The Cohen–Macaulayness conjecture for generalized power-sum varieties

Let $\boldsymbol{\tau}=(\tau_1,\tau_2,\tau_3,\ldots)$ be a partition, and let $X_\lambda$ denote the variety associated with the partition $\lambda$ in the algebra of generalized power sums. Cohen–Macaulayness conjecture. $X_\lambda$ is Cohen–Macaulay if and only if one of the following holds: (1) $\lambda=(m^{(r)},1^{(s)})$ with $r\geq 1$ and $m>s\geq 0$; (2) $\lambda=(2^{(r)},1^{(s)})$ with $r\geq 1$ and $s\geq 0$; (3) $\lambda=(2m,m^{(s)})$ with $m\geq 1$. This conjecture summarizes the results of earlier work, the results established in the paper, and computer calculations; the converse classification remains open in the absence of a proof for all remaining partitions.