On Cohen–Macaulayness of Sn-Invariant Subspace Arrangements
On Cohen–Macaulayness of Sn-Invariant Subspace Arrangements
复制标题
关于 Sn 不变子空间排列的 Cohen-Macaulayness
DOI：
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
Steven V. Sam
中科院分区：
文献类型：
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作者：
Aaron Brookner;David Corwin;P. Etingof;Steven V. Sam
给定n的一个划分$lambda$，考虑$C^n$的子空间$E_lambda$，其中第一个$E_lambda_1 $坐标相等，第二个$E_lambda_2 $坐标相等，等等。本文研究了由对称群的平移联合组成的子空间排列$X_lambda$。特别地，我们专注于确定何时$X_lambda$是Cohen-Macaulay。这是受到以前的工作的第三作者来自研究的理性Cherednik代数，并回答了这个问题的积极时，所有部分的$lambda$是平等的。我们证明了当$lambda$至少有4个不同的部分时，$X_lambda$不是Cohen-Macaulay，并处理了大量的情况下，当$lambda$有2或3个不同的部分。沿着，我们还解决了Sergeev和Veselov关于变形牛顿和生成的代数的Cohen-Macaulay性的一个猜想。我们的技术结合了联合收割机经典的技术，从交换代数和不变理论，在许多情况下，我们可以减少一个无限的家庭，有时可以处理的计算机代数有限检查。
Given a partition $lambda$ of n, consider the subspace $E_lambda$ of $C^n$ where the first $lambda_1$ coordinates are equal, the next $lambda_2$ coordinates are equal, etc. In this paper, we study subspace arrangements $X_lambda$ consisting of the union of translates of $E_lambda$ by the symmetric group. In particular, we focus on determining when $X_lambda$ is Cohen-Macaulay. This is inspired by previous work of the third author coming from the study of rational Cherednik algebras and which answers the question positively when all parts of $lambda$ are equal. We show that $X_lambda$ is not Cohen-Macaulay when $lambda$ has at least 4 distinct parts, and handle a large number of cases when $lambda$ has 2 or 3 distinct parts. Along the way, we also settle a conjecture of Sergeev and Veselov about the Cohen-Macaulayness of algebras generated by deformed Newton sums. Our techniques combine classical techniques from commutative algebra and invariant theory, in many cases we can reduce an infinite family to a finite check which can sometimes be handled by computer algebra.