The set of all irreducible polynomial representations of general linear group GL????(ℱ)of degree ????is described by the module {ℒλ(ℱ)}; where λruns over all partitions λ=(λ1,λ2,...,λs). There are a number of classical formulas that express the formal character of the representation ℒλ(ℱ)in terms of standard symmetric polynomials. Such formulas are also valid for the more general representation modules {ℒλμ⁄(ℱ)}associated to skew partition λμ⁄; where μ⊆λ.Let ℛbe a commutative ring with identity and ℱa free ℛ-module. The Weyl module resolution studied by Buchsbaum where the Weyl module ????λμ⁄(ℱ)is the image of the Weyl map ????′λμ⁄(ℱ)for the skew-partition λμ⁄.Reduction the terms of theresolution of the characteristic-free of Weyl module to the terms of theresolution of Lascoux by employing the boundary mapsfor the partition (9,6,3)and prove that the sequence of the reduction terms is exact.1.INTRODUCTIONThe precise definitions of the boundary maps are given in [1]; where it is proved that the complex resolution ℬ•in characteristic-zero of ℒ????(????)is exact, where ℬ•:0→ℬ(????2)????(????2)→...→ℬ1????1→ℬ0→ℒ????????⁄(ℱ)→0Note that the terms of the resolution ℬ•of ℒ????????⁄(ℱ)are direct sums of tensor products of the fundamental representations of ????????????(ℱ).Hassan generalized the techniques in [2]for the partitions (3,3,3), and (4,4,3) in [3,4]respectively, alsoauthors in [5-7] studied the cases (8,7,3), (6,6,4;0,0), (7,7,4;0,0).The reduction resolution termsof Weyl modulefrom characteristic-freeto Lascouxfound in this work and prove that the sequence of these terms is exact.2.THE TERMS OF CHARACTERISTIC-FREE RESOLUTIONWe stratify the following formula for the case of partition (????,????,????)to obtain the terms of the resolution for the partition (9,6,3), [2].????????????([????,????;0])⨂????????⨁∑????32(ℯ+1)????????????????([????,????+ℯ+1;ℯ+1])⨂????????−ℯ−1⨁ℯ≥0∑????32(ℯ2+1)????????31(ℯ1+1)????????????????([????+ℯ1+1,????+ℯ2+1;ℯ2−ℯ1])ℯ1≥0,ℯ2≥ℯ1⨂????????−(ℯ1+ℯ2+2);Babylonian Journal of MathematicsVol. 2024, pp. 117–131DOI:https://doi.org/10.58496/BJM/2024/015;ISSN:3006-113Xhttps://mesopotamian.press/journals/index.php/mathematics

Title

Babylonian Journal of Mathematics

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Mesopotamian Academic Press

Publisher Country

Iraq

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Prtinted only

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