171 | Mathematics 2015) عام 1العدد ( 28مجلة إبن الھيثم للعلوم الصرفة و التطبيقية المجلد Ibn Al-Haitham J. for Pure & Appl. Sci. Vol. 28 (1) 2015 Complex of Lascoux in Partition (3,3,2) Haytham R. Hassan Alaa Omer Aziz Received in :26 October 2014 , Accepted in:5 January 2015 Abstract In this paper, the complex of Lascoux in the case of partition (3,3,2) has been studied by using diagrams ,divided power of the place polarization )( kij ,Capelli identites and the idea of mapping Cone . Key words: Divided power algebra, Resolution of Weyl module, Place polarization, Mapping Cone. 172 | Mathematics 2015) عام 1العدد ( 28مجلة إبن الھيثم للعلوم الصرفة و التطبيقية المجلد Ibn Al-Haitham J. for Pure & Appl. Sci. Vol. 28 (1) 2015 1.Introduction Let R be commutative ring with 1, F be a free module and FDS be the divided power of degree s . In [1] Buchsbaum used another type of maps whose images define Schur and Weyl modules which sends an element ba  of kqkp DD   to   baa kp , where  kp aa is the component of the diagonal of a in kp DD  , the generalization of this map to ones, where there more factors were called in the 'box map'. In [3] ,[4] and [5] the author studied the complex of characteristic zero in the partition (2,2,2) ,(3,3,3) and (4,4,3), using this modified and the letter place methods [3] , In this paper we study the complex of Lascoux in the case of partition (3,3,2) as a diagram by using the idea of the mapping Cone [6] , and the map )( kij which means the thk divided power of the place polarization ij where j must be less than i with it's Capelli identities [1] , specificly in this work we used only the following identities (1.1)     0 )( 31 )( 21 )( 32 )( 32 )( 21 )1(   kllk (1.2) )3.1()1(32 )1( 31 )1( 31 )1( 32 )1( 21 )1( 31 )1( 31 )1( 21   and Where ij is the place polarization from place j to place i . 2.The terms of Lascoux complex in the case of partition (3, 3, 2) The terms of the Lascaux complex are obtained from the determinantal expansion of the Jocobi-Trudi matrix of the partition. The positions of the terms of the complex are determined by the length of the permutation to which they correspond [2], [3]. Now in the case of the partition )2,3,3( , we have the following matrix:           FDFDFD FDFDFD FDFDFD 245 134 023 Then the Lascoux complex has the correspondence between it's terms as follows: identityFDFDFD  233     0 )( 31 )( 32 )( 21 )( 21 )( 32   lkkl 173 | Mathematics 2015) عام 1العدد ( 28مجلة إبن الھيثم للعلوم الصرفة و التطبيقية المجلد Ibn Al-Haitham J. for Pure & Appl. Sci. Vol. 28 (1) 2015 )32(413  FDFDFD )321(512  FDFDFD )32(242  FDFDFD )31(530  FDFDFD )231(440  FDFDFD So, the complex of Lascoux in the case of the partition )2,3,3( has the form:- FDFDFD FDFDFD FDFDFD FDFDFD FDFDFD FDFDFD 233 224 143 125 044 035          3.The complex of Lascoux as a diagram Consider the following diagram : FDFDFDFDFDFDFDFDFD pp 143044035 21  1k S 2k H 3k FDFDFDFDFDFDFDFDFD qq 233224125 21  So, if we define FDFDFDFDFDFDp 0440351 :  as )()( 211 vvp  where ; FDFDFDv 035  FDFDFDFDFDFDk 1250351 :  as )()( 321 vvk  where ; FDFDFDv 035  FDFDFDFDFDFDk 2240442 :  as   )()( 2322 vvk  where ; FDFDFDv 044  . Now, we have to define the following map which makes the diagram S commutative: FDFDFDFDFDFDq 2241251 :  so we have: 1211 pkkq   which implies that   21 2 32321  q 174 | Mathematics 2015) عام 1العدد ( 28مجلة إبن الھيثم للعلوم الصرفة و التطبيقية المجلد Ibn Al-Haitham J. for Pure & Appl. Sci. Vol. 28 (1) 2015 Now we use Capelli identities from (1.1), (1.2):                  1 32 1 31 1 322 11 21 1 32 2 31 2 32 1 2121 2 32 )(     Thus ,      131 1 32 1 212 1 1  q . On the other hand, if we define FDFDFDFDFDFDq 2332242 :  as )()( 212 vvq  where ; FDFDFDv 224  and FDFDFDFDFDFDk 2331433 :  as )()( 323 vvk  where ; FDFDFDv 143  . Now we need to define 2p to make the diagram H commute: FDFDFDFDFDFDp 1430442 :  . Such that 2223 kqpk   i.e.  2 3221232   p again by using Caplli identities we get                   )( 131 1 21 1 322 1 32 1 31 1 32 1 21 2 32 2 32 1 21     then     )1(31 1 21 1 322 1 2  p . Now consider the following diagram: FDFDFDFDFDFDFDFDFD pp 143044035 21  1k E  F 3k FDFDFDFDFDFDFDFDFD qq 233224125 21  Define FDFDFDFDFDFD 142125:  by   )()( 221 vv  where ; FDFDFDv 125  . Proposition 3.1: The diagram E is commutative. 175 | Mathematics 2015) عام 1العدد ( 28مجلة إبن الھيثم للعلوم الصرفة و التطبيقية المجلد Ibn Al-Haitham J. for Pure & Appl. Sci. Vol. 28 (1) 2015 proof:- To prove E is commutative, we need to prove 112 kpp                                         1 1 32 2 21 1 21 1 31 1 31 1 21 1 32 2 21 1 21 1 31 2 21 1 32 1 21 1 31 1 21 1 322 1 12 k pp       Proposition 3.2: The diagram F is commutative . proof:-                                            3 2 21 1 32 1 31 1 21 1 31 1 21 2 21 1 32 1 31 1 21 1 32 2 21 1 31 1 32 1 212 11 2112 k qq Finally by using the mapping Cone we can define the maps 21, and 3 where: FDFDFD FDFDFD FDFDFD 125 044 0353 :     FDFDFD FDFDFD FDFDFD FDFDFD 224 143 125 044 2 :         and FDFDFD FDFDFD FDFDFD 233 224 143 1 :      FDFDFD FDFDFD FDFDFD FDFDFD FDFDFD FDFDFD 233 224 143 125 044 035 1230           by 176 | Mathematics 2015) عام 1العدد ( 28مجلة إبن الھيثم للعلوم الصرفة و التطبيقية المجلد Ibn Al-Haitham J. for Pure & Appl. Sci. Vol. 28 (1) 2015  FDFDFDxxkxpx 035113 ));(),(()(   ;)()()),()(()),(( 1221212212 xkxkxxpxx   FDFDFD FDFDFD xx 125 044 21 ),(      ;))()(()),(( 2213211 xqxkxx  FDFDFD FDFDFD xx 224 143 21 ),(     Proposition 3.3 : FDFDFD FDFDFD FDFDFD FDFDFD FDFDFD FDFDFD 233 224 143 125 044 035 1230           is complex. proof:- Since  132 and  1 21 are injective from it is definition (see [1]), then we get 3 is injective. Now    )))(())((,))(()((( ))(),(( ))(),(()( 21232132212 32212 11232 xkxqxxp xx xkxpx     Now                                 0 ))(( ))(( )()()())(())(( 3121 2 21 1 32 1 31 1 21 2 21 1 32 32 2 2121 1 31 2 21 1 32 32 2 2121 1 31 1 21 1 322 1 32212 x x xxxxp                                0 ))(( ))(( )()()())(())(( 21 2 3231 1 323132 1 21 2 32 21 2 3231 1 32 2 32 1 21 21 2 3231 1 31 1 322 11 21212321 x x xxxgxk    177 | Mathematics 2015) عام 1العدد ( 28مجلة إبن الھيثم للعلوم الصرفة و التطبيقية المجلد Ibn Al-Haitham J. for Pure & Appl. Sci. Vol. 28 (1) 2015 so we get  0))(( 32 x  and                                                                       ))(( ))(( )())(( )())(( ))())((,)())((( )()()),()((),)(( 2 2 21 1 32 1 31 1 21 1 32 2 21 1 2 32 1 21 1 31 1 32 1 21 2 32 1 2 32 1 212 1 31 1 322 11 21 1 21 2 2 21 1 321 1 31 1 21 1 322 11 32 1 2 322 1 31 1 32 1 212 1 2 2 211 1 31 1 21 1 322 1 1 122221212121 x x xx xx xxxx xkxqxxpxx         from then (1.1) and (1.2) we get                                    0 ))(( ))((),)(,( 2 2 21 1 32 1 31 1 21 1 31 1 21 2 21 1 32 1 2 32 1 21 1 31 1 32 1 31 1 32 2 32 1 212121 x xxx   References [1] Buchsbaum D.A. and Rota G.C., (2001), Aprroches to resolution of Weyl modules, Adv. In applied Math. 27 , 82-191. [2] Akin K., Buchshaum D.A. and Weyman J., (1982), Schur functors and complexes, Adv. Math. 44, 207-278. [3] Buchsbaum D.A., (1986) A characteristic-free realizations of the Giambelli and Jacoby-Trudi determinatal identities, proc. of K.I.T. workshop on Algebra and Topology, Springer-Verlag. [4] 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". [5] Haytham R. Hassan, (2012), The Reduction of Resolution of Weyl Module from Characteristic-Free Resolution in Case (4,4,3), J. Ibn Al-Haitham for pur and applied science, 25, 341-355. [6] Rotman J.J., (1979), Introduction to homological algebra, Academic Press, INC. 178 | Mathematics 2015) عام 1العدد ( 28مجلة إبن الھيثم للعلوم الصرفة و التطبيقية المجلد Ibn Al-Haitham J. for Pure & Appl. Sci. Vol. 28 (1) 2015 3,3,2)سلسلة السكو في التجزئة ( ھيثم رزوقي حسن الء عمر عزيزآ قسم الرياضيات/كلية العلوم/الجامعة المستنصرية 2015كانون الثاني 5, قبل البحث في : 2014تشرين االول 26أستلم البحث في : الخالصة بأعتماد المخططات ، والقوى المقسمة (3,3,2)درست في ھذا البحث سلسلة السكو في حالة التجزئة )(الستقطاب مكان kij مع مشخصات كابلي وتطبيقات كون. القوى المقسمة الجبرية ،تحلل مقاس وايل ،مكان االستقطاب ،تطبيق كون . :الكلمات المفتاحية