Telecommunications Board. Embedded, Everywhere: A research agenda for networked Systems of Embedded Computers. National Academy

We study the quotient complex ∆(B lm)/S l ≀ Sm as a means of deducing facts about the ring k[x 1 ,. .. , x lm ] S l ≀Sm. It is shown in [He] that ∆(B lm)/S l ≀ Sm is shellable when l = 2, implying Cohen-Macaulayness of k[x 1 ,. .. , x 2m ] S 2 ≀Sm for any field k. We now confirm for all pairs (l, m) with l > 2 and m > 1 that ∆(B lm)/S l ≀ Sm is not Cohen-Macaulay over Z Z/2Z 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 ≀Sm. We also prove that ∆(B lm)/S l ≀ Sm and the links of many of its faces are collapsible, and we give a partitioning for ∆(B lm)/S l ≀ Sm.