  We study the quotient complex $\Delta (B_{lm})/S_l\wr S_m$ as a means of
deducing facts about the ring $k[x_1,..., x_{lm}]^{S_l\wr S_m}$. It is shown in
[He] that this quotient complex is shellable when $l=2$, implying
Cohen-Macaulayness of $k[x_1,..., x_{2m}]^{S_2\wr S_m}$ 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 $\integ /2\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
$k[x_1,..., x_{lm}]^{S_l\wr S_m}$. 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.
