Outcome for the Case (11,10; 1)

Authors

Waleed Khalid Jaber, Department of Mathematics, College of Education for Pure Science Ibn-Al-Haitham, University of Baghdad, Baghdad, Iraq AND Department of Mathematics, College of Computer Sciences and Mathematics, University of Thi-Qar, Nassiriyah, IraqFollow
Niran Sabah Jasim, Department of Mathematics, College of Education for Pure Science Ibn-Al-Haitham, University of Baghdad, Baghdad, Iraq

Abstract

Through this work R is a commutative ring with 1, ℱ is a free R-module and 𝒟_𝒾 is the divided power algebra of degree 𝒾. A partition of length ℓ(λ) = n is a sequence ( λ ) = (λ₁, λ₂, …, λ_n) of non-negative integers in non-increasing order λ_1≥λ₂ ≥ … ≥ λn > 0. The weight of a partition λ is |λ| =λ ₁ + λ₂ + … + λ_n. A relative sequence is a pair ( λ,μ ) of sequences such that μ ≤ λ and denoted by λ /μ . If both ( λ ) and μ are partitions then the relative sequence ( λ /μ ) is called a skew partition. Let 𝒵₂₁ be the free generator of divided power algebra 𝒟(𝒵₂₁) in one generator. The divided power element 𝒵₂₁^(ℯ) of degree ℯ of the free generator 𝒵₂₁ acts on 𝒟_p + e + 𝒟_q − e by place polarization of degree ℯ from place 1 to place 2. The graded algebra 𝒜 = 𝒟(𝒵₂₁) acts on the graded module ℳ = 𝒟_p + e + 𝒟_q − e = ∑ℳ_q − e, the degree of the second factor determines the grading. M is a graded left $A = - $odule, where for w = 𝒵₂₁^(ℯ)∈ $ $ and 𝓋 ∈ 𝒟_p + e + 𝒟_q − e, so we have: w(𝓋) = 𝒵₂₁^(ℯ) = (𝓋) = ∂₂₁^(ℯ). This article surveys the exactness of the Weyl resolution of the sequence for the case (11,10; 1) after getting the contracting homotopy of {𝒮_𝒾}; where 𝒾 = 0,1, …, 8, and the terms of the characteristic-free resolution for it.

Keywords

Divided power algebra, Letter place, Overlap, Place polarizations, Resolution, Weyl module

Subject Area

Mathematics

First Page

664

Last Page

676

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

Receive Date

10-11-2023

Revise Date

3-4-2023

Accept Date

3-6-2024

How to Cite this Article

Jaber, Waleed Khalid and Jasim, Niran Sabah (2025) "Outcome for the Case (11,10; 1)," Baghdad Science Journal: Vol. 22: Iss. 2, Article 25.
DOI: https://doi.org/10.21123/bsj.2024.9905