A partitioning and related properties for the quotient complex [formula omitted

Abstract

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