Some properties of symmetry classes of tensors

Transcription

1 The d Aual Meetg Mathematcs (AMM 07) Departmet of Mathematcs, Faculty of Scece Chag Ma Uversty, Chag Ma Thalad Some propertes of symmetry classes of tesors Kulathda Chmla, ad Kjt Rodtes Departmet of Mathematcs, Faculty of Scece Naresua Uversty, Phtsaulok 65000, Thalad Abstract I ths artcle, we vestgate a property of orbtal subspace o symmetry classes of tesors assocated wth a fte group ad rreducble character We dscuss some propertes of operator o -module correspodg to the exstece of o*-bass o tesor space We also show that there s a subspace of -module beg somorphc to a subspace of -folds tesor space Keywords: Symmetry classes of tesors, Irreducble character, Orbtal subspace, o*-bass 00 MSC: 5A69, 5A03, 5A8 Itroducto Symmetry classes of tesors assocated to a fte group, ts rreducble character ad fte dmesoal er product space V, V( ), s a subspace of -fold tesor space V Precsely, t s the mage of the cetral dempotet elemet T (, ) of the group algebra Symmetry classes of tesors have varous applcatos mathematcs ad appled mathematcs For stace, [] Jafar ad Madad studed symmetry classes of tesors as a - module ad the obtaed the correspodg character They appled ther results to geeralze fermat's lttle theorem I [] Darafsheh ad Pourak showed that the dmeso of symmetry classes of tesors assocated wth a certa cyclc subgroup of S ca be expressed as the Euler fucto ad the Mobus fucto I [3] Cheug, Duffer ad L studed duced operator ad characterzed multplcatve map : M M preservg geeralzed matrx fuctos o M, where M s the set of all complex matrces I [4] Torres ad Slva exploted the structure of the crtcal orbtal sets of symmetry classes of tesors assocated to sg uform parttos ad they establshed ew coectos betwee symmetry classes of tesors, matchg o bpartte graphs ad codg theory Normally, symmetry classes of tesors have bee vestgated may aspects The duced operator was studed by may authors for example, [5] by Merrs ad Watks, [6] by L ad Zahara, [3] by Cheug, Duffer ad L etc The dmesos of symmetry classes of tesors have bee studed [7],[8],[9],[] Oe of the most actve topcs s the problems of fdg a ecessary ad suffcet codto for the exstece of a specal bass (orthogoal *-bass or o*-bass) for the symmetry classes of tesors assocated wth a fte group ad a rreducble character Correspodg author Speaker E-mal address: kulathda_u@hotmalcom (K Chmla), kjtr@uacth (K Rodtes) Proceedgs of AMM 07 AL 03

2 The o*-bass has bee studed for several classes of groups: for example, dcyclc groups [9], certa groups [8], dhedral groups [0], sem-dhedral groups [] ad youg subgroups [] By [3] olmes studed assumpto o a group that esures that o matter how the group s embedded a symmetrc group, the correspodg symmetrzed tesor space has a o*-bass e troduced operator -module ad studed o-bass group va these operator Later he used o- bass group lk to the exstece of o*-bass o a tesor space Motvated by [3] we provded a addtoal structure to the -module Also, uder the ew structure, we show that there s a subspace of -module that s somorphc to a subspace of -folds tesor space Prelmares Let V be a m -dmesoal vector space over, a subgroup of the full symmetrc group S, ad V V V V For ay, defe the operator P : V V by P ( v v ) v v () ( ) ad P s called the permutato operator assocated wth o V Now f s ay complex rreducble character of, the the operator o V, () T (, ) ( ) P s called the symmetrzer assocated wth ad ad the mage of V( ): T(, )( V ) s called the symmetry class of tesors assocated wth ad Note that the elemets V( ) of the form T (, ), v v T(, )( v v ) are called decomposable symmetrzed tesors Such a oto does ot reflect ts depedece o ad Sce V V( ), Irr( ) we have V( ) s spaed by the decomposable symmetrzed tesors v v [4] Fx postve tegers m, ad put There s a rght acto of o, m, : ( (),, ( )), m,,, m gve by (),, ( ) ;, m, Suppose that e,, e m s a orthogoal bass of V To avod trvaltes, we assume that m For, put : () ( ) e e e The, by [4], t s well kow that e : m, The er product o V duces a er product o V( ) s gve by ad () (*,*) u v () u, v s a bass for V V By [4] the duced er product o t () t t * * () ( e, e) ( ),,, m,, m, Proceedgs of AMM 07 AL 03

3 If W s a subspace of V, the a orthogoal bass of W cosstg etrely of stadard symmetrzed tesors, s called a o-bass, orthogoal*-bass or o*-bass of W (relatve to ) Let A( ) ( a ( )) be ay complex rreducble represetato of affordg For ay j j j (), we defe the subspace V ( ) of V as the mage of the followg operator By [5], t s proved that j rs jr s these relatos that j V( ) V( ) ad () Tj aj( ) P * TT T, T T ad T (, ) T I [7] they deduced from () V( ) V ( ) () * For ay, we ow subdvde V to subspaces For ay (), let V T ( e ): j () So, by [7], j () * V V V ( ) V If T j ( e): j J s a bass for V for some subset J of bass for V, ad hece the set s a bass for V ( ), [7] j( ): T e j J,, (), the j ( ): T e j J s a 3 Ma Results 3 Relato betwee tesor space ad coset space Let be a fte group ad let be a subgroup of Let Irr( ) deote the set of rreducble character of, Irr( ) Suppose that A( ) ( a j ( )) s a utary represetato of that affords Frst we cosder the operator T o V ( ) Lemma 3 Let, m, Let be a rreducble character of ad fx () The the statemets below hold: () T ( e), T ( e ) a ( ),, () T ( e ), T ( e ) a ( ) Proof * T ( e), T( e ) TT( e), e T( e), e () () a ( ) P ( e ), e a ( ) P ( e ), e Proceedgs of AMM 07 AL 03 3

4 By () () () T e T e a a ( ) e, e a ( ), ( ), ( ) ( ),,, ad summg over the set () follows Lemma 3 Let be a represetatve of the orbt of the acto o m, The where T ( ): T ( e ), for all, V ( ) T, ( ), m Proof Let Orb( ) We set T ( ): T ( e ) ad T ( ): T ( e ) ', for all, m, Clam that T ( ) T ( ) Note that dm dm T ( e ) a ( ) P ( e ) a ( ) e ad dm T( e' ) a ( ) e ' () Sce, are the same orbt, we obta that g 0 for some g 0 (3) dm Substtutg (3) (), we have T( e' ) a( ) eg0' (4) Let g ad g g 0 ' The g ad g g 0 ' So, dm dm T( e) a( g ) eg ad T ( e' ) a ( g g0') eg g g Thus T ( e ) T ( e ), where ' g Ths mples that T ( ) T ( ) Therefore ' 0 V ( ) T, ( ) For each subgroup of ad ab,, we defe a form B o by puttg dm B(,) a b a( b ha), h extedg learly the frst compoet ad atlearly the secod Proposto 33 Proof all B s a well-defed -varat hermta form Frst ote that sce s a utary represetato, we have a () a a ( a ) for a ad dm Clearly c ad ab, ; B s well-defed To show that () B s -varat, let Proceedgs of AMM 07 AL 03 4

5 Fally, we show that dm B( ca, cb) a( b c hca) h dm a( b h' a), h' c h' B (,) a b B s hermta form For ab,, dm B(, b a) a( a hb) h dm a( b h a) h dm a( b ka) k B (,) a b B s a hermta form Therefore Put C : ker a well-defed form B o for B, where ker (, ) 0, B gve by B x B x y y The B x y B x y, (, ) (, ) xy,, where here ad below we use x to deote s closed uder the acto of ad so we have a well-defed acto of o -varat Theorem 34 ) ) The form Proof Let (, j-etry ) dm C a g hg, dm : ( ) g h B s postve defte ad let,, B duces x ker B By Proposto 33, ker B C Clearly B s a g g be a bass of Let M be the matrx wth B( g g j) We clam that M M The ( j-etry, ) of dm M s B ( g, g ) B ( g, g ) a ( g cg ) a ( g lg ) k k j k j k k k c l dm a g cg a g lc l c k ( k ) ( j g k ) (5) Sce k j k k j k k k a ( g c g ) a ( g lc g ) a ( g c g ) a ( g lc g ) for ay c, c, we have c k Replacg g g cg, we have to k a( gk cg ) a( gj lc g k ) a( gk cg ) a ( gj lc g k ) k g g c g k ad so j k j a g lc g a g lg g The (5) reduces Proceedgs of AMM 07 AL 03 5

6 dm dm dm a () g a ( gj lgg ) a ( ) (( ) ) g a g gj lg l g l g l a ( g lg ) dm j dm a ( g j lg ) l dm dm a ( gj lg ) l B( g gj) Thus M M as clamed As M s hermta, t s smlar to a dagoal matrx wth the egevalues of M alog the ma dagoal Sce M M, a egevalue of M s ether or 0 ece the rak of M s equal to the trace of M, dm tr M B( gk, gk) a( g hg) k g h Sce dm C C rakm, () follows Fally sce egevalue of M s ether or 0, the form B o s postve semdefte, so that the duced form B o C s postve defte Ths proves () Theorem 35 Let m,, fxed () ad each utary represetato The T ( e) 0 f ad oly f ker B I partcular, T ( ) kerb Proof ( ) Assume that Ths mples that T ( e ) 0 The () () a() P( e ) a() e 0 a () e 0 Set we obta that For, we defe o by f a e ( ) 0 relato Thus s parttoed to equvalet classes, ad we have [ ] Let S be a set of represetatve of rght coset of by So Clearly s a equvalet S t We assume S,,, t ad the (dsjot uo), for all,, t Furthermore g h for all gh, [ ] Therefore, 0 a ( ) e a ( h) e a ( h ) e a ( t h ) e t h h h Sce e ' s are learly depedet, ths mples that a ( h ) 0 for all j,, t Thus j h a g h g mples that ( ) 0, h j B (, g) 0, g So, ker B dm ( ) Assume that ker B The B (, ) a( h ) 0, h Proceedgs of AMM 07 AL 03 6

7 Thus, h Settg t a( h) 0, Sce l h l ad h a h, we have ( l ) 0 a( l h) e 0 Note that [ ] l l ad l ( l h), h Thus a ( h) e 0, l,,, t, h l Corollary 36 If h, we have l h, the l a ( ) e 0 Therefore dm( T ( )) () T ( e ) 0 3 O-bass group ad the exstece of o # -bass We call s a o-bass group f for every ad for every rreducble utary represetato ( a ( )) for all (), the vector space C has a bass that s orthogoal j relatve to B cosstg etrely of elemets of the form a( a ) Such a bass shall be called a o-bass of C I ths secto, we say that V ( ) has a o # -bass f for every m,, there exsts g,, gt wth dm A A t C such that ( g j ): T e j t s a orthogoal bass of T ( ) Now we study relatos betwee o-bass group ad the exsteces of o # -bass of symmetrzed tesor spaces Theorem 3 If s a o-bass group ad : S ( ) s a homomorphsm, the V has a o # -bass relatve to ( ) Proof Assume that s a o-bass group ad : S ( ) s a homomorphsm Put J ( ) ad fx beg a utary represetato wth : J L ( ) It s eough to show that V( J ) has a o # -bass (relatve to J ) Set ( J ), where J s the stablzer of uder the rght acto of J o m, Also put A beg a utary represetato of By the defto of o-bass group, C has a o-bass a,, a t ad by Theorem 34, dm t a( g hg) g h Put ( ) we have a S We clam that ( ): j j j T e j t s a o # -bass of V ( J ) For jk, t, dm B( aj, ak) a( ak haj) h dm a( k j ) J J dm a( ) J k Jj T ( e ), T ( ) j e k J The equatos above wth j k shows that each T ( e ) s ozero j Proceedgs of AMM 07 AL 03 7

8 O the other had, the equatos above wth j k show that the vectors T( e j ) are mutually orthogoal I partcular, the set T( e ): j t j s learly depedet By Theorem 35, the theorem follows I the ext theorem, we characterze o-bass group expressed terms of symmetrzed tesors The Cayley embeddg : S s the homomorphsm that takes g to the permutato ( ) o gve by ()() g h gh,( h ) Ths permutato s vewed as a elemet of S, where Theorem 3 Let be a fte group, let The s a o-bass group f ad oly f Proof ( ) Follows by Theorem 3 ( ) Assume that m as the set of fuctos from to, ad let S be the cayley embeddg : V has a o # -bass relatve to ( ) V has a o # -bass relatve to ( ) Fx ad Irr( ) We vew,,,m usg the same oe-to-oe correspodece,, by whch we detfy the symmetrc group o wth S Defe m, by, g, g, g Them clearly the stablzer of s Put A ( ) beg a utary represetato of ( ) By assumpto, T ( ) has a o # -bass, that s, there exsts g,, gt wth dm t a( g hg) g h such that T ( e( g j )): j t s a orthogoal bass of T ( ) The computatos the proof of Theorem 3 show that for jk, t, B ( g, g ) T ( e ), T ( e ), j k j k J ad by Theorem 35 mples that gj : j t s a o-bass group of C Refereces [] Jafar M ad Madad AR(03), Symmetry classes of tesors as group modules, Joural of Algebra, 393, [] Darafsheh, MR ad Pourak MR(998), O the dmeso of cyclc symmetry classes of tesors, Joural of Algebra, 05, [3] Cheug, WS, Duffer, A ad L CK(005), Multplcatve preservers ad duced operators, Lear Algebra ad ts Applcatos, 40, [4] Torres, MM ad Slva, PC(0), Tesors, matchgs ad codes, Lear Algebra ad ts Applcatos, (6)436, [5] Russell, M ad Wllam, W(98), Elemetary dvsors of duced Trasformatos o symmetry classes of tesors, Lear Algebra ad ts applcatos, 38, 7-6 [6] Ch-Kwog, L ad Alexadru, Z(00), Iduced operators o symmetry classes of tesors, IEICE Trasactos of the Amerca Mathematcal Socety, ()354, Proceedgs of AMM 07 AL 03 8

9 [7] Bo-Yg, W ad Mg-Peg, (99), The subspaces ad orthomormal bases of symmetry classes of tesors, Lear ad Multlear Algebra, 30, [8] olmes, RR ad T-Yau T(99), Symmetry classes of tesors assocated wth certa groups, Lear ad Multlear Algebra, 30, [9] Darafsheh, MR ad Pourak MR(995), O the orthogoal bass of the symmetry classes of tesors assocated wth the dcyclc group, Mathematcs Subject Classfcato, - [0] Poursalavat NS(04), O the symmetry classes of tesors assocated wth certa frobeus groups, Pure add Appled Mathematcs Joural, 3, 7-0 [] ormoz ad Rodtes, K(03), Symmetry classes of tesors assocated wth the semdhedral groups SD 8, Colloquum Mathematcum, 3, [] Shahryar M ad Zama Y(0), Symmetry classes of tesors assocated wth youg subgroups, Asa-Europea Joural of Mathematcs, 4, [3] olmes, RR(004), Orthogoalty of cosets relatve to rreducble characters of fte groups, Lear ad Multlear Algebra, 5, [4] Tam, TY(0), Multlear Algebra, Departmet of Mathematcs ad Statstcs, Aubur Uversty [5] Merrs, R(997), Multlear Algebra, ordo ad Breach Scece Publshers Proceedgs of AMM 07 AL 03 9