arXiv subjects

Houshan Fu

Publications and source records attributed to Houshan Fu.

24 records · Page 2Linked to original sources

arXiv2024

Deformed Intersections of Half-spaces

This paper is devoted to the classification problems concerning extended deformations of convex polyhedra and real hyperplane arrangements in the following senses: combinatorial equivalence of face posets, normal equivalence on normal fans of convex polyhedra, and sign equivalence on half-spaces. The extended deformations of convex polyhedra arise from parallel translations of given half-spaces and hyperplanes, whose normal vectors give rise to the so-called ``derived arrangement'' proposed by Rota as well as Crapo in different forms. We show that two extended deformations of convex polyhedra are normally (combinatorially, as a consequence) equivalent if they are parameterized by the same open face of the derived arrangement. Note that these extended deformations are based on parallel translations of the given hyperplanes. It allows us to study three deformations of real hyperplane arrangements: parallel translations, conings, and elementary lifts, whose configuration spaces are parameterized by open faces of the derived arrangement. Consequently, it gives a characterization of the normal, combinatorial, and sign equivalences of those three deformations via the derived arrangement. Additionally, the relationships among these three equivalence relations are discussed, and several new descriptions of real derived arrangements associated with faces and sign vectors of real hyperplane arrangements are provided.

math.CO↗

arXiv2023

One-element Extensions of Hyperplane Arrangements

We classify one-element extensions of a hyperplane arrangement by the induced adjoint arrangement. Based on the classification, several kinds of combinatorial invariants including Whitney polynomials, characteristic polynomials, Whitney numbers and face numbers, are constants on those strata associated with the induced adjoint arrangement, and also order-preserving with respect to the intersection lattice of the induced adjoint arrangement. As a byproduct, we obtain a convolution formula on the characteristic polynomials $χ(\mathcal{A}+H_{\bmα,a},t)$ when $\mathcal{A}$ is defined over a finite field $\mathbb{F}_q$ or a rational arrangement.

math.CO↗

arXiv2023

Adjoints of Matroids

We show that an adjoint of a loopless matroid is connected if and only if it itself is connected. Our first goal is to study the adjoint of modular matroids. We prove that a modular matroid has only one adjoint (up to isomorphism) which can be given by its opposite lattice, and proceed to present some alternative characterizations of modular matroids associated to adjoints and opposite lattices. The other purpose is to investigate the adjoint sequence $ad^0M,adM,ad^2M,\ldots$ of a connected matroid $M$. We classify such adjoint sequences into three types: finite, cyclic and convergent. For the first two types, the adjoint sequences eventually stabilize at the finite projective geometries except for free matroids. For the last type, the infinite non-repeating adjoint sequences are convergent to the infinite projective geometries.

math.CO↗

arXiv2020

Parallel Translates of Represented Matroids

Given an $\Bbb{F}$-represented matroid $(M,ρ)$ with the ground set $[m]$, the representation $ρ$ naturally defines a hyperplane arrangement $\mathcal{A}_ρ$. We will study its parallel translates $\mathcal{A}_{ρ,{g}}$ of $\mathcal{A}_ρ$ for all ${ g}\in \mathbb{F}^m$. Its intersection semi-lattices $L(\mathcal{A}_{ρ,{ g}})$ and the characteristic polynomials $χ(\mathcal{A}_{ρ,{ g}},t)$ will be classified by the intersection lattice of the derived arrangement $\mathcal{A}_{δρ}$, which is a hyperplane arrangement associated with the derived matroid $(δM,δρ)$ and also known as the discriminantal arrangement in the literature. As a byproduct, we obtain a comparison result and a decomposition formula on the characteristic polynomials $χ(\mathcal{A}_{ρ,{ g}},t)$.

math.CO↗

arXiv2020

Bijections on $r$-Shi and $r$-Catalan Arrangements

Associated with the $r$-Shi arrangement and $r$-Catalan arrangement in $\Bbb{R}^n$, we introduce a cubic matrix for each region to establish two bijections in a uniform way. Firstly, the positions of minimal positive entries in column slices of the cubic matrix will give a bijection from regions of the $r$-Shi arrangement to $O$-rooted labeled $r$-trees. Secondly, the numbers of positive entries in column slices of the cubic matrix will give a bijection from regions of the $r$-Catalan arrangement to pairings of permutation and $r$-Dyck path. Moreover, the numbers of positive entries in row slices of the cubic matrix will recover the Pak-Stanley labeling, a celebrated bijection from regions of the $r$-Shi arrangement to $r$-parking functions.

math.CO↗

arXiv2020

Truncated Homogeneous Symmetric Functions

Extending the elementary and complete homogeneous symmetric functions, we introduce the truncated homogeneous symmetric function $h_λ^{\dd}$ in $(\ref{THSF})$ for any integer partition $λ$, and show that the transition matrix from $h_λ^{\dd}$ to the power sum symmetric functions $p_λ$ is given by \[M(h^{\dd},p)=M'(p,m)z^{-1}D^{\dd},\] where $D^{\dd}$ and $z$ are nonsingular diagonal matrices. Consequently, $\{h_λ^{\dd}\}$ forms a basis of the ring $Λ$ of symmetric functions. In addition, we show that the generating function $H^{\dd}(t)=\ssum_{n\ge 0}h_n^{\dd}(x)t^n$ satisfies \[ω(H^{\dd}(t))=\left(H^{\dd}(-t)\right)^{-1},\] where $ω$ is the involution of $Λ$ sending each elementary symmetric function $e_λ$ to the complete homogeneous symmetric function $h_λ$.

math.CO↗