Parallel translates of represented matroids

Abstract

Given an $\mathbb{F}$-represented matroid $(M,\rho )$ with the ground set $[m]$, the representation ρ naturally defines a hyperplane arrangement ${\mathcal{A}}_{\rho }$. We will study its parallel translates ${\mathcal{A}}_{\rho ,\mathit{g}}$ of ${\mathcal{A}}_{\rho }$ for all $\mathit{g}\in {\mathbb{F}}^m$. Its intersection semi-lattices $L({\mathcal{A}}_{\rho ,\mathit{g}})$ and the characteristic polynomials $\chi ({\mathcal{A}}_{\rho ,\mathit{g}},t)$ will be classified by the intersection lattice of the derived arrangement ${\mathcal{A}}_{\delta \rho }$, which is a hyperplane arrangement associated with the derived matroid $(\delta M,\delta \rho )$ 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 $\chi ({\mathcal{A}}_{\rho ,\mathit{g}},t)$.

Introduction

Our matroid terminology and notation will follow Oxley [11]. For a field $\mathbb{F}$, let M be an $\mathbb{F}$-represented matroid of rank r with the ground set $[m]=\{1,2,\dots ,m\}$, and let $\rho :[m]\to {\mathbb{F}}^n$ be a matroid representation of M and write $\rho (i)={\mathit{\rho }}_i$, $i\in [m]$. Denote by the pair $(M,\rho )$ an $\mathbb{F}$-represented matroid. Associated with $(M,\rho )$, the hyperplane arrangement ${\mathcal{A}}_{\rho }$ consists of hyperplanes $H_i:{\mathit{\rho }}_i\mathit{x}=0$ for all $i\in [m]$. We are interested in classifying all parallel translates ${\mathcal{A}}_{\rho ,\mathit{g}}=\{H_{{\mathit{\rho }}_1,g_1},\dots ,H_{{\mathit{\rho }}_m,g_m}\}$ of ${\mathcal{A}}_{\rho }$, where $\mathit{g}=(g_1,\dots ,g_m)$ is a vector in ${\mathbb{F}}^m$ and the hyperplanes $H_{{\mathit{\rho }}_i,g_i}$ are defined by the equations

$$H_{{\mathit{\rho }}_i,g_i}:{\mathit{\rho }}_i\mathit{x}=g_i,1\le i\le m.$$

A vector is understood as either a row vector or a column vector, depending on its meaning in the context. Our motivation is the work of Athanasiadis [2] on deformations of hyperplane arrangements and the work of Oxley-Wang [12] on derived matroids.

Given a hyperplane arrangement $\mathcal{A}$ in a vector space V. The characteristic polynomial of $\mathcal{A}$ is

$$\chi (\mathcal{A},t):=\sum _{X\in L(\mathcal{A})}\mu (V,X)t^{\dim X},$$

where $L(\mathcal{A})$ is the intersection semi-lattice whose members are all possible nonempty intersections of hyperplanes of $\mathcal{A}$, including V, ordered by the reverse of set inclusion, and μ is the Möbius function on $L(\mathcal{A})$; see [10], [15], [16]. Our classification about the parallel translates ${\mathcal{A}}_{\rho ,\mathit{g}}$ with $\mathit{g}\in {\mathbb{F}}^m$ is to classify the semi-lattices $L({\mathcal{A}}_{\rho ,\mathit{g}})$ and the characteristic polynomials $\chi ({\mathcal{A}}_{\rho ,\mathit{g}},t)$. When $\mathit{g}=\mathbf{0}$, denote ${\mathcal{A}}_{\rho }={\mathcal{A}}_{\rho ,\mathbf{0}}$ whose characteristic polynomial $\chi ({\mathcal{A}}_{\rho },t)$ is the characteristic polynomial $\chi (M,t)$ of the matroid M (see [13], [16], [17]). So ${\mathcal{A}}_{\rho ,\mathit{g}}$ is a parallel translate of ${\mathcal{A}}_{\rho }$.

Rota first introduced the concept of derived matroids to investigate “dependencies among dependencies” of matroids at the Bowdoin College Summer 1971 NSF Conference on Combinatorics. In 1980, Longyear [8] studied the derived matroids δM when M is ${\mathbb{F}}_2$-represented. For any field $\mathbb{F}$, Oxley and Wang [12] gave a specific definition of the derived matroid of an $\mathbb{F}$-represented matroid $(M,\rho )$. Let $\mathcal{C}(M)$ denote the set of all circuits of M. The derived matroid $(\delta M,\delta \rho )$ is an $\mathbb{F}$-represented matroid of rank $r(\delta M)=m-r(M)$ with the ground set $\mathcal{C}(M)$ such that

$$(\delta \rho )(I)={\mathbf{c}}_I\text{ for }I\in \mathcal{C}(M),$$

where the circuit vector ${\mathbf{c}}_I=(c_1,c_2,\dots ,c_m)$ in ${\mathbb{F}}^m$ is unique (up to a non-zero scalar multiple) and defined by ${\sum }_{i=1}^m{c}_i{\mathit{\rho }}_i=0$, where $c_i\ne 0$ if and only if $i\in I$. Similar as ${\mathcal{A}}_{\rho }$, the hyperplane arrangement ${\mathcal{A}}_{\delta \rho }$ in ${\mathbb{F}}^m$ is defined by

$${\mathcal{A}}_{\delta \rho }=\{H_I|I\in \mathcal{C}(M)\},\mathrm{where}{H}_I:{\mathit{c}}_I\mathit{x}=0$$

and called the derived arrangement of $(M,\rho )$.

In 1989, Manin and Schechtman [9] introduced the discriminantal arrangement to characterize the general position parallel translates of an affine hyperplane arrangement in general position, known as Manin-Schechtman arrangement. In 1994, Falk [6] shown that neither combinatorial nor topological structure of the discriminantal arrangement is independent of the original arrangement, that is, in matroid language, the derived matroid $(\delta M,\delta \rho )$ depends on the representation ρ of the original matroid M, confirmed in [12, page 4-6]. In 1997, Bayer and Brandt [3] studied the discriminantal arrangements without general position assumptions on the original arrangement. It is easily seen from [3, Theorem 2.4] that the discriminantal arrangement is exactly the derived arrangement defined above. Please see [1], [7], [14] for more study on discriminantal arrangements.

For each $X\in L({\mathcal{A}}_{\delta \rho })$, consider the restriction of ${\mathcal{A}}_{\delta \rho }$ to X, denoted ${\mathcal{A}}_{\delta \rho }/X$, which is a hyperplane arrangement in X consisting of the hyperplanes $H_I\cap X$ of X, where $H_I\in {\mathcal{A}}_{\delta \rho }$ and $X⊈H_I$. Let $M({\mathcal{A}}_{\delta \rho }/X)=X\setminus {\bigcup }_{H_I:X⊈H_I\in {\mathcal{A}}_{\delta \rho }}H$ be the complement of ${\mathcal{A}}_{\delta \rho }/X$. We have the disjoint decomposition [16, page 410]

$${\mathbb{F}}^m=\underset{X\in L({\mathcal{A}}_{\delta \rho })}{⨆}M({\mathcal{A}}_{\delta \rho }/X).$$

Indeed, for each $\mathit{x}\in {\mathbb{F}}^m$, assuming $X={\bigcap }_{\mathit{x}\in H_I\in {\mathcal{A}}_{\delta \rho }}{H}_I$, we have $\mathit{x}\in M({\mathcal{A}}_{\delta \rho }/X)$. Below are our main results.

Theorem 1.1 Geometric Characterization

If vectors $\mathit{g},\mathit{h}\in M({\mathcal{A}}_{\delta \rho }/X)$ for some $X\in L({\mathcal{A}}_{\delta \rho })$, then

$$L({\mathcal{A}}_{\rho ,\mathit{g}})\cong L({\mathcal{A}}_{\rho ,\mathit{h}})\mathit{\text{and}}\chi ({\mathcal{A}}_{\rho ,\mathit{g}},t)=\chi ({\mathcal{A}}_{\rho ,\mathit{h}},t).$$

When $\mathit{g}\in M({\mathcal{A}}_{\delta \rho })$, the arrangement ${\mathcal{A}}_{\rho ,\mathit{g}}$ is in general position, exactly the same as the arrangement studied by Bayer and Brandt [3]. Theorem 1.1 says that $\chi ({\mathcal{A}}_{\rho ,\mathit{g}},t)$ are the same polynomial for all $\mathit{g}\in M({\mathcal{A}}_{\delta \rho }/X)$, denoted by $\chi (X,t)$. For convenience let us write

$$\chi ({\mathcal{A}}_{\rho ,\mathit{g}},t)=\chi (X,t)=\sum _{k=0}^r{(-1)}^k{a}_k(X)t^{n-k},\mathit{g}\in M({\mathcal{A}}_{\delta \rho }/X).$$

Theorem 1.2 Uniform Comparison Theorem

Under the notations of (4), if $X\subseteq Y$ for $X,Y\in L({\mathcal{A}}_{\delta \rho })$, then

$$a_k(X)\le a_k(Y),0\le k\le r.$$

Theorem 1.3 Decomposition Formula

Under the notations of (4), if $\mathbb{F}$ is a finite field of q elements, then

$$\sum _{X\in L({\mathcal{A}}_{\delta \rho })}\chi (X,q)\chi ({\mathcal{A}}_{\delta \rho }/X,q)=q^n{(q-1)}^m.$$

If $\mathbb{F}$ is an infinite field, then

$$\sum _{X\in L({\mathcal{A}}_{\delta \rho })}\chi (X,t)\chi ({\mathcal{A}}_{\delta \rho }/X,t)=t^n{(t-1)}^m.$$

To end this section, we give a small example to illustrate the above theorems.

Example 1.4

Consider the uniform matroid $U_{2,4}$ represented over $\mathbb{R}$ by

$${\mathit{\rho }}_1=(1,0),{\mathit{\rho }}_2=(0,1),{\mathit{\rho }}_3=(1,1),{\mathit{\rho }}_4=(1,-1).$$

The associated arrangement ${\mathcal{A}}_{\rho }$ consists of hyperplanes $H_i:{\mathit{\rho }}_i\mathit{x}=0$ in ${\mathbb{R}}^2$. For convenience we assume that the ground set of $U_{2,4}$ is $E=\{{\mathit{\rho }}_i|i=1,2,3,4\}$. It is obvious that for each i, $I_i=E\setminus \{{\mathit{\rho }}_i\}$ is a circuit. The corresponding circuit vectors are

$${\mathit{c}}_{I_1}=(0,2,-1,1),{\mathit{c}}_{I_2}=(2,0,-1,-1),{\mathit{c}}_{I_3}=(1,-1,0,-1),{\mathit{c}}_{I_4}=(1,1,-1,0),$$

and the derived arrangement is ${\mathcal{A}}_{\delta \rho }=\{H_{I_i}:{\mathit{c}}_{I_i}\mathit{x}=0|i=1,2,3,4\}$ in ${\mathbb{R}}^4$. The intersection lattice $L({\mathcal{A}}_{\mathit{\delta \rho }})$ consists of flats $V={\mathbb{R}}^4,H_{I_1},H_{I_2},H_{I_3},H_{I_4}$, $W=\bigcap _{i=1}^4{H}_{I_i}$ with $W\subseteq H_{I_i}\subseteq V$. By routine calculations, we can obtain the characteristic polynomials

$$\chi ({\mathcal{A}}_{\delta \rho }/V,t)=t^4-4t^3+3t^2,\chi ({\mathcal{A}}_{\delta \rho }/H_{I_i},t)=t^3-t^2,\chi ({\mathcal{A}}_{\delta \rho }/W,t)=t^2,$$

and

$$\chi (V,t)=t^2-4t+6,\chi (H_{I_i},t)=t^2-4t+5,\chi (W,t)=t^2-4t+3.$$

As Theorem 1.2, Theorem 1.3, we have the coefficient comparison $a_k(W)\le a_k(H_{I_i})\le a_k(V)$ for $k=0,1,2,i=1,2,3,4$, and the decomposition formula

$$\sum _{X\in L({\mathcal{A}}_{\delta \rho })}\chi (X,t)\chi ({\mathcal{A}}_{\delta \rho }/X,t)=t^2{(t-1)}^4.$$