8. 43

47 10. WEYL S CHAACTE FOMULA Notato. V s a vector space of deso. If =(,..., ) ad 1..., the the character of the L(V)-odule 1 s deoted by. D (V) (= h T V f 0) 1,..., Lea 1. If s a edoorphs of V the ( ) s a syetrc ratoal fucto of the egevalues of. POOF. The fucto P(x,...,x ) = (dag(x,...,x )) s a ratoal 1 1 fucto of x,...,x, ad t s syetrc sce dag(x,...,x ) s 1 1 cojugate to dag(x,...,x ) for S. (1) () Now choose a bass of V so that the atrx A of s Jorda Noral 1 For, ad for t, let A be the atrx obtaed fro A by chagg the t 1 1 s o the upper dagoal to t s. Let be the edoorphs t correspodg to A. For t 0, A s cojugate to A, so ( ) = ( ). t t 1 t Sce s a ratoal fucto t s cotuous (where defed), so ( ) = l ( ) = l = ( ) = P(x,...,x ). t t 0 1 t 0 t 0 Exercse. Phrase ths usg the Zarsk topology, by eas of the dscrat of the characterstc polyoal. 1 Lea. Let be a cojugacy class S wth cycle type...1 ad let be a edoorphs of V wth egevalues x,...,x. If s =x +...+x, 1 1 the 1 s...s = ( ) ( ) 1 wth suato over parttos of wth parts POOF. Let g. We ay suppose that has atrx dag(x,...,x ) wth 1 respect to the stadard bass e,...,e of V. Cosder the edoorphs of 1 T V sedg x to g x = gx. We copute ts trace two ways. Cosderg T V as a S -odule, by 1 Lea 3, we have 44

48 ad the by 1 Lea 6 ths becoes T V = S h Ho ( S h, T V) S T V ( S h ) (h T V). Now ths s a soorphs as both a S -odule ad a Ed (T V)-odule, S ad sce the acto of L(V) o T V coutes wth that of S, the correspodg acto of g o the rght had sde s gve by the acto of g o S h ad of o h T V, so the trace of ths acto s ( ) ( ). O the other had we ca copute the trace drectly: g (e... e ) = x...x e... e g (1) g () so the trace s x...x sued over 1 (,..., ) 1,..., 1 ad -1 = for each j 1 1 g (j) j Now the codto that -1 = for each j s equvalet to requrg g (j) j that the fucto j be costat o the cycles volved g. It j 1 follows that the trace s equal to s...s. 1 Equatg the two calculatos of the trace gves the requred equato. Theore. Let be a partto of wth parts ad let be a edoorphs of V wth egevalues x,...,x. If s =x +...+x, the 1 1 ( ) s s 1 1 ( ) =....!...! 1 coj class 1 POOF. Take the forula Lea, ultply by, usg the orthogoalty of the. ( ), ad su over Theore. (Weyl s Character Forula for the geeral lear group). If L(V) has egevalues x,...,x the 1 l 1 l x,...,x ( ) = -1 x,..., 1 where l =