A partitioning and related properties for the quotient complex $\mathrm{\Delta }(B_{lm})\mathrm{/}{S}_l\wr S_m$

We study the quotient complex $\mathrm{\Delta }(B_{lm})\mathrm{/}{S}_l\wr S_m$ as a means of deducing facts about the ring $k[x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_{lm}{]}^{S_l\wr S_m}$. It is shown in [He] that this quotient complex is shellable when $l=2$, implying Cohen-Macaulayness of $k[x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_{2m}{]}^{S_2\wr S_m}$ for any field $k$. We now confirm for all pairs $(l,m)$ with $l>2$ and $m>1$ that this quotient complex is not Cohen-Macaulay over $\text{integ}\mathrm{/}2\text{integ}$, but it is Cohen-Macaulay over fields of characteristic $p>m$ (independent of $l$). This yields corresponding characteristic-dependent results for the ring of invariants $k[x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_{lm}{]}^{S_l\wr S_m}$. We also prove that this quotient complex and the links of many of its faces are collapsible, and we give a partitioning for this quotient complex.