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Given a partition   $\lambda $   of   $n$  , consider the subspace   $E_\lambda $   of   ${\mathbb {C}}^n$   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 (CM). 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 CM when   $\lambda $   has at least four distinct parts, and handle a large number of cases when   $\lambda $   has two or three 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.

On Cohen–Macaulayness of S n -Invariant Subspace Arrangements