arXiv subjects

Shaheen Nazir

Publications and source records attributed to Shaheen Nazir.

15 recordsLinked to original sources

arXiv2025

On partitions associated with elementary symmetric polynomials

The elementary symmetric partition function is a map on the set of partitions. It sends a partition lambda to the partition whose parts are the summands in the evaluation of the elementary symmetric function on the parts of lambda. These elementary symmetric partition functions have been studied before, and are related to plethysm. In this note, we study properties of the elementary symmetric partition functions, particularly related to injectivity and the number of parts appearing in their image partitions.

math.CO↗

arXiv2023

Constructions and deformations of Calabi--Yau 3-folds in codimension 4

We construct polarized Calabi--Yau 3-folds with at worst isolated canonical orbifold points in codimension 4 that can be described in terms of the equations of the Segre embedding of $\mathbb P^2 \times \mathbb P^2$ in $\mathbb P^8$. We investigate the existence of other deformation families in their Hilbert scheme by either studying Tom and Jerry degenerations or by comparing their Hilbert series with those of existing low codimension Calabi--Yau 3-folds. Among other interesting results, we find a family of Calabi--Yau 3-fold with five distinct Tom and Jerry deformation families, a phenomenon not seen for $\mathbb Q$-Fano 3-folds. We compute the Hodge numbers of $\mathbb P^2 \times \mathbb P^2 $ Calabi--Yau 3-folds and corresponding manifolds obtained by performing crepant resolutions. We obtain a manifold with a pair of Hodge numbers that does not appear in the famously known list of 30108 distinct Hodge pairs of Kruzer--Skarke, in the list of 7890 distinct Hodge pairs corresponding to complete intersections in the product of projective spaces and in Hodge paris obtained from Calabi--Yau 3-folds having low codimension embeddings in weighted projective spaces.

math.AG↗

arXiv2023

Dimension of splines on graphs in the case of degree two and smoothness one in two variables

Continuous spline functions are defined as piecewise polynomials on the faces of a polyhedral complex that agree on the intersections of two faces. Splines are used in approximation theory and numerical analysis, with applications in data interpolation, to create smooth curves in computer graphics and to find numerical solutions to partial differential equations. Gilbert, Tymoczko, and Viel generalized the classical splines combinatorially and algebraically: a generalized spline is a vertex labeling of a graph $G$ by elements of the ring so that the difference between the labels of any two adjacent vertices lies in the ideal generated by the corresponding edge label. We study the generalized splines on the planar graphs whose edges are labeled by two-variable polynomials of the form $(ax+by+c)^2$ and whose vertices are labeled by polynomials of degree at most two. In this paper we address the upper-bound conjecture for the dimension of degree-2 splines of smoothness 1. The dimension is expressed in terms of the rank of the extended cycle basis matrix. We also provide a combinatorial algorithm on graphs to compute the rank by contracting certain subgraphs.

math.CO↗

arXiv2022

On the $f$-vectors of $r$-multichain subdivisions

For a poset $P$ and an integer $r\geq 1$, let $P_r$ be a collection of all $r$-multichains in $P$. Corresponding to each strictly increasing map $ı:[r]\rightarrow [2r]$, there is an order $\preceq_ı$ on $P_r$. Let $\D(G_ı(P_r))$ be the clique complex of the graph $G_ı$ associated to $P_r$ and $ı$. In a recent paper \cite{NW}, it is shown that $\D(G_ı(P_r))$ is a subdivision of $P$ for a class of strictly increasing maps. In this paper, we show that all these subdivisions have the same $f$-vector. We give an explicit description of the transformation matrices from the $f$- and $h$-vectors of $Δ$ to the $f$- and $h$-vectors of these subdivisions when $P$ is a poset of faces of $\D$. We study two important subdivisions Cheeger-Müller-Schrader's subdivision and the $r$-colored barycentric subdivision which fall in our class of $r$-multichain subdivisions.

math.CO↗

arXiv2021

On the homeomorphism and homotopy type of complexes of multichains

In this paper we define and study for a finite partially ordered set P a class of simplicial complexes on the set P_r of r-element multichains from P. The simplicial complexes depend on a strictly monotone function from [r] to [2r]. We show that there exactly 2^r such functions which yield subdivisions of the order complex of P of which 2^{r-1} are pairwise different. Within this class are for example the order complexes of the interval and the zig-zag poset of P and the rth edgewise subdivision of the order complex of P. We also exhibit a large subclass for which our simplicial complexes are order complexes and homotopy equivalent to the order complex of P.

math.CO↗

arXiv2019

The $f$- and $h$-vectors of Interval Subdivisions

The interval subdivision Int$(Δ)$ of a simplicial complex $Δ$ was introduced by Walker. We give the complete combinatorial description of the entries of the transformation matrices from the $f$- and $h$-vectors of $Δ$ to the $f$- and $h$-vectors of Int$(Δ)$. We show that if $Δ$ has non-negative $h$-vector then the $h$-polynomial of its interval subdivision has only real roots. As a consequence, we prove the Charney-Davis conjecture for Int$(Δ)$, if $Δ$ has non-negative reciprocal $h$-vector.

math.AC↗

arXiv2019

The equivariant cohomology of weighted flag orbifolds

We describe the torus-equivariant cohomology of weighted partial flag orbifolds ${\mathrm{w}}Σ$ of type $A$. We establish counterparts of several results known for the partial flag variety that collectively constitute what we refer to as ``Schubert Calculus on ${\mathrm{w}}Σ$''. For the weighed Schubert classes in ${\mathrm{w}}Σ$, we give the Chevalley's formula. In addition, we define the weighted analogue of double Schubert polynomials and give the corresponding Chevalley--Monk's formula.

math.AT↗

arXiv2018

On $\g$- and local $\g$-Vectors of the Interval Subdivision

We show that the $\g$-vector of the interval subdivision of a simplicial complex with a nonnegative and symmetric $h$-vector is nonnegative. In particular, we prove that such $\g$-vector is the $f$-vector of some balanced simplicial complex. Moreover, we show that the local $\g$-vector of the interval subdivision of a simplex is nonnegative; answering a question by Juhnke-Kubitzke et al.

math.AC↗

arXiv2017

An Efficient Algebraic Criterion for Shellability

In this paper, we give a new and efficient algebraic criterion for the pure as well as non-pure shellability of simplicial complex $Δ$ over [n]. We also give an algebraic characterization of a leaf in a simplicial complex (defined in [8]). Moreover, we introduce the concept of Gallai-simplicial complex $Δ_Γ(G)$ of a finite simple graph G. As an application, we show that the face ring of the Gallai simplicial complex associated to tree is Cohen-Macaulay.

math.AC↗

arXiv2017

Linear Residuals and Gallai-Simplicial Complexes

In this paper, we give a new algebraic criterion for the {\em shellability} of (non-pure) simplicial complex $Δ$ over $[n]$, shellable in the sense of Björner and Wachs \cite{BW}. We show that the spanning simplicial complex of doubly uni-cyclic graph is non-pure shellable. Moreover, we introduce the concept of Gallai-simplicial complex $Δ_Γ(G)$ of a finite simple graph $G$. We applied the obtained criterion to discuss the shellability of Gallai simplicial complexes associated to various classes of graphs..

math.AC↗

arXiv2013

On the admissibility of certain local system

A rank one local system on the complement of a hyperplane arrangement is said to be admissible if it satisfies certain non-positivity condition at every resonant edges. It is known that the cohomology of admissible local system can be computed combinatorially. In this paper, we study the structure of the set of all non-admissible local systems in the character torus. We prove that the set of non-admissible local systems forms a union of subtori. The relations with characteristic varieties are also discussed.

math.AG↗

arXiv2011

Generalizations of Nekrasov-Okounkov Identity

Nekrasov-Okounkov identity gives a product representation of the sum over partitions of a certain function of partition hook length. In this paper we give several generalizations of the Nekrasov-Okounkov identity using the cyclic symmetry of the topological vertex.

math.CO↗

arXiv2011

On the connectivity of the realization spaces of line arrangements

We prove that under certain combinatorial conditions, the realization spaces of line arrangements on the complex projective plane are connected. We also give several examples of arrangements with eight, nine and ten lines which have disconnected realization spaces.

math.AG↗

arXiv2008

Admissible local systems for a class of line arrangements

A rank one local system $\LL$ on a smooth complex algebraic variety $M$ is admissible roughly speaking if the dimension of the cohomology groups $H^m(M,\LL)$ can be computed directly from the cohomology algebra $H^*(M,\C)$. We say that a line arrangement $\A$ is of type $\CC_k$ if $k \ge 0 $ is the minimal number of lines in $\A$ containing all the points of multiplicity at least 3. We show that if $\A$ is a line arrangement in the classes $\CC_k$ for $k\leq 2$, then any rank one local system $\LL$ on the line arrangement complement $M$ is admissible. Partial results are obtained for the class $\CC_3$.

math.AG↗

arXiv2008

On the intersection of rational transversal subtori

We show that under a suitable transversality condition, the intersection of two rational subtori in an algebraic torus $(\C^*)^n$ is a finite group which can be determined using the torsion part of some associated lattice. Applications are given to the study of characteristic varieties of smooth complex algebraic varieties. As an example we discuss A. Suciu's line arrangement, the so-called deleted $B_3$-arrangement.

math.AG↗