For a polynomial ring S in n variables, we consider the natural action of the symmetric group ${\mathfrak{S}}_n$ on S by permuting the variables. For an ${\mathfrak{S}}_n$-invariant monomial ideal $I\subseteq S$ and $j\ge 0$, we give an explicit recipe for computing the modules ${\mathrm{Ext}}_S^j(S/I,S)$, and use this to describe the projective dimension and regularity of I. We classify the ${\mathfrak{S}}_n$-invariant monomial ideals I that have a linear free resolution, and also characterize those which are Cohen–Macaulay. We then consider two settings for analyzing the asymptotic behavior of regularity: one where we look at powers of a fixed ideal I, and another where we vary the dimension of the ambient polynomial ring and examine the invariant monomial ideals induced by I. In the first case we determine the asymptotic regularity for those ideals I that are generated by the ${\mathfrak{S}}_n$-orbit of a single monomial by solving an integer linear optimization problem. In the second case we describe the behavior of regularity for any I, recovering a recent result of Murai.

对于一个多项式环Š在ñ变量，我们考虑对称群的自然动作${\mathfrak{小号}}_{ñ}$通过置换变量在S上。为${\mathfrak{小号}}_{ñ}$不变单项式理想 $\mathrm{一世}\subseteq \mathrm{小号}$ 和 $Ĵ\ge 0$，我们给出了计算模块的明确方法 ${\mathrm{分机}}_{\mathrm{小号}}^{Ĵ}（\mathrm{小号}/\mathrm{一世}，\mathrm{小号}）$，并以此描述I的射影维和规律性。我们将${\mathfrak{小号}}_{ñ}$不变的单项式理想I具有线性自由分辨率，并且也表征那些是Cohen–Macaulay。然后，我们考虑两种设置来分析正则性的渐近行为：一种是查看固定理想I的幂，另一种是更改环境多项式环的维数并检查由I引起的不变单项理想。在第一种情况下，我们确定由I生成的那些理想I的渐近正则性。${\mathfrak{小号}}_{ñ}$通过求解整数线性优化问题，确定一个单项式的-轨道。在第二种情况下，我们描述了任意I的规律行为，从而恢复了Murai的最新结果。