arXiv subjects

Dharm Veer

Publications and source records attributed to Dharm Veer.

10 recordsLinked to original sources

arXiv2025

$t$-Young complexes and squarefree powers of $t$-path ideals

We introduce a new class of simplicial complexes, called \emph{$t$-Young complexes}, arising from a Young diagram and a positive integer~$t$. We show that every $t$-Young complex is either contractible or homotopy equivalent to a wedge of spheres. A complete characterization of their vertex-decomposability is provided, and in several cases, we establish explicit formulas for their homotopy types. Interestingly, $t$-Young complexes naturally appear as the Alexander dual complexes of squarefree powers of $t$-path ideals of path graphs, as well as of certain ideals generated by subsets of their minimal generators. As an application, we derive formulas for the projective dimension and Krull dimension of these squarefree powers.

math.AC↗

arXiv2025

When are Morse resolutions polyhedral?

It is known that the chain complex of a simplex on $q$ vertices can be used to construct a free resolution of any ideal generated by $q$ monomials, and as a direct result, the Betti numbers always have binomial upper bounds, given by the number of faces of a simplex in each dimension. It is also known that for most monomials the resolution provided by the simplex is far from minimal. Discrete Morse theory provides an algorithm called \say{Morse matchings} by which faces of the simplex can be removed so that the chain complex on the remaining faces is still a free resolution of the same ideal. An immediate positive effect is an often considerable improvement on the bounds on Betti numbers. A caveat is the loss of the combinatorial structure of the simplex we started with: the output of the Morse matching process is a cell complex with no obvious structure besides an \say{address} for each cell. The main question in this paper is: which Morse matchings lead to Morse complexes that are polyhedral cell complexes? We prove that if a monomial ideal is minimally generated by up to four generators, then there is a maximal Morse matching of the simplex such that the resulting cell complex is a polyhedral cell complex. We then give an example of a monomial ideal minimally generated by six generators whose minimal free resolution is supported on a Morse complex and the Morse complex cannot be polyhedral no matter what Morse matching is chosen, and we go further to show that this ideal cannot have any polyhedral minimal free resolution.

math.AC↗

arXiv2024

Polyominoes and Knutson ideals

In this article, we study two fundamental questions on polyomino ideals which are radicality and primality. In order to study the question of radicality, we initiate the study of Knutson ideals among polyominoes. Knutson ideals were introduced by Conca and Varbaro after the work of Knutson on compatibly split ideals. Knutson ideals are known to have nice properties, for example, they are well behaved with Gr\"{o}bner bases, and it has square-free initial ideals; hence they are radical. We show that polyomino ideals associated with closed path, weakly closed path, simple thin, and ladder polyominoes are Knutson. We also show that polyomino ideals associated with a class of thin polyominoes are Knutson; hence they are radical. In fact, we show that these polyomino ideals are prime and the reduced Gr\"{o}bner basis is computed. Furthermore, we prove that under a certain condition, if a parallelogram polyomino is extracted from another parallelogram polyomino, the resulting collection of cells is Knutson. We also compute their Gr\"{o}bner basis.

math.AC↗

arXiv2023

Cohen-Macaulay permutation graphs

In this article, we characterize Cohen-Macaulay permutation graphs. In particular, we show that a permutation graph is Cohen-Macaulay if and only if it is well-covered and there exists a unique way of partitioning its vertex set into $r$ disjoint maximal cliques, where $r$ is the cardinality of a maximal independent set of the graph. We also provide some sufficient conditions for a comparability graph to be a uniquely partially orderable (UPO) graph.

math.AC↗

arXiv2023

Green-Lazarsfeld property $N_p$ for Segre product of Hibi rings

In this article, we prove that if a Hibi ring satisfies property $N_2$, then its Segre product with a polynomial ring in finitely many variables also satisfies property $N_2$. When the polynomial ring is in two variables, we also prove the above statement for $N_3$. Moreover, we study the minimal Koszul relations of the second syzygy module of Hibi rings.

math.AC↗

arXiv2023

On Cohen-Macaulay posets of dimension two and permutation graphs

We characterize Cohen-Macaulay posets of dimension two; they are precisely the shellable and strongly connected posets of dimension two. We also give a combinatorial description of these posets. Using the fact that co-comparability graph of a 2-dimensional poset is a permutation graph, we characterize Cohen-Macaulay permutation graphs.

math.CO↗

arXiv2023

Polyocollection ideals and primary decomposition of polyomino ideals

In this article, we study the primary decomposition of some binomial ideals. In particular, we introduce the concept of polyocollection, a combinatorial object that generalizes the definitions of collection of cells and polyomino, that can be used to compute a primary decomposition of non-prime polyomino ideals. Furthermore, we give a description of the minimal primary decomposition of non-prime closed path polyominoes. In particular, for such a class of polyominoes, we characterize the set of all zig-zag walks and show that the minimal prime ideals have a very nice combinatorial description.

math.AC↗

arXiv2022

The Charney-Davis conjecture for simple thin polyominoes

Let $\mathcal{P}$ be a simple thin polyomino and $\Bbbk$ a field. Let $R$ be the toric $\Bbbk$-algebra associated to $\mathcal{P}$. Write the Hilbert series of $R$ as $h_{R}(t)/(1-t)^{\dim(R)}$. We show that $$(-1)^{\left\lfloor{\frac{\mathrm{deg} h_R(t)}{2}}\right\rfloor}h_{R}(-1) \geq 0$$ if $R$ is Gorenstein. This shows that the Gorenstein rings associated to simple thin polyominoes satisfy the Charney-Davis conjecture.

math.AC↗

arXiv2021

The $h$-polynomial and the rook polynomial of some polyominoes

Let $X$ be a convex polyomino such that its vertex set is a sublattice of $\mathbb{N}^2$. Let $\Bbbk[X]$ be the toric ring (over a field $\Bbbk$) associated to $X$ in the sense of Qureshi, \emph{J. Algebra}, 2012. Write the Hilbert series of $\Bbbk[X]$ as $(1 + h_1 t + h_2 t^2 + \cdots )/(1-t)^{\dim(\Bbbk[X])}$. For $k \in \mathbb{N}$, let $r_k$ be the number of configurations in $X$ with $k$ pairwise non-attacking rooks. We show that $h_2 < r_2$ if $X$ is not a thin polyomino. This partially confirms a conjectured characterization of thin polyominoes by Rinaldo and Romeo, \emph{J. Algebraic Combin.}, 2021.

math.AC↗

arXiv2021

On the linearity of the syzygies of Hibi rings

In this article, we prove necessary conditions for Hibi rings to satisfy Green-Lazarsfeld property $N_p$ for $p=2$ and $3$. We also show that if a Hibi ring satisfies property $N_4$, then it is a polynomial ring or it has a linear resolution. Therefore, it satisfies property $N_p$ for all $p\geq 4$ as well. As a consequence, we characterize distributive lattices whose comparability graph is chordal in terms of the subposet of join-irreducibles of the distributive lattice. Moreover, we characterize complete intersection Hibi rings.

math.AC↗