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 $\chi ({\mathcal{A}}_{\rho ,\mathit{g}},t)$.

中文翻译：

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代表拟阵的平行翻译

给定一个 $\mathbb{F}$代表的拟阵 $（\mathrm{中号}，\rho ）$ 与地面设置 $[米]$，表示ρ自然定义了超平面排列${\mathcal{一种}}_{\rho }$。我们将研究其平行翻译${\mathcal{一种}}_{\rho ，\mathit{G}}$ 的 ${\mathcal{一种}}_{\rho }$ 对所有人 $\mathit{G}\in {\mathbb{F}}^{米}$。它的交点半格$\mathrm{大号}（{\mathcal{一种}}_{\rho ，\mathit{G}}）$ 和特征多项式 $\chi （{\mathcal{一种}}_{\rho ，\mathit{G}}，Ť）$ 将根据派生排列的相交格子进行分类 ${\mathcal{一种}}_{\delta \rho }$，这是与派生的拟阵相关的超平面布置 $（\delta \mathrm{中号}，\delta \rho ）$并且在文献中也称为判别式排列。作为副产品，我们获得了特征多项式的比较结果和分解公式$\chi （{\mathcal{一种}}_{\rho ，\mathit{G}}，Ť）$。