Mathematics - 341 الصرفة و التطبيقيةمجلة إبن الهيثم للعلوم 2012 السنة 25 المجلد 3 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 3 Vol. 25 Year 2012 The Reduction of Resolution of Weyl Module from Characteristic-Free Resolution in Case (4,4,3) H. R. Hassan Department of Mathematics, College of Science, Al-Mustansiriya University Received in: 8 January 2012 Accepted in: 22 April 2012 Abstract In this paper we study the relation between the resolution of Weyl Module FK )3,4,4( in characteristic-free mode and in the Lascoux mode (characteristic zero), more precisely we obtain the Lascoux resolution of FK )3,4,4( in characteristic zero as an application of the resolution of FK )3,4,4( in characteristic-free. Key word : Resolution, Weyl module, Lascoux mode, divided power, characteristic-free. Introduction Let R be a commutative ring with 1 and F be free R-module by FDn we mean the divided power of degree n. The resolution Res[ p, q, r, 21,tt ] of Weyl module FK µλ / associated to the three-rowed skew-shape 1 2 2 1 2 2( , , ) / ( , , 0)p t t q t r t t t+ + + + , namely , the shape represented by the diagram 1t p 2t q r In general, the Weyl module µλ /K is presented by the "box" map FDDFD KFDFDFD FDDFD trtqp d rqp rktq k ktp   −−++ > ′ −− > ++ ⊗⊗  →⊗⊗→ ⊗⊗ ∑ ∑ 22 / 11 0 / 0 µλ µλ where the maps FDFDFDFDDFD rqprktq k ktp ⊗⊗→⊗⊗∑ −− > ++ 11 0 may be interpreted as Kth divided power of the place polarization from place 1 to place2(i.e. )(32 k∂ ),the maps FDFDFDFDDFD rqptrtqp ⊗⊗→⊗⊗ −−++∑  22 may be place 2 interpreted as th divided power of the place polarization from place 2 to 3 (i.e. )(32 ∂ ) [1]. Buchsbaum in [1], Hassan in [2] studied the resolution of Weyl module in the case of the partition (2,2,2) and (3,3,3) respectively. In section two of this paper we review the terms of characteristic-free resolution of Weyl module in the case of the partition (4,4,3). Mathematics - 342 الصرفة و التطبيقيةمجلة إبن الهيثم للعلوم 2012 السنة 25 المجلد 3 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 3 Vol. 25 Year 2012 In section three we apply this resolution to the Lascoux resolution in the same case by using the way in [1] and [2] with capelli identities [3]. Note: We have to mention that we shall use Dn instead of DnF to refer to divided power algebra of degree n. Characteristic-Free Resolution of the Partition (4,4,3) In this section, we find the terms of the resolution of Weyl module in the case of the partition (4,4,3). In general the terms of the resolution of Weyl module in the case of a three- rowed partition (p,q,r) which appeared in [3] are ⊕⊗+++∑⊕⊗ −− ≥ + 1 0 )1( 32 ])1;1,([sRe]);,([sRe     rr DqpyZDoqp ⊗−++++∑ ≥≥ ++ ]);1,1([sRe 1221 ,0 )1( 31 )1( 32 121 12    qpzZZ )2( 21 ++− rD where x, y and z stand for the separator variables, and the boundary map is zyx ∂+∂+∂ . Let again Bar(M,A,S) be the free bar module on the set },,{ zyxS = consisting of three separators x, y and z where A is the free associative (non-commutative) algebra generated by 3221, ZZ and 31Z and their divided powers with the following relations: )( 32 )( 31 )( 31 )( 32 abba ZZZZ = and )(21 )( 31 )( 31 )( 21 abba ZZZZ = and the module M is the direct sum of tensor product of divided power module rq DD ⊗ for suitable p, q and r with the action of 3221, ZZ and 31Z and their divided powers. We'll consider the case when 4,4 == qp and 3=r . In this case, we see that the index  in the first sum runs from 0 to 2, while the indices 1 and 2 in the second sum are very restricted : 1 must be 0, and 2 can be only 0 and 1 (since 12  ≥ and 121 ≤+  ). These comments are true whatever the values of p and q, then in this case we have ⊕⊗+++∑⊕⊗ −− ≥ + 13 0 )1( 323 ])1;14,4([sRe])0;4,4([sRe     DyZD ⊗−++++∑ ≥≥ + ]);14,14([sRe 1221 ,0 )1( 32 121 2    yZ )2(3 21 ++− D So =⊗+++∑ −− ≥ + 13 0 )1( 32 ])1;14,4([sRe     DyZ 0 )3( 321 )2( 32232 ])3;7,4([sRe])2;6,4([sRe])1;5,4([sRe DZDyZDyZ ⊗⊕⊗⊕⊗ Where yZ )2(32 is the complex 00 )2( 323232 →→→ yZyZZ and yZ )3(32 is the complex 00 )3(32 )2( 323232 )2( 32323232 →→⊕→→ yZyyZZyyZZyyZyZZ Then in this case we have the following terms  In dimension zero (M0) we have 344 DDD ⊗⊗ Mathematics - 343 الصرفة و التطبيقيةمجلة إبن الهيثم للعلوم 2012 السنة 25 المجلد 3 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 3 Vol. 25 Year 2012  In dimension one (M1) we have 344 )( 21 DDDxZ bb b ⊗⊗ −+ with b=1,2,3,4 and bb b DDDyZ −+ ⊗⊗ 344 )( 32 with b =1,2,3  In dimension two (M2) we have the sum of the following terms • ;344 )( 21 )( 21 21 DDDxZxZ bb bb ⊗⊗ −+ with .4,3,221 =+= bbb • ;254 )( 2132 DDDxZyZ bb b ⊗⊗ −+ with b= 2,3,4,5. • ;344 )( 32 )( 32 21 bb bb DDDxZyZ −+ ⊗⊗ with .3,221 =+= bbb • ;164 )( 21 )2( 32 DDDxZyZ bb b ⊗⊗ −+ with b= 3,4,5,6. • ;074 )( 21 )3( 32 DDDxZyZ bb b ⊗⊗ −+ with b= 4,5,6,7. • ;24531 )( 32 bb b DDDzZyZ −+ ⊗⊗ with b= 1,2. In dimension three (M3) we have the sum of the following terms: • ;344 )( 21 )( 21 )( 21 321 DDDxZxZxZ bb bbb ⊗⊗ −+ with .4,3321 =++= bbbb • ;254 )( 21 )( 2132 21 DDDxZxZyZ bb bb ⊗⊗ −+ with 5,4,321 =+= bbb and 21 ≥b • ;164 )( 21 )( 21 )2( 32 21 DDDxZxZyZ bb bb ⊗⊗ −+ with 6,5,421 =+= bbb and 31 ≥b • ;164 )( 213232 DDDxZyZyZ bb b ⊗⊗ −+ with b= 3,4,5,6. • ;074 )( 21 )( 21 )3( 32 21 DDDxZxZyZ bb bb ⊗⊗ −+ with 7,6,521 =+= bbb and 41 ≥b • ;074 )( 21 )( 32 )( 32 21 DDDxZxZyZ kk kbb ⊗⊗ −+ with 321 =+ bb and 7,6,5,4=k • 074323232 DDDyZyZyZ ⊗⊗ • ;155 )( 213132 DDDxZzZyZ bb b ⊗⊗ −+ with b= 1,2,3,4,5. • ;065 )( 2131 )2( 32 DDDxZzZyZ bb b ⊗⊗ −+ with b= 2,3,4,5,6. • 065313232 DDDzZyZyZ ⊗⊗ . In dimension four (M4) we have the sum of the following terms: • 30821212121 DDDxZxZxZxZ ⊗⊗ • ;254 )( 21 )( 21 )( 2132 321 DDDxZxZxZyZ bb bbb ⊗⊗ −+ with 5,4321 =++= bbbb and 21 ≥b • ;164 )( 21 )( 21 )( 21 )2( 32 321 DDDxZxZxZyZ bb bbb ⊗⊗ −+ with 6,5321 =++= bbbb and 31 ≥b . Mathematics - 344 الصرفة و التطبيقيةمجلة إبن الهيثم للعلوم 2012 السنة 25 المجلد 3 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 3 Vol. 25 Year 2012 • ;164 )( 21 )( 213232 21 DDDxZxZyZyZ bb bb ⊗⊗ −+ with 6,5,4321 =++= bbbb and 31 ≥b . • ;074 )( 21 )( 21 )( 21 )3( 32 321 DDDxZxZxZyZ bb bbb ⊗⊗ −+ with 7,6321 =++= bbbb and 41 ≥b . • ;074 )( 21 )( 21 )( 32 )( 32 2121 DDDxZxZyZyZ bb bbcc ⊗⊗ −+ with ,321 =+ cc 7,6,5321 =++= bbbb and 41 ≥b . • ;074 )( 21323232 DDDxZyZyZyZ bb b ⊗⊗ −+ with b= 4,5,6,7. • ;155 )( 21 )( 213132 21 DDDxZxZzZyZ bb bb ⊗⊗ −+ with .5,4,3,221 =+= bbb • ;065 )( 21 )( 2131 )2( 32 21 DDDxZxZzZyZ bb bb ⊗⊗ −+ with 6,5,4,321 =+= bbb and 21 ≥b . • ;065 )( 21313232 DDDxZzZyZyZ bb b ⊗⊗ −+ with b= 2,3,4,5,6. In dimension five (M5) we have the sum of the following terms: • 1010212121 )3( 21 )2( 32 DDDxZxZxZxZyZ ⊗⊗ • ;164 )( 21 )( 21 )( 213232 321 DDDxZxZxZyZyZ bb bbb ⊗⊗ −+ with 6,5321 =++= bbbb and 31 ≥b . • 0011212121 )4( 21 )3( 32 DDDxZxZxZxZyZ ⊗⊗ • ;074 )( 21 )( 21 )( 21 )( 32 )( 32 32121 DDDxZxZxZyZyZ bb bbbcc ⊗⊗ −+ with ,321 =+ cc 7,6321 =++= bbbb and 41 ≥b . • ;074 )( 21 )( 21323232 21 DDDxZxZyZyZyZ bb bb ⊗⊗ −+ with 7,6,5321 =++= bbbb and 41 ≥b . • ;155 )( 21 )( 21 )( 213132 321 DDDxZxZxZzZyZ bb bbb ⊗⊗ −+ with .5,4,3321 =++= bbbb • ;065 )( 21 )( 21 )( 2131 )2( 32 321 DDDxZxZxZzZyZ bb bbb ⊗⊗ −+ with 1 2 3 4,5, 6b b b b= + + = and 21 ≥b . • ;065 )( 21 )( 21313232 21 DDDxZxZzZyZyZ bb bb ⊗⊗ −+ with 6,5,4,321 =+= bbb and 21 ≥b . In dimension six (M6) we have the sum of the following terms: • ;0011212121 )4( 21 )( 32 )( 32 21 DDDxZxZxZxZyZyZ cc ⊗⊗ with .321 =+ cc Mathematics - 345 الصرفة و التطبيقيةمجلة إبن الهيثم للعلوم 2012 السنة 25 المجلد 3 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 3 Vol. 25 Year 2012 • ;074 )( 21 )( 21 )( 21323232 321 DDDxZxZxZyZyZyZ bb bbb ⊗⊗ −+ with 7,6321 =++= bbbb and 41 ≥b . • ;155 )( 21 )( 21 )( 21 )( 213132 4321 DDDxZxZxZxZzZyZ bb bbbb ⊗⊗ −+ with 5,4 4 1 =∑= =i ibb . • ;065 )( 21 )( 21 )( 21 )( 2131 )2( 32 4321 DDDxZxZxZxZzZyZ bb bbbb ⊗⊗ −+ with 6,5 4 1 =∑= =i ibb and 21 ≥b . • ;065 )( 21 )( 21 )( 21313232 321 DDDxZxZxZzZyZyZ bb bbb ⊗⊗ −+ with 6,5,4321 =++= bbbb and 21 ≥b . In dimension seven (M7) we have the sum of the following terms: • 101021212121213132 DDDxZxZxZxZxZzZyZ ⊗⊗ . • 001121212121 )2( 2131 )2( 32 DDDxZxZxZxZxZzZyZ ⊗⊗ . • ;065 )( 21 )( 21 )( 21 )( 21313232 4321 DDDxZxZxZxZzZyZyZ bb bbbb ⊗⊗ −+ with 6,5 4 1 =∑= =i ibb and 21 ≥b . And finally in dimension eight ( M8)we have • 001121212121 )2( 21313232 DDDxZxZxZxZxZzZyZyZ ⊗⊗ . As in [3], it is necessary to introduce a quotient of Bar complex modulo the Capelli identities relations; the proof these relation are compatible with the boundary map zyx ∂+∂+∂ is complicated and proved in [3]. Lascoux resolution for (4,4,3) The Lascoux resolution of the Weyl module associated to the partition (4,4,3) looks like this 335236 344146 254155 00 DDDDDD DDDDDD DDDDDD ⊕⊗⊕⊗ →⊕⊗→⊕→⊕→⊕⊗→ ⊕⊗⊕⊗ where the position of the terms of the complex determined by the length of the permutations to which they correspond. The correspondence between the terms of the resolution above and permutations is as follows ⇔⊕⊗ 344 DDD identity ⇔⊕⊗ 425 DDD (1 2) ⇔⊕⊗ 353 DDD (2 3) ⇔⊕⊗ 155 DDD (1 2 3) Mathematics - 346 الصرفة و التطبيقيةمجلة إبن الهيثم للعلوم 2012 السنة 25 المجلد 3 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 3 Vol. 25 Year 2012 ⇔⊕⊗ 326 DDD (2 1 3) ⇔⊕⊗ 146 DDD (1 3) Now, the terms can be presented as below, following Buchsbaum method [1]. 00 AM = 111 BAM ⊕= 222 BAM ⊕= 333 BAM ⊕= and jj BM = ; for j=4,5,6,7,8. Where the sA are the sums of the Lascoux terms, and the sB are the sums of the others. Now, we define the map 1σ from 1B to 1A as follows • );( 2 1 )( 2121 )2( 21 vxZvxZ ∂ where 326 DDDv ⊕⊗∈ • );( 3 1 )( )2(2121 )3( 21 vxZvxZ ∂ where 317 DDDv ⊕⊗∈ • );( 4 1 )( )3(2121 )4( 21 vxZvxZ ∂ where 308 DDDv ⊕⊗∈ • );( 2 1 )( 3232 )2( 32 vyZvyZ ∂ where 164 DDDv ⊕⊗∈ • );( 3 1 )( )2(3232 )3( 32 vyZvyZ ∂ where 074 DDDv ⊕⊗∈ We should point out that the map 1σ satisfies the identity: 0101 1 BBAA δσδ = …(3.1) where by 01 AA δ we mean the component of the boundary of the fat complex which conveys A1 to A0. We will use notation iiii BAAA 11 , ++ δδ etc. Then we can define 011 : AA →∂ as 011 AA δ=∂ . It is easy to show that 1∂ which we defined above satisfies the condition (3.1), for example: ))( 3 1 ())()(( )2(2121 )3( 211 0101 vxZvxZ AAAA ∂=δσδ  ))((3 1 )2( 2121 v∂∂= )( )3( 21 v∂= ))(( )3(2101 vxZBBδ= . At this point we are in position to define 122 : AA →∂ By 1212 12 BAAA δσδ +=∂ 001 01 BAA AA = → δ 1σ Mathematics - 347 الصرفة و التطبيقيةمجلة إبن الهيثم للعلوم 2012 السنة 25 المجلد 3 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 3 Vol. 25 Year 2012 Proposition (3.1): The composition 021 =∂∂  [1],[2] . Now, we have to define a map 222 : AB →σ Such that 211 )( 12121212 σδσδδσδ  BAAABBAB +=+ ………(3.2) We define this map as follows: • ;02121 vxZxZ where 326 DDDv ⊕⊗∈ • ;021 )2( 21 vxZxZ where 317 DDDv ⊕⊗∈ • ;0)2(2121 vxZxZ where 317 DDDv ⊕⊗∈ • ;021 )3( 21 vxZxZ where 308 DDDv ⊕⊗∈ • ;0)2(21 )2( 21 vxZxZ where 308 DDDv ⊕⊗∈ • ;0)3(2121 vxZxZ where 308 DDDv ⊕⊗∈ • );( 3 1 21 )3( 2132 )3( 2132 vxZyZvxZyZ ∂ where 227 DDDv ⊕⊗∈ • );( 6 1 )2( 21 )2( 2132 )4( 2132 vxZyZvxZyZ ∂ where 218 DDDv ⊕⊗∈ • );( 10 1 )3( 21 )2( 2132 )5( 2132 vxZyZvxZyZ ∂ where 209 DDDv ⊕⊗∈ • ;03232 vyZyZ where 164 DDDv ⊕⊗∈ • )( 3 1 )( 3 1 )2( 21313231 )2( 2132 )3( 21 )2( 32 vzZyZvxZyZvxZyZ ∂−∂ ;where 137 DDDv ⊕⊗∈ • )( 12 1 )( 6 1 32 )2( 21 )2( 21323121 )2( 2132 )4( 21 )2( 32 vxZyZvxZyZvxZyZ ∂∂−∂∂ ;where 128 DDDv ⊕⊗∈ • )( 5 1 )( 30 1 )4( 21313231 )2( 21 )2( 2132 )5( 21 )2( 32 vzZyZvxZyZvxZyZ ∂−∂∂ ;where 119 DDDv ⊕⊗∈ • );( 20 1 )4( 2132 )2( 2132 )6( 21 )2( 32 vxZyZvxZyZ ∂∂ where 1010 DDDv ⊕⊗∈ • )( 12 1 )( 6 1 312132 )2( 2132 )2( 31 )2( 3132 )4( 21 )3( 32 vxZyZvZyZvxZyZ ∂∂∂+∂ );( 12 1 32 )3( 213132 vzZyZ ∂∂− where 038 DDDv ⊕⊗∈ Mathematics - 348 الصرفة و التطبيقيةمجلة إبن الهيثم للعلوم 2012 السنة 25 المجلد 3 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 3 Vol. 25 Year 2012 • )( 18 1 )( 9 1 3132 )2( 21 )2( 2132 )2( 3121 )2( 2132 )5( 21 )3( 32 vxZyZvxZyZvxZyZ ∂∂∂+∂∂ );( 30 1 )2( 32 )3( 21 )2( 2132 vxZyZ ∂∂+ where 029 DDDv ⊕⊗∈ • )( 36 1 )( 9 1 32 )5( 21313231 )4( 213132 )6( 21 )3( 32 vzZyZvzZyZvxZyZ ∂∂−∂∂− where 0110 DDDv ⊕⊗∈ • );( 30 1 )2( 31 )3( 21 )2( 2132 )7( 21 )3( 32 vxZyZvxZyZ ∂∂ where 0011 DDDv ⊕⊗∈ • ;032 )2( 32 vyZyZ where 074 DDDv ⊕⊗∈ • ;0)2(3232 vyZyZ where 074 DDDv ⊕⊗∈ • );( 3 1 32313231 )3( 32 vzZyZvzZyZ ∂ where 065 DDDv ⊕⊗∈ It is easy to show that 2σ which is defined above satisfies the condition (3.2), for example we chose one of them ))(( )4(21 )3( 321 1212 vxZyZBBAB δσδ + ))(())(()( )2(3132 )2( 21131 )2( 32 )3( 211 )3( 32 )4( 211 vxZvxZvxZ ∂∂+∂∂+∂= σσσ ))(()( )4(21 )3( 321 )31( 3121 vxZvxZ ∂−∂+ σ )( 2 1 )( 3 1 )( 4 1 )2( 31322121 )2( 32 )4( 2121 )3( 32 )3( 2121 vxZvxZvxZ ∂∂∂+∂∂+∂∂= )( 3 1 )( )4(21 )2( 3232 )3( 3121 vyZvxZ ∂∂−∂+ )( 2 1 )( 3 1 )( 4 1 )2( 3132212131 )2( 32 )2( 2121 )3( 32 )3( 2121 vxZvxZvxZ ∂∂∂+∂∂∂+∂∂= )( 3 1 )( )2(32 )4( 2132 )3( 3121 vyZvxZ ∂∂−∂+ )( 3 1 )( 3 1 )( 3 1 )2( 31 )2( 21323132 )3( 2132 )2( 32 )4( 2132 vyZvyZvyZ ∂∂−∂∂∂−∂∂− and )( 12 1 )( 6 1 )(( 2131 )2( 2132 )2( 31 )2( 21321 1212 vxZyZvxZyZBAAA ∂∂+∂+ δσδ  ))( 12 1 31 )3( 213132 vzZyZ ∂∂− )( 6 1 )( 2 1 ))( 6 1 ( )2(31 )2( 2132 )3( 3121 )2( 3132 )2( 211 vyZvxZvxZ ∂∂−∂+∂∂= σ )( 12 1 )( 6 1 ))( 6 1 ( 312132 )2( 2121 )2( 312132213121 )2( 32 )2( 211 vxZvxZvxZ ∂∂∂∂−∂∂∂+∂∂∂+σ Mathematics - 349 الصرفة و التطبيقيةمجلة إبن الهيثم للعلوم 2012 السنة 25 المجلد 3 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 3 Vol. 25 Year 2012 )( 12 1 )( 12 1 ))( 3 1 ( 3132 )3( 213232 )3( 21 )2( 322132 )4( 21 )2( 321 vyZvxZvyZ ∂∂∂+∂∂∂+∂∂−σ )( 6 1 )( 12 1 )( 12 1 )2( 31 )2( 2132 )3( 3121 )2( 31322121 vyZvxZvxZ ∂∂−∂+∂∂∂= )( 6 1 )( 6 1 )( 6 1 )2( 31322121 )2( 3132212131 )2( 32 )2( 2121 vxZvxZvxZ ∂∂∂+∂∂∂+∂∂∂+ )( 6 1 )( 4 1 )( 2 1 )2( 31 )2( 21323132 )3( 2132 )3( 3121 vyZvyZvxZ ∂∂−∂∂∂−∂+ )( 4 1 )( 6 1 )( 3 1 )3( 32 )3( 21213132 )3( 2132 )2( 32 )4( 2132 vxZvyZvyZ ∂∂+∂∂∂−∂∂− )( 12 1 )( 12 1 )( 6 1 3132 )3( 2132 )2( 3132212131 )2( 32 )2( 2121 vyZvxZvxZ ∂∂∂+∂∂∂+∂∂∂+ )( 3 1 )()( 2 1 )2( 31 )2( 2132 )3( 3121 )2( 31322121 vyZvxZvxZ ∂∂−∂+∂∂∂= )( 3 1 )( 3 1 3132 )3( 213231 )2( 32 )2( 2121 vyZvxZ ∂∂∂−∂∂∂+ )( 4 1 )( 3 1 )3( 32 )3( 2121 )2( 32 )4( 2132 vxZvyZ ∂∂+∂∂−  Proposition (3.2) [1],[2] : we have exactness at Ai . Now by using 2σ we can also define 233 : AA →∂ by 2323 23 BAAA δσδ +=∂ Proposition (3.3) [1],[2] : 032 =∂∂  . We need to define 333 : AB →σ which satisfying 222 )( 23232323 σδσδδσδ  BAAABBAB +=+ …(3.3) As follows • ;0212121 vxZxZxZ where 317 DDDv ⊕⊗∈ • ;02121 )2( 21 vxZxZxZ where 308 DDDv ⊕⊗∈ • ;021 )2( 2121 vxZxZxZ where 308 DDDv ⊕⊗∈ • ;0)3(212121 vxZxZxZ where 308 DDDv ⊕⊗∈ • ;021 )2( 2132 vxZxZyZ where 227 DDDv ⊕⊗∈ • ;021 )3( 2132 vxZxZyZ where 218 DDDv ⊕⊗∈ • ;0)2(21 )2( 2132 vxZxZyZ where 218 DDDv ⊕⊗∈ Mathematics - 350 الصرفة و التطبيقيةمجلة إبن الهيثم للعلوم 2012 السنة 25 المجلد 3 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 3 Vol. 25 Year 2012 • ;021 )4( 2132 vxZxZyZ where 209 DDDv ⊕⊗∈ • ;0)2(21 )3( 2132 vxZxZyZ where 209 DDDv ⊕⊗∈ • ;0)2(21 )2( 2132 vxZxZyZ where 209 DDDv ⊕⊗∈ • vxZxZyZ 21 )3( 21 )2( 32 )(3 1 )2( 21213132 vxZzZyZ ∂− ; where 128 DDDv ⊕⊗∈ • )( 4 1 )3( 21213132 )4( 21 )2( 32 vxZzZyZvxZyZ ∂ ; where 119 DDDv ⊕⊗∈ • 0)2(21 )3( 21 )2( 32 vxZxZyZ where 119 DDDv ⊕⊗∈ • vxZxZyZ 21 )5( 21 )2( 32 )(5 1 )4( 2121 )5( 31 )2( 32 vxZzZyZ ∂− where 1010 DDDv ⊕⊗∈ • vxZxZyZ )2(21 )4( 21 )2( 32 )(4 1 )4( 21213132 vxZzZyZ ∂− where 1010 DDDv ⊕⊗∈ • vxZxZyZ )3(21 )3( 21 )2( 32 )(3 2 )4( 21213132 vxZzZyZ ∂− where 1010 DDDv ⊕⊗∈ • vxZyZyZ )3(213232 )(3 1 21213132 vxZzZyZ ∂− where 137 DDDv ⊕⊗∈ • 0)4(213232 vxZyZyZ where 218 DDDv ⊕⊗∈ • vxZyZyZ )5(213232 );(10 1 )3( 21 )2( 213132 vxZzZyZ ∂ with 119 DDDv ⊕⊗∈ • 0)6(213232 vxZyZyZ with 1010 DDDv ⊕⊗∈ • vxZxZyZ 21 )4( 21 )2( 32 vxZxZyZ 21 )5( 21 )2( 32 +∂∂ )(6 1 31 )3( 21213132 vxZzZyZ )( 30 1 32 )4( 21213132 vxZzZyZ ∂∂ ; with 0110 DDDv ⊕⊗∈ • ;0)2(21 )4( 21 )2( 32 vxZxZyZ with 1010 DDDv ⊕⊗∈ • vxZxZyZ )3(21 )3( 21 )2( 32 )(3 2 213132 vxZzZyZ with 1010 DDDv ⊕⊗∈ • vxZyZyZ )3(213232 );(3 1 21213132 vxZzZyZ ∂− with 137 DDDv ⊕⊗∈ • 0)4(213232 vxZyZyZ ; with 128 DDDv ⊕⊗∈ • vxZyZyZ )5(213232 );(10 1 )3( 21213132 vxZzZyZ ∂− with 119 DDDv ⊕⊗∈ • 0)6(213232 vxZyZyZ with 1010 DDDv ⊕⊗∈ Mathematics - 351 الصرفة و التطبيقيةمجلة إبن الهيثم للعلوم 2012 السنة 25 المجلد 3 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 3 Vol. 25 Year 2012 • vxZxZyZ 21 )4( 21 )3( 32 −∂∂− )(36 1 31 )2( 21213132 vxZzZyZ )( 12 1 32 )3( 21213132 vxZzZyZ ∂∂ ; with 029 DDDv ⊕⊗∈ • vxZxZyZ 21 )5( 21 )3( 32 +∂∂ )(6 1 31 )3( 21213132 vxZzZyZ )( 30 1 32 )4( 21213132 vxZzZyZ ∂∂ ; with 0110 DDDv ⊕⊗∈ • vxZxZyZ )2(21 )4( 21 )3( 32 )(12 1 )4( 2132213132 vxZzZyZ ∂∂ ; 0110 DDDv ⊕⊗∈ • vxZxZyZ 21 )6( 21 )3( 32 )(60 7 31 )4( 21213132 vxZzZyZ ∂∂ − ; with 0011 DDDv ⊕⊗∈ • 0)2(21 )5( 21 )3( 32 vxZxZyZ ; with 0011 DDDv ⊕⊗∈ • vxZxZyZ )3(21 )4( 21 )3( 32 )(6 1 31 )4( 21213132 vxZzZyZ ∂∂ − ; with 0011 DDDv ⊕⊗∈ • vxZyZyZ )4(2132 )3( 32 −∂∂ − )( 6 1 3121213132 vxZzZyZ )( 12 1 31 )2( 21213132 vxZzZyZ ∂∂ ; with 038 DDDv ⊕⊗∈ • vxZyZyZ )6(2132 )2( 32 ))()(( 60 1 )4( 213121313231 )3( 21213132 vxZzZyZvxZzZyZ ∂∂+∂∂ − ; with 0110 DDDv ⊕⊗∈ • 0)7(2132 )2( 32 vxZyZyZ ; with 0011 DDDv ⊕⊗∈ • vxZyZyZ )6(21 )2( 2132 )( 60 1 )( 12 1 31 )4( 2121313231 )3( 21213132 vxZzZyZvxZzZyZ ∂∂−∂∂ − ; with 0110 DDDv ⊕⊗∈ • 0)7(21 )2( 2132 vxZyZyZ ; with 0011 DDDv ⊕⊗∈ • 0213232 vyZyZyZ ; with 074 DDDv ⊕⊗∈ • )()2(213132 vxZzZyZ )(3 1 21213132 vxZzZyZ ∂ ; with 137 DDDv ⊕⊗∈ • )()3(213132 vxZzZyZ )(3 1 )2( 21213132 vxZzZyZ ∂ ; with 128 DDDv ⊕⊗∈ Mathematics - 352 الصرفة و التطبيقيةمجلة إبن الهيثم للعلوم 2012 السنة 25 المجلد 3 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 3 Vol. 25 Year 2012 • )()4(213132 vxZzZyZ )(10 1 )3( 21213132 vxZzZyZ ∂ ; with 119 DDDv ⊕⊗∈ • )()5(213132 vxZzZyZ )(5 1 )4( 21213132 vxZzZyZ ∂ ; with 1010 DDDv ⊕⊗∈ • )()2(2131 )2( 32 vxZzZyZ )(3 1 31213132 vxZzZyZ ∂ ; with 047 DDDv ⊕⊗∈ • )()3(2131 )2( 32 vxZzZyZ ))()(( 12 1 )2( 21322131323121213132 vxZzZyZvxZzZyZ ∂∂+∂∂ ; with 038 DDDv ⊕⊗∈ • )()4(2131 )2( 32 vxZzZyZ )( 30 1 )( 9 1 32 )3( 2121313231 )2( 21213132 vxZzZyZvxZzZyZ ∂∂+∂∂ ; with 038 DDDv ⊕⊗∈ • )()5(2131 )2( 32 vxZzZyZ );(20 1 32 )4( 21213132 vxZzZyZ ∂∂ with 0110 DDDv ⊕⊗∈ • )()7(2131 )2( 32 vxZzZyZ );(15 1 32 )4( 21213132 vxZzZyZ ∂∂ with 0011 DDDv ⊕⊗∈ • 0)(213232 vzZyZyZ ; with 065 DDDv ⊕⊗∈ • )()5(2132 )2( 32 vxZyZyZ );(20 1 32 )3( 21213132 vxZzZyZ ∂∂ with 029 DDDv ⊕⊗∈ Again easily we can show that 3σ which defined above satisfies the condition (3.3), and here we chose one of them as an example )( 2323 2 BBAB δσδ + =))(( )4(2131 )2( 32 vxZzZyZ =∂+∂+= ))(())(())((2 )2(2131 )3( 32232 )5( 21 )3( 322 )5( 21 )3( 322 vzZyZvxZyZvxZyZ σσσ )( 15 1 )( 9 1 )( 9 2 )2( 32 )3( 21 )2( 21323132 )2( 21 )2( 2132 )2( 3121 )2( 2132 vxZyZvxZyZvxZyZ ∂∂−∂∂∂+∂∂= )( 3 1 )( 5 1 )( 30 1 32 )4( 21313232 )4( 21 )2( 21313232 )2( 21 )2( 2132 vzZyZvxZzZyZvxZyZ ∂∂+∂∂−∂∂+ =∂∂+ )( 3 1 31 )3( 213132 vzZyZ )( 90 13 )( 9 2 3132 )2( 21 )2( 213132 )2( 3221 )2( 2132 vxZzZyZvxZyZ ∂∂∂+∂∂ )( 15 1 )2( 32 )3( 21 )2( 2132 vxZyZ ∂∂+ )(3 1 )( 15 2 31 )3( 21313232 )4( 213132 vzZyZvzZyZ ∂∂+∂∂+ and Mathematics - 353 الصرفة و التطبيقيةمجلة إبن الهيثم للعلوم 2012 السنة 25 المجلد 3 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 3 Vol. 25 Year 2012 )( 2323 2 BAAA δσδ + )( 9 1 ( 3132 )2( 21213132 vxZzZyZ ∂∂∂ ))( 30 1 3232 )3( 21213132 vxZzZyZ ∂∂∂+ )2( 3121 )2( 21323132 )2( 21 )2( 2132 9 2 )( 9 1 ∂∂+∂∂∂= xZyZvxZyZ )( 30 1 )( 3 1 32 )3( 2132 )2( 213231 )3( 213132 vxZyZvzZyZ ∂∂∂+∂∂+ )(15 2 32 )4( 213132 vzZyZ ∂∂+ )( 9 2 )( 9 1 )2( 3121 )2( 21323132 )2( 21 )2( 2132 vxZyZvxZyZ ∂∂+∂∂∂= )( 15 1 )( 3 1 )2( 32 )3( 21 )2( 213231 )3( 213132 vxZyZvzZyZ ∂∂+∂∂+ )( 15 2 )( 30 1 32 )4( 2131323132 )2( 21 )2( 2132 vzZyZvxZyZ ∂∂+∂∂∂ )( 9 2 )( 90 13 )2( 3121 )2( 21323132 )2( 21 )2( 2132 vxZyZvxZyZ ∂∂+∂∂∂= )( 15 1 )( 3 1 )2( 32 )3( 21 )2( 213231 )3( 213132 vxZyZvzZyZ ∂∂+∂∂+ ).(15 2 32 )4( 213132 vzZyZ ∂∂+  So from all we have done above we have the complex 0123 1230 AAAA →→→→ ∂∂∂ …(3.4) where i∂ defined as follows : • );())(( 21211 vvxZ ∂=∂ with 335 DDDv ⊕⊗∈ • );())(( 32321 vvyZ ∂=∂ with 254 DDDv ⊕⊗∈ • );()()( 2 1 ))(( )2(21323121322121 )2( 21322 vyZvxZvxZvxZyZ ∂−∂+∂∂=∂ with 236 DDDv ⊕⊗∈ . • );()()( 2 1 ))(( )2(3232 )2( 322121323231322 vyZvxZvyZvzZyZ ∂−∂−∂∂=∂ with 155 DDDv ⊕⊗∈ . and the map 3∂ is defined as • );()())(( 21313232 )2( 21322131323 vzyZZvxyZZvxZzyZZ ∂+∂=∂ with 146 DDDv ⊕⊗∈ . Proposition (3.4) [1],[2] : The complex Mathematics - 354 الصرفة و التطبيقيةمجلة إبن الهيثم للعلوم 2012 السنة 25 المجلد 3 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 3 Vol. 25 Year 2012 )3,4,4(0123 1230 KAAAA →→→→→ ∂∂∂ is exact. References 1. Buchsbaum, D. A. (2004), Characteristic free example of Lascoux resolution and letter place mathods for intertwining numbers, European Journal of Gombinatorics, 25, 1169- 1179. 2. Haytham, R. Hassan, (2006), Application of the characteristic free resolution of Weyl module to the Lascoux resolution in the case (3,3,3). Ph. D. thesis, universita di Roma "Tor Vergata". 3. Buchsbaum, D. A. and Rota,G. C. (2001), Approaches to resolution of Weyl modules, Adv. In applied math. 27, 82-191. 4. Buchsbaum, D. A. and Rota,G. C. (1994), A new construction in homological algebra, Natl acad. Sci, USA 91, 4115-4119. 5. Akin, K.; Buchsbaum, D. A. and Weyman, J. (1982), Schur functors and Schur complexes, Adv. Math. 44, 207-278. 6. Vermani, L. R.(2000). An elementary approach to homological algebra, chapman Applied mathematics 130. Mathematics - 355 الصرفة و التطبيقيةمجلة إبن الهيثم للعلوم 2012 السنة 25 المجلد 3 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 3 Vol. 25 Year 2012 اختزال تحلل مقاس وايل من صيغة المميز الحر الى تحلل السكو في حالة (4,4,3) هيثم رزوقي حسن قسم الرياضيات، كلية العلوم، الجامعة المستنصرية 2012نيسان 22قبل البحث في: 2012كانون الثاني 8استلم البحث في: الخالصة FKمقاس وايل ( في هذا البحث ندرس العالقة بين تحلل ) في صيغة المميز الحر والتحلل لالسكو في )3,4,4( FKصيغة المميز صفر و بدقة اكثر سوف نحصل على تحلل السكو لـ ا لتحلل تطبيقفي صيغة المميز صفر )3,4,4( FK صيغة المميز الحر. في )3,4,4( : تحلل، مقاس وايل، تحلل السكو، تقسيم القوة، المميز الحر. الكلمات المفتاحية Received in: 8 January 2012 Accepted in: 22 April 2012