A partitioning and related properties for the quotient complex Δ(Blm)/Sl Sm

We study the quotient complex Δ(Blm)/Sl Sm as a means of deducing facts about the ring k[x1,…,xlm]Sl Sm. It is shown in Hersh (preprint, 2000) that Δ(Blm)/Sl Sm is shellable when l=2, implying Cohen–Macaulayness of k[x1,…,x2m]S2 Sm for any field k. We now confirm for all pairs (l,m) with l>2 and m>1 that Δ(Blm)/Sl Sm is not Cohen–Macaulay over , but it is Cohen–Macaulay over fields of characteristic p>m (independent of l). This yields corresponding characteristic-dependent results for k[x1,…,xlm]Sl Sm. We also prove that Δ(Blm)/Sl Sm and the links of many of its faces are collapsible, and we give a partitioning for Δ(Blm)/Sl Sm.