On Free Resolution of Weyl Module and Zero Characteristic Resolution In The Case of Partition (7,5,3)

Abstract

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.

Keywords

• characteristic zero resolution
• weyl module
• free resolution
• characteristic-free resolution
• resolution