论文标题
亚历山大对称简单络合物和史丹利 - 赖斯纳理想的双重
Alexander duals of symmetric simplicial complexes and Stanley-Reisner Ideals
论文作者
论文摘要
给定一个上升链$（i_n）_ {n \ in \ mathbb {n}} $的$ \ sym $ -invariant squarefree单元理想，我们研究了亚历山大·迪斯（Alexander duals）$（i_n^\ vee）_ {n \ in \ mathbb in \ mathbb {使用一种新颖的组合工具，我们称\ emph {避免到对称性为对称性}，我们将根据原始发电机的方式提供对最小生成到对称性的最小生成的明确描述。将此结果与离散几何形状的方法相结合，这使我们能够证明$ i_n^\ vee $的轨道发生器数量由$ n $中的多项式给出，以实现足够大的$ n $。对于最小程度的轨道发生器的数量也是如此，最终是$ n $中的线性功能。前者的结果暗示，$ i_n $的主要组件的$ \ sym $ - 孔数量在$ n $中以$ n $的$ n $生长。作为另一个应用程序，我们表明，对于每个$ i \ geq 0 $，$ i_n $的相关stanley-reisner综合体的$ i $二维面数也由$ n $的$ n $用于$ n $。
Given an ascending chain $(I_n)_{n\in\mathbb{N}}$ of $\Sym$-invariant squarefree monomial ideals, we study the corresponding chain of Alexander duals $(I_n^\vee)_{n\in\mathbb{N}}$. Using a novel combinatorial tool, which we call \emph{avoidance up to symmetry}, we provide an explicit description of the minimal generating set up to symmetry in terms of the original generators. Combining this result with methods from discrete geometry, this enables us to show that the number of orbit generators of $I_n^\vee$ is given by a polynomial in $n$ for sufficiently large $n$. The same is true for the number of orbit generators of minimal degree, this degree being a linear function in $n$ eventually. The former result implies that the number of $\Sym$-orbits of primary components of $I_n$ grows polynomially in $n$ for large $n$. As another application, we show that, for each $i\geq 0$, the number of $i$-dimensional faces of the associated Stanley-Reisner complexes of $I_n$ is also given by a polynomial in $n$ for large $n$.
