arXiv subjects

Chwas Ahmed

Publications and source records attributed to Chwas Ahmed.

6 recordsLinked to original sources

arXiv2026

Regularity of Symbolic Powers of Co-Chordal Edge Ideals

Let $G$ be a finite simple co-chordal graph with at least one edge, and let $I(G)$ be its edge ideal. Over an arbitrary field, we prove that the symbolic powers of $I(G)$ satisfy $\operatorname{reg} I(G)^{(s)}=2s$ for every $s\ge1$. Thus every symbolic power of $I(G)$ has a degree resolution. Using Takayama's formula and clique trees, we reduce the regularity problem to a topological one. Finite convex geometry then shows that nontrivial homology implies a suitable set of leaves, and a weighted counting argument gives the required regularity formula. Finally, we show that symbolic powers of co-chordal edge ideals are not necessarily componentwise linear.

math.AC↗

arXiv2022

The graded Betti numbers of truncation of ideals in polynomial rings

Let $R=\mathbb{K}[x_1,\dots,x_n]$, a graded algebra $S=R/I$ satisfies $N_{k,p}$ if $I$ is generated in degree $k$, and the graded minimal resolution is linear the first $p$ steps, and the $k$-index of $S$ is the largest $p$ such that $S$ satisfies $N_{k,p}$. Eisenbud and Goto have shown that for any graded ring $R/I$, then $R/I_{\geq k}$, where $I_{\geq k}=I\cap M^k$ and $M=(x_1,\dots,x_n)$, has a $k$-linear resolution (satisfies $N_{k,p}$ for all $p$) if $k\gg0$. For a squarefree monomial ideal $I$, we are here interested in the ideal $I_k$ which is the squarefree part of $I_{\geq k}$. The ideal $I$ is, via Stanley-Reisner correspondence, associated to a simplicial complex $\Delta_I$. In this case, all Betti numbers of $R/I_k$ for $k>\min\{\text{deg}(u)\mid u\in I\}$, which of course is a much finer invariant than the index, can be determined from the Betti diagram of $R/I$ and the $f$-vector of $\Delta_I$. We compare our results with the corresponding statements for $I_{\ge k}$. (Here $I$ is an arbitrary graded ideal.) In this case we show that the Betti numbers of $R/I_{\ge k}$ can be determined from the Betti numbers of $R/I$ and the Hilbert series of $R/I_{\ge k}$.

math.AC↗

arXiv2022

Quasi-forest simplicial complexes and almost Cohen-Macaulay

In this paper we study the quasi-forest simplicial complexes and we define the concept of simplicial $k$-cycle (denoted by $\mathcal{S}_k$) and simplicial $k$-point (denoted by $\mathcal{P}_k$). We show that a simplicial complex $Δ$ is quasi-forest if and only if it does not have any $\mathcal{P}_k$ and any $\mathcal{S}_k$ for $k\geq 3$. Furthermore we characterize almost Cohen-Macaulay quasi-forest simplicial complexes. In the end we show that the cycle graph $G=C_n$ is almost Cohen-Macaulay if and only if $n=3,4,5,6,7,8,9,11$.

math.AC↗

arXiv2022

Sequentially Cohen-Macaulay Co-Chordal Graphs: Structure and Projective Dimension

We introduce a class of chordal graphs called ($d_1$,$d_2$,$\dots$,$d_q$)-trees. A graph belongs to this class if and only if its clique complex is sequentially Cohen-Macaulay, providing a complete classification of all sequentially Cohen-Macaulay co-chordal graphs. This class also yields a classification of bi-sequentially Cohen-Macaulay graphs. We study the relationship between the projective dimension of a graph and its maximum vertex degree. We show that the projective dimension is always at least the maximum vertex degree, although this bound is not always tight, even for co-chordal graphs. However, equality holds when the graph is sequentially Cohen-Macaulay co-chordal or has a full vertex.

math.AC↗

arXiv2019

On the number of principal ideals in d-tonal partition monoids

For a positive integer $d$, a non-negative integer $n$ and a non-negative integer $h\leq n$, we study the number $C_{n}^{(d)}$ of principal ideals; and the number $C_{n,h}^{(d)}$ of principal ideals generated by an element of rank $h$, in the $d$-tonal partition monoid on $n$ elements. We compute closed forms for the first family, as partial cumulative sums of known sequences. The second gives an infinite family of new integral sequences. We discuss their connections to certain integral lattices as well as to combinatorics of partitions.

math.CO↗

arXiv2019

Tonal partition algebras: fundamental and geometrical aspects of representation theory

For $l,n \in \mathbb{N}$ we define tonal partition algebra $P^l_n$ over $\mathbb{Z}[δ]$. We construct modules $\{ Δ_{\underlineμ} \}_{\underlineμ}$ for $P^l_n$ over $\mathbb{Z}[δ]$, and hence over any integral domain containing $\mathbb{Z}[δ]$ that is a $\mathbb{Z}[δ]$-algebra (such as $\mathbb{C}[δ]$), that pass to a complete set of irreducible modules over the field of fractions. We show that $P^l_n$ is semisimple there. That is, we construct for the tonal partition algebras a modular system in the sense of Brauer [6]. (The aim is to investigate the non-semisimple structure of the tonal partition algebras over suitable quotient fields of the natural ground ring, from a geometric perspective.) Using a `geometrical' index set for the $Δ$-modules, we give an order with respect to which the decomposition matrix over $\mathbb{C}$ (with $δ\in \mathbb{C}^{\times}$) is upper-unitriangular. We establish several crucial properties of the $Δ$-modules. These include a tower property, with respect to $n$, in the sense of Green [20, §6] and Cox $\textit{ et al}$ [8]; contravariant forms with respect to a natural involutive antiautomorphism; a highest weight category property; and branching rules.

math.RT↗