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1 O Represetatio Theory i Coputer Visio Probles Ao Shashua School of Coputer Sciece ad Egieerig Hebrew Uiversity of Jerusale Jerusale 91904, Israel eail: shashua@cs.huji.ac.il Roy Meshula Departet of Matheatics The Techio Haifa, Israel Lior Wolf School of Coputer Sciece ad Egieerig Hebrew Uiversity of Jerusale Jerusale 91904, Israel Aat Levi School of Coputer Sciece ad Egieerig Hebrew Uiversity of Jerusale Jerusale 91904, Israel Gil Kalai The Istitute of Matheatics Hebrew Uiversity of Jerusale Jerusale 91904, Israel Abstract We itroduce the followig geeral questio: Let V be a coplex -diesioal space ad for k cosider the GL(V )-odule V ( k) V defied by V ( k) =f v 1 v 2 V : di Spafv 1 ::: v gk g : We would like to deterie di V ( k) for ay choice of k. This questio appears i various disguises i coputer visio probles where the costraits of a ulti-liear proble occupy a lowdiesioal subspace. We discuss two such probles: aalysis of costraits i sigle view idexig fuctios (the 8-poit shape tesor), ad the aalysis of the costraits i dyaic P! P aligets, i.e., where the poit sets are allowed to ove withi a k-diesioal subspace while the -diesioal space is beig ultiply projected (ultiple views) oto copies of the -diesioal space. We the derive the solutio to the geeral proble usig tools fro represetatio theory. 1 Itroductio Multiliear costraits i coputer visio applicatios are of growig iterest i Structure for Motio (SFM), Idexig ad Graphics. May of the applicatios where ultiple ea- The referece to this auscript is Techical Report , Leibiz Ceter for Research, School of Coputer Sciece ad Eg., the Hebrew Uiversity of Jerusale.

2 sureets are ivolved like ultiple-view geoetry of static ad dyaic scees, idexig fuctios ito 3D data-sets, separatio of various attribute/odalities such as cotet ad style have a ultiliear for. As a result, a growig aout of work has bee published o the various aspects of those algebraic fuctios ad their applicatios see [10, 6] for the recet suary of various ulti-liear aps ad their associated tesors. I this paper we raise a geeral questio ad deostrate its relevace to the curret research i ultiliearity i coputer visio. The questios takes the followig for: Let V be a coplex -diesioal space ad for k cosider the GL(V )-odule V ( k) V defied by V ( k) =f v 1 v 2 V : di Spafv 1 ::: v gk g : We would like to deterie di V ( k) for ay choice of k. We will show that this questio appears i a oe disguised for or aother i a uber of visio probles ad, for exaple, focus o two of those probles: (i) aalysis of costraits i sigle view idexig fuctios (the 8-poit shape tesor), ad (ii) the aalysis of the costraits i dyaic P!P appigs, i.e., where the poit sets are allowed to ove withi a k- diesioal subspace while the -diesioal space is beig ultiply projected (ultiple views) oto copies of the -diesioal space. We the derive the solutio to the geeral proble usig tools fro represetatio theory. We will describe the geeral otatios i the ext sectio (ad provide a brief prier o represetatio theory i the appedix), followed by the detailed descriptio of the two probles etioed above ad the way the are apped to the questio of di V ( k), ad followed by the derivatio of the structure ad diesio of the GL(V ) odule V ( k) by coutig irreducibles followed with exaples of its applicatio to soe istaces of dyaic P!P appigs. 2 Notatios We will describe below the otatios ad sybols we will be usig later i the paper. A brief accout of the relevat facts cocerig the represetatio theory of the geeral liear group ca be foud i the Appedix. Let V be a fiite -diesioal vector space over the coplex ubers, ad let the group of autoorphiss of V deoted by GL(V ). We deote the exterior powers of V by ^ V ad the syetric powers by Sy P V. A partitio of is deoted by =( 1 ::: k ) such that 1 ::: k 1 ad i =. A partitio is represeted by its Youg diagra (or shape ) which cosists of k left aliged rows of boxes with i boxes i row i. We deote by i the uber of ters i that are greater tha or equal to i, ad =( 1 ::: r ) is called the cojugate partitio of. We deote by T the set of stadard tableaux o ad f the uber of stadard tableaux o,! f = Q h (i j) ij where h ij = i + j ; i ; j +1is called the hook legth of a box i positio (i j), ad the product of the hook-legths is over all boxes of the diagra. We deote by d () the uber of sei-stadard tableaux: Y ; i + j d () = : h ij (i j) Let t be a tableau o (a uberig of the boxes of the diagra) ad let P (t) deote the group of all perutatios 2 S which perute oly the rows of t. Siilarly, let Q(t)

3 deote the group of perutatios that preserve the colus of t. Let a t b t be two eleets i the group algebra C S defied as: a t = g2p (t) g b t = g2q(t) ad we deote Schur s Module by S t (V )=V a t b t. 3 The 8-poit Shape Tesor Proble sg(g)g I this sectio we will ake the coectio betwee the questio of di V ( k) ad a riddle regardig the iteral structure of the 8-poit shape tesor. Shape tesors were first itroduced i [2, 16, 3] with the basic idea that sigle-view ivariats of a 3D scee ca be obtaied by algebraically eliiatig the viewig positio (caera) paraeters give a sufficiet uber of poits. Later, the sae aalysis was coducted i a reduced (but practical i visio applicatios) settig where a referece plae is idetified i advace [11, 12, 5, 4] which is the case we will focus o here. The proble settig is as follows. Let P i = ( i Y i Z i W i ) > 2 P 3, i = 1 ::: 8, deote 8 poits i 3D projective space ad let M be a 3 4 projectio atrix, thus p i = MPi where p i 2 P 2 be the correspodig iage poits i the 2D projective plae. We wish to algebraically eliiate the caera paraeters (atrix M) byhav- ig a sufficiet uber of poits. This could be doe succictly if we first ake a chage of basis: Let the coplaar poits be deoted by P 1 ::: P 4 with the coordiates ( ) ( ) ( ) ( ) which is appropriate whe P 1 ::: P 4 are ideed coplaar. Let the iage udergo a projective chage of coordiates such that the correspodig poits p1 ::: p4 be assiged e 1 =(1 0 0) e 2 =(0 1 0) e 3 =(0 0 1) e 4 = (1 1 1), respectively. Give this setup the caera atrix M cotais oly 4 o-vaishig etries: M = " Let ^M =( ) 2P 3 be a poit (represetig the caera) ad let ^P i be the projectio atrix: " Wi 0 0 i # 0 0 W i Z i ^P = 0 W i 0 Y i Ad as i the geeral case we have the duality p i = MPi = ^P ^M i where the role of the otio (the caera) ad shape have bee switched. Let l i li 0 be two distict lies passig through the iage poit p i, i.e., p > i l i =0ad p > i l0 i =0, ad therefore we have l i > ^P ^M i =0 ad li 0> ^P ^M i =0.Fori =5 ::: 8 we have therefore E ^M =0where: E = l > ^P 5 5 l > ^P 8 8 l 0> 5 ^P 5 l 0> 8 ^P 8 Therefore the deteriat of ay 4 rows of E ust vaish. The choice of the 4 rows ca iclude 2 poits, 3 poits, or 4 poits (o top of the 4 basis poits P 1 ::: P 4 ) ad each such choice deteries a ultiliear costrait whose coefficiets are arraged i a tesor. The 8-poit tesor is whe 4 poits are chose: by choosig oe row fro each poit we # (1)

4 obtai a vaishig deteriat ivolvig 4 poits which provides 16 costraits (per view) l 5 i l6 j l7 k l8 t Qijkt =0for the 81 coefficiets of the tesor Q ijkt. The idices i j k l follow the covariat-cotravariat otatios (upper idex represets poits, lower represet lies) ad follow the suatio covetio (cotractio) u i v i = u 1 v 1 + u 2 v 2 + ::: + u v. The tesor cotais 81 coefficiets, however, they satisfy iteral ( sythetic borrowig fro [10]) liear costrait. Exactly how ay is a ope proble which we will show boils dow to the questio of di V ( k). Sice P 1 ::: P 4 are coplaar we have the costrait P i > =0, i =1 ::: 4 ad, due to our choice of coordiates, =( ) T. Cosider the faily of caera atrices M = u > for all choices of u =(u 1 u 2 u 3 ) >. I other words, the 4 th colu of M cosists of the arbitrary vector u ad all other etries vaish. Thus we have that MP either vaishes or is equal to u (up to scale) for all P. Let l i li 0 be lies through u, therefore l i > M jp = l i > ^P ^M j =0 li 0> M j P = li 0> ^P ^M j =0 for all poits P, ad dually for all projectio atrices ^P. Therefore the 4 4 deteriats of E vaish regardless of ^P i. We have a sigle tesor Q ijkt resposible for the 16 quadliear costraits li 5l6 j l7 k l8 t Qijkt =0(we have a choice of 2 lies for each poit, thus 16 costraits). Fro the discussio above, the four lies cotracted by the tesor are all coicidet with the arbitrary poit u. Therefore, the questio is what is the diesio of the set of costraits li 5l6 j l7 k l8 t Qijkt =0where the lies are arbitrary but for a 2-diesioal subspace? Recall the defiitio of V ( k) ad set =3 =4 k =2: V (3 4 2) = fv 1 v 2 v 3 v 4 j di Spafv 1 ::: v 4 g2g where v 1 ::: v 4 are vectors i R 3. Our questio regardig the uber of sythetic costraits is equivalet to the questio of what is the diesio of V (3 4 2)? 4 Dyaic P!P Mappigs Cosider a cofiguratio of poits i Q i 2 P ;1, i = 1 ::: q udergoig a projective appig Q i! Q 0 i. The it is well kow that Q0 i = AQ i where A 2 GL() is soe ivertible atrix. However, cosider the followig coplicatio where each poit Q i ay chage its positio up to a k-diesioal subspace (k =1eas that Q i is fixed, k =2eas that Q i ay chage its positio alog soe lie i P, ad so forth), ad we are give >2 observatios Q (j) i where j =1 :::. I other words, the observatios are geerated by a cobiatio of global (ukow) trasforatios A i 2 GL() ad local (ukow) oveets withi (ukow) subspaces of diesio up to k<. The task is to recover the global trasforatios A i fro the observatios. Q (j) i The defiitio above is a geeralizatio of particular cases which were itroduced i the past uder the ae of dyaic SFM, or SFM of ultiply ovig poits, ad the relevat literature icludes [1, 15, 19, 13, 17, 8, 14, 9, 18]. For istace, [15] cosider the case where =3(poits Q i belog to the 2D projective plae), =3ad k =2. I other words, a cofiguratio of coplaar poits are viewed by a ovig caera ad the poits ove alog arbitrary straight lies (k =2) or stay fixed ( static, k =1) while the caera chages positios. It was show there that the iage observatios (across three views) satisfy a tesorial costrait, where i the case where all poits are ovig alog alog lies, 26 observatios are sufficiet for a uique solutio to the tesor, whe all poits are static (without beig labeled as such) the those observatios fill a 10 diesioal subspace (thus at least 16 poits should be dyaic for a uique solutio for observatios).

5 I a later paper [19] the case of dyaic 3D to 3D aliget was itroduced, where =4 =3 k =2. I that case, the observatios are govered by a tesor, where the observatios fro ovig poits fill a 60-diesioal space (thus there 4 tesors satisfyig the costraits), ad static poits fill a 20-diesioal space. Aog the various aspects of those tesors, oe iportat aspect is the coutig of ecessary costraits for a solutio. Soe of those coutig issues, eve i the particular low diesio exaples give above, are ot obvious. The atter becoes fairly subtle whe dealig with the geeral dyaic P!P appigs where the issue of coutig costraits is a ope proble. We observe that sice tesor products coute with liear trasforatios, the issue of diesio coutig is idepedet of the atrices A i 2 GL(). Therefore, the geeral proble of coutig the costraits of a dyaic P ;1!P ;1 appig is isoorphic to the questio of di V ( k), where i this case k. Whe we copute the costraits of dyaic appigs we have other liitatios which are ot described i [15, 19] ad ca be also described i the V ( k) fraework. For exaple, i the case of dyaic P 2! P 2 aliget the collectio of easureets arisig fro triplets of atchig poits ust spa the 2D plae. We ay ask what is the largest uber of colliear poits allowed? (which beyod that the solutio becoes degeerate). I other words, the questio is how ay poits ovig o the sae striaght lie path will geerate liearly idepedet costraits. The aswer is di V (2 3 2) ote that = 2 because the effective diesio of the vector space is 2 eve though the poits are i defied i the 2D projective plae (i.e., = 3). Likewise, i the case of dyaic P 2! P 2 aliget the axial uber of poits allowed o a sigle lie is also di V (2 3 2) ad out of these poits di V (2 3 1) static poits will give us liearly idepedet costraits (i both cases). Fro the exaples above we have that di V (3 3 2)=26ad di V (4 3 2) = 60 (poit ovig alog straight lie paths) ad di V (3 3 1)=10ad di V (4 3 1) = 20 (static poits) for the 2D ad 3D cases, respectively. I the followig sectio we aalyze the structure of V ( k) ad as a result deterie di V ( k) for ay choice of k. 5 The Structure of V ( k) So far we have preseted two (urelated) Visio probles which are isoorphic to the di V ( k) questio. We will provide below the stateet ad proof about the structure of V ( k). The stateet appears very siilar to the classic result (see Appedix) of decoposig of V ito irreducible GL(V )-odules: V = M M ` t2t S t (V ) with the differece that ot all diagras are icluded oly those diagras for which k+1 =0. Clai 1 I particular V ( k) = M k+1 =0 di V ( k) = S (V ) f : k+1 =0 f s :

6 Proof: suppose ` ad k+1 =0. Let t be the tableau give by t(i j) = P i;1 l=1 l + j. Notig that V ( r 1) = Sy r V it follows that Therefore, hece, V a t = Sy 1 V Sy k V = V ( 1 1) V ( k 1) V ( k) : S t (V )=V a T b T V ( k) b T V ( k) M k+1 =0 S (V ) f V ( k): To show the other directio let ( ) be a heritia for o V ad let the iduced for o V be give by Note that (u 1 u v 1 v )= Y i=1 (u 1 ^^u v 1 v ) (u i v i ) : = 1! (u 1 ^^u v 1 ^^v ) = 1! det[(u i v j )] i j=1 : Let ` with k+1 6=0, the the cojugate partitio =( 1 2 ::: t ) satisfies 1 k +1. Let l j = P j r=1 r ad let t be the tableau give by t(i j) =l j;1 + i. The S t (V ) = V a t b t V b t = ^1 V ^l V : Suppose ow that v 1 ::: v 2 V satisfy di Spafv 1 ::: v gk. The v 1 ^^ v 1 =0therefore for ay u 1 ::: u 2 V ((u 1 u ) b T v 1 v )= ly r=1 1 r! ( l r^ i=l r;1+1 It follows that V ( k) is orthogoal to M hece, k+1 6=0 u i di V ( k) di l r^ i=l r;1+1 S (V ) f M k+1 =0 v i )=0: S (V ) f : Clai 1 ca be used to give explicit forulas for di V ( k) whe either k or ; k are sall. I the later case we write di V ( k) = ; k+1 6=0 f d () ad ote that the partitios of with k+1 6=0correspod to all partitios of all ubers up to ; k ; 1.

7 5.1 Exaples To calculate di V ( ; 1) ote that oly =(1 ) ust be excluded, thus: f (1 ) =1 d (1 ) () = hece, di V ( ; 1) = ; : To calculate di V ( ; 2) we ust exclude, i additio to the above, the partitio (2 1 ;2 ), thus: +1 f (2 1 ;2 ) = ; 1 d (2 1 ;2 )() =( ; 1) hece, di V ( ; 2) = +1 ; [ +( ; 1) 2 ]: To calculate di V ( ; 3) we ust exclude, i additio to the above, the partitios (3 1 ;3 ) ad (2 2 1 ;4 ), thus: ; 1 ; 1 +2 f (3 1 ;3 ) = d (3 1 2 ;3 )() = 2 Hece, ( ; 3) f (2 2 1 ;4 ) = 2 ( ; 3) +1 d (22 1 ;4 )() = 2 ; 1 di V ( ; 3) = +1 ; [ +( ; 1) 2 ; ( ; 3)2 +1 ]: 2 4 ; 1 + With these i id, we ca easily resolve the first of the ope probles which is the uber of sythetic costraits of the 8-poit shape tesor with 4 coplaar poits. We have see that the aswer is di V ((3 4 2): di V ((3 4 2) = 3= 4=0 f d where =( 1 ::: 4 ), is a partitio of 4, i.e., ad P i i =4.We have therefore oly three partitios which satisfy 3 = 4 =0: = (4) (2 2) (3 1) to cosider. Thus, f (4) = 1 d (4) =15 f (2 2) =2 d (2 2) =6 f (3 1) =3ad d (3 1) =15. Therefore, di V (3 4 2)= =72. We ca also verify the special cases of dyaic P 2!P 2 ad P 3!P 3 by substitutig the values of k i the forulas above. For exaple: di V (3 3 2) = 27 ; 1 = 26 ad di V (4 3 2) = 64 ; 4 = 60 (poit ovig alog straight lie paths) ad di V (3 3 1) = 27 ; ( ) = 10 ad di V (4 3 1) = 64 ; ( ) = 20 (static poits). Also di V (2 3 2) = 8 ; 0=8poits ovig alog oe lie path out of which up to di V (2 3 1) = 8 ; [0 + 4] = 4 are static poits o this lie will give us liearly idepedet costraits.

8 6 Suary We have show that certai o-obvious coutig probles exist i SFM literature such as the uber of sythetic costraits of the 8-poit shape tesor with 4 coplaar poits, ad the uber of costraits ecessary for the geeral dyaic P!P aliget proble ad i geeral i probles where the costraits of a ulti-liear proble occupy a low-diesioal subspace. We have show that a certai geeral questio lies at the heart of those coutig probles: Let V be a coplex -diesioal space ad for k cosider the GL(V )-odule V ( k) V defied by V ( k) =f v 1 v 2 V : di Spafv 1 ::: v gk g : We would like to deterie di V ( k) for ay choice of k. Thus, for istace we showed that di V (3 4 2) is the uber of sythetic costraits of the 8-poit shape tesor with 4 coplaar poits, ad di V ( k) stads for the uber of costraits of a P ;1!P ;1 aliget proble with appigs ad where the poits ove i a k ; 1 diesioal subspaces. We have the show that the questios of di V ( k) is aturally addressed i the cotext of represetatio theory by coutig the irreducibles of V over a subset of diagras of the partitio of. It is worthwhile otig that represetatio theory tools have ot bee used so far i the coputer visio literature, thus the fact that such probles exist i the cotext of visio tasks suggest that soe failiarity with these kid of tools would bear fruits also i future research. Refereces [1] S. Avida ad A. Shashua. Trajectory triagulatio: 3D recostructio of ovig poits fro a oocular iage sequece. IEEE Trasactios o Patter Aalysis ad Machie Itelligece, 22(4): , [2] S. Carlsso. Duality of recostructio ad positioig fro projective views. I Proceedigs of the workshop o Scee Represetatios, Cabridge, MA., Jue [3] S. Carlsso ad D. Weishall. Dual coputatio of projective shape ad caera positios fro ultiple iages. Iteratioal Joural of Coputer Visio, 27(3), [4] C. Rother ad S. Carlsso. Liear Multi View Recostructio ad Caera Recovery. I Proceedigs of the Iteratioal Coferece o Coputer Visio, Vacouver, Caada, July [5] A. Criiisi, I. Reid, ad A. Zissera. Duality, rigidity ad plaar parallax. I Proceedigs of the Europea Coferece o Coputer Visio, Frieburg, Geray, Spriger, LNCS [6] O. Faugeras ad Q.T. Luog with cotributios fro T. Papadopoulo. The geoetry of ultiple iages MIT Press, [7] W.Fulto ad J.Harris. Represetatio Theory: a First Course. Spriger-Verlag, [8] M. Ha ad T. Kaade. Recostructio of a Scee with Multiple Liearly Movig Objects.I Proc. of Coputer Visio ad Patter Recogitio, Jue, 2000.

9 [9] M. Ha ad T. Kaade. Multiple Motio Scee Recostructio fro Ucalibrated Views. I Proceedigs of the Iteratioal Coferece o Coputer Visio, Vacouver, Caada, July [10] R.I. Hartley ad A. Zissera. Multiple View Geoetry. Cabridge Uiversity Press, [11] M. Irai ad P. Aada. Parallax geoetry of pairs of poits for 3D scee aalysis. I Proceedigs of the Europea Coferece o Coputer Visio, LNCS 1064, pages 17 30, Cabridge, UK, April Spriger-Verlag. [12] M. Irai, P. Aada, ad D. Weishall. Fro referece fraes to referece plaes: Multiview parallax geoetry ad applicatios. I Proceedigs of the Europea Coferece o Coputer Visio, Frieburg, Geray, Spriger, LNCS [13] R.A. Maig ad C.R. Dyer. Iterpolatig view ad scee otio by dyaic view orphig. I Proceedigs of the IEEE Coferece o Coputer Visio ad Patter Recogitio, pages , Fort Collis, Co., Jue [14] D. Segal ad A. Shashua 3d recostructio fro taget-of-sight easureets of a ovig object see fro a ovig caera. I Proceedigs of the Europea Coferece o Coputer Visio, Dubli, Irelad, Jue [15] A. Shashua ad Lior Wolf. Hoography tesors: O algebraic etities that represet three views of static or ovig plaar poits. I Proceedigs of the Europea Coferece o Coputer Visio, Dubli, Irelad, Jue [16] D. Weishall, M. Wera, ad A. Shashua. Duality of ulti-poit ad ulti-frae geoetry: Fudaetal shape atrices ad tesors. I Proceedigs of the Europea Coferece o Coputer Visio, LNCS 1065, pages , Cabridge, UK, April Spriger-Verlag. [17] Y. Wexler ad A.Shashua. O the sythesis of dyaic scees fro referece views. I Proceedigs of the IEEE Coferece o Coputer Visio ad Patter Recogitio, South Carolia, Jue [18] Lior Wolf ad A. Shashua. O projectio atrices P k ;! P 2, k =3 6, ad their applicatios i coputer visio. I Proceedigs of the Iteratioal Coferece o Coputer Visio, Vacouver, Caada, July [19] Lior Wolf, A. Shashua, ad Y. Wexler. Joi tesors: o 3d-to-3d aliget of dyaic sets. I Proceedigs of the Iteratioal Coferece o Patter Recogitio, Barceloa, Spai, Septeber A A Represetatio Theory Digest I this sectio we briefly recall soe relevat facts cocerig the represetatio theory of the geeral liear group. For a thorough itroductio see [7]. Let V be a fiite -diesioal vector space over the coplex ubers. The collectio of ivertible atrices is deoted by GL() which is the group of autoorphiss of V deoted by GL(V ). The vector space V (-fold tesor product) is spaed by decoposable tesors of the for v 1 v, where the vectors v i are i V. Hece the diesio of V is. The vector space V is the -fold direct su of V, thus is of diesio. The exterior powers ^V of V,, is the vector space spaed by the iors of the atrix [v 1 ::: v ] where the vectors v i are i V. Hece the diesio of ^V is ;. The exterior powers are the iages of the ap V! V give by (v 1 v )! 2S sg()v (1) v ()

10 where S deotes the syetric group (of perutatios of letters). The syetric powers Sy V are the iages of the ap V! V give by (v 1 v )! v (1) v () 2S ; Hece the vector space Sy +;1 V is of diesio. Note that, V V ; =Sy 2 V ^2V ; with the appropriate diesio: 2 +1 = This decopositio ito irreducibles (see later) is ot true for V, > 2. The reaider of this sectio is devoted to the ecessary otatio for represetig V as a decopositio of irreducibles. A represetatio of a group G o a coplex fiite diesioal space U is a hooorphis G to GL(U ) - the group of liear autoorphiss of U. The actio of g 2 G o u 2 U is deoted by g u. The G;odule U is irreducible if it cotais o o-trivial G;ivariat subspaces. Ay fiite diesioal represetatio of a copact group G ca be decoposed as a direct su of irreducible represetatios. This basic property called coplete reducibility also holds for all holoorphic represetatios of the geeral liear group GL(V ). The ai focus of this paper is the space V ( k) = Spafv 1 v 2 V : di Spafv 1 ::: v gk g : Sice V ( k) is ivariat uder the GL(V ) actio give by g v 1 v = g(v 1 ) g(v ) it is atural to study its structure by decoposig it ito irreducible GL(V )- odules. The descriptio of the fiite diesioal irreducible represetatios (irreps) of GL(V ) depeds o the Cobiatorics of partitios ad Youg diagras which we ow describe: AP partitio of is a ordered set = ( 1 ::: k ) such that 1 ::: k 1 ad i =. A partitio is represeted by its Youg diagra (also called shape) which cosists of k left aliged rows of boxes with i boxes i row i. The cojugate partitio = ( 1 ::: r ) to a partitio is defied by iterchagig rows ad colus i the Youg diagra or without referece to the diagra, i is the uber of ters i that are greater tha or equal to i. A assiget of the ubers f1 ::: g to each of the boxes of the diagra of, oe uber to each box, is called a tableau. A tableau i which all the rows ad colus of the diagra are icreasig is called a stadard tableau. We deote by f the uber of stadard tableaux o, i.e., the uber of ways to fill the youg diagra of with the ubers fro 1 to, such that all rows ad colus are icreasig. Let (i j) deote the coordiates of the boxes of the diagra where i =1 :: k deotes the row uber ad j deotes the colu uber, i.e., j =1 ::: i i the i th row. The hook legth h ij of a box at positio (i j) i the diagra is the uber of boxes directly below plus the uber of boxes to the right plus 1 (without referece to the diagra, h ij = i + j ; i ; j +1). The,! f = Q h (i j) ij where the product of the hook-legths is over all boxes of the diagra. We deote by d () the uber of sei-stadard tableaux which is the uber of ways to fill the diagra with the ubers fro 1 to, such that all rows are o-decreasig ad all colus are icreasig. We have: Y ; i + j d () = : h ij (i j)

11 Let S deote the syetric group o f1 ::: g. The group algebra C S is the algebra spaed by the eleets of S C G = f j 2 C g 2S where additio ad ultiplicatio are defied as follows: ( )+( )= ( + ) 2S 2S 2S ad ( )( )= ( )g 2S 2S g2s g= for 2 C. Let t be a tableau o (a uberig of the boxes of the diagra) ad let P (t) deote the group of all perutatios 2 S which perute oly the rows of t. Siilarly, let Q(t) deote the group of perutatios that preserve the colus of t. Let a t b t be two eleets i the group algebra C S defied as: a t = g2p (t) g b t = g2q(t) sg(g)g: The group algebra C S acts o V o the right by perutig factors, i.e., (v 1 v ) = v (1) v (). For a geeral shape ad a tableau t o the iage of a t, V a t, is the subspace: V a t =Sy 1 V Sy k V V ad the iage of b t is V b t = ^1 V ^r V V where is the cojugate partitio to. The Youg syetrizer is defied by c t = a t b t 2 C S. The iage of the Youg syetrizer S t (V )=V c t is the Schur Module associated to t ad is a irreducible GL(V )- odule. The isoorphis type of S t (V ) depeds oly o the shape so we ay write S t (V ) = S (V ). It turs out that all the polyoial irreps of GL(V ) are of the for S (V ) for soe ad a partitio `. Let T deote the set of stadard tableaux o the the direct su decopositio of V ito irreducible GL(V )-odules is give by V = M ` Sice d () = di S (V ) it follows that M M ` t2t S (V ) f : di V = = S t (V ) = ` d ()f : For exaple, cosider = =3, i.e., V V V where di V =3. There are three possible partitios of 3 these are (3) (1 1 1) ad (2 1). Fro the above, S (3) (V )= Sy 3 V ad S (1 1 1) V = ^3V. There are two, f (2 1) =2, stadard tableaux for =(2 1) ad these are 123 ad 132 (uberig of boxes left to right ad top to botto). There are eight, d (2 1) (3) = 8, sei-stadard tableaux which are: , ,223 ad 233. We have the decopositio: V V V =Sy 3 V ^3V (S (2 1) V ) 2 with the appropriate diesios: 27=10+1+(8+8).

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