arXiv subjects
Publications and source records attributed to Fayadh Kadhem.
Continuing our previous work, this paper closely studies the relationship between the cluster algebra structures on the coordinate ring of Schubert cells and those on the coordinate ring of partial flag varieties. We give a finite-type classification for these cluster structures and point out several results that were left open in our previous work.
This paper addresses the question of how able the current trends of Artificial Intelligence (AI) are in managing to take the responsibility of a full course of mathematics at a college level. The study evaluates this ability in four significant aspects, namely, creating a course syllabus, presenting selected material, answering student questions, and creating an assessment. It shows that even though the AI is strong in some important parts like organization and accuracy, there are still some human aspects that are far away from the current abilities of AI. There is still a hidden emotional part, even in science, that cannot be fulfilled by the AI in its current state. This paper suggests some recommendations to integrate the human and AI potentials to create better outcomes in terms of reaching the target of creating a full course of mathematics, at a university level, as best as possible.
This study aims to shed light on new (sub)classes of matroids originating from cluster algebras and investigate their properties. We focus on what we call cluster matroids and build some results on them. Then, we point out a relationship between these kinds of matroids and uniform matroids and study their minors.
In the construction of a cluster algebra on the homogeneous coordinate ring of a partial flag variety by Geiß, Leclerc and Schr{ö}er, they defined a special map denoted by ``tilde". This map lifts each element $f$ of the coordinate ring of a Schubert cell uniquely to an element $\widetilde{f}$ of the (multi-homogeneous) coordinate ring of the corresponding partial flag variety. The significance of this map appears from its essential role; it lifts the cluster algebra of the coordinate ring of a cell to a cluster algebra living in the coordinate ring of the corresponding partial flag variety. This paper takes a closer look at this map and gives an explicit algorithm to calculate it for the \textit{generalized minors}.
The main goal of this paper is to show that the (multi-homogeneous) coordinate ring of a partial flag variety $\mathbb{C} [G / P_K^{-}]$ admits a cluster algebra structure if $G$ is any simply-connected semisimple complex algebraic group. We use derivation properties and a special lifting map to prove that the cluster algebra structure $\mathcal{A}$ of the coordinate ring $\mathbb{C}[N_K]$ of a Schubert cell constructed by Goodearl and Yakimov can be lifted, in an explicit way, to a cluster structure $\hat{\mathcal{A}}$ living in the coordinate ring of the corresponding partial flag variety. Then we use a minimality condition to prove that the cluster algebra $\hat{\mathcal{A}}$ is indeed equal to $\mathbb{C}[G / P_K^{-}]$.