We study the asymptotic behavior of the Castelnuovo-Mumford regularity along chains of graded ideals in increasingly larger polynomial rings that are invariant under the action of symmetric groups. A linear upper bound for the regularity of such ideals is established. We conjecture that their regularity grows eventually precisely linearly. We establish this conjecture in several cases, most notably when the ideals are Artinian or squarefree monomial.

中文翻译：

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Castelnuovo-Mumford 正则性直至对称

我们研究了 Castelnuovo-Mumford 正则性在越来越大的多项式环中沿着梯度理想链的渐近行为，这些环在对称群的作用下是不变的。建立了这种理想规律的线性上限。我们推测它们的规律性最终会精确地线性增长。我们在几种情况下建立了这个猜想，最显着的是当理想是 Artinian 或无平方单项式时。