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

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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 [He] that this quotient complex 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 this quotient complex is not Cohen-Macaulay over /2, 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[x₁,..., xlm]Sₗ Sₘ. 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.