Document Type : Research Paper


1 Department of Mathematics, College of Science, Mustansiriyah University, Baghdad, Iraq

2 Ministry of Education, Directorate General of Education in Baghdad, karkh 1, Baghdad, Iraq;


The characteristic-free resolution of K_((7,5,3) ) is applied to the Lascoux resolution of K_((7,5,3) ) F (characteristic zero resolution) in this paper. This research teaches us about the relationship between the resolution of the weyl module K_((7,5,3) ) F in the characteristic-free mode and the Lascoux mode.
In this work, let R be a commutative ring with 1, Ӻ be a free R-module and Ḍ_į be "the divided power algebra" of degree į. Ṃ is a left-graded module with for Ѡ = Ƶ_21^ҝ⋵ Å and Ѵ ⋵〖 Ḍ〗_(ᴯ_1 )⊗ Ḍ_(ᴯ_2 ) . We have Ѡ(Ѵ)= Ƶ_(21 )^ҝ (Ѵ)=〖 ∂〗_(21 )^ҝ (Ѵ). where the separator ẋ vanishes between Ƶ_ḁᵬ^((ț))and ∂_ḁᵬ^((ț)). Also by using Capelli identities we prove the sequences and the subsequences of the terms of characteristic zero satisfy the commutative for each diagram in these sequences and subsequences. Finally we get the reduction of the terms of the resolution of the Weyl module for characteristic free to the terms of the resolution of the Weyl module for characteristic 0.


Main Subjects

[1].David AB, Taylor B.D. Homotopies for Resolution of Skew-Hook Shapes. Adv Appl Math.2003; 30(0196-8858):26-43.
[2]. Haytham RH. The Reduction of Resolution of Weyl Module From Characteristic-Free Resolution in Case (4,4,3). IHJPAS. 2012; 25(3)(2521-3407):341-355.
[3].Giandomenico B, David AB. Threading Homology Through Algebra: Selected Patterns. Clorendon Press. Oxford; 2006.
[4].Haytham RH, Mays MM. Application of the resolution of the characteristic-free resolution of Weyl module to Lascoux resolution in case (6,6,3). Int J Eng Appl Sci. 2016; 3(2394- 3661):57-63.
[5].  Najah MM. Resolution of Wely Module in the Case of the Partition (7,6,3). M.Sc. Thesis: College of Science, Al-Mustansiriyah University, Iraq; 2017.
[6].Haytham RH, Niran SJ. Application of Weyl Module in the Case of Two Rows. J.Phys.:Conf.Ser. 2018; 1003 (012051)(1742 – 6588, 742 – 6596): 1-15, doi :10.1088/1742- 6596/1003/1/012051.
[7].Haytham RH, Niran SJ. A Complex of Characteristic Zero in the Case of the Partition (8,7,3). Sci. Int. (Lahore). 2018; 30(4)(1013- 5316):639-641.
[8].David AB. A Characteristic Free Example of Lascoux Resolution, and Letter Place Methods for Intertwining Numbers. Eur J Comb.2004; 25(0195-6698):1169-1179.
[9].Haytham RH. Application of the characteristicfree resolution of Weyl Module to the Lascoux resolution in the case (3,3,3). Ph.D. Thesis: Universita di Roma "Tor Vergata", Italian (Roma); 2006.
[10].Haytham RH, Najah MM. Complex of Lascoux in the Case of Partition (7,6,3). Aust J Basic Appl Sci. 2016; 10(18)(1991-8178):89-93.