ON MODULAR HOMOLOGY IN THE BOOLEAN-ALGEBRA, II

Let R be an associative ring with identity and Ohm an n-element set. F or less than or equal to n consider the R-module M-k with k-element su bsets of Ohm as basis. The r-step inclusion map partial derivative(r): M-k-->Mk-r is the linear map defined on this basis through partial ati ve(r)(Delta):=Gamma(1)+Gamma(2)+...+Gamma((rk)) where the Gamma(i) are the (k-r)-element subsets of Delta. For m < r one obtains chains M:0 <--(partial derivative r) M-m <--(partial derivative r) Mm+r <--(parti al derivative r) Mm+2r <--(partial derivative r) Mm+3r <--(partial der ivative r) <--(partial derivative r) ... <--(partial derivative r) <-- 0 of inclusion maps which have interesting homological properties if R has characteristic p > 0. V. B. Mnukhin and J. Siemens (J. Combin. T heory 74, 1996 287-300; J. Algebra 179, 1995, 191-199) introduced the notion of p-homology to examine such sequences when r = 1 and here we continue this investigation when r is a power of p. We show that any s ection of M not containing certain middle terms is p-exact and we dete rmine the homology modules for such middle terms. Numerous infinite fa milies of irreducible modules for the symmetric groups arise in this f ashion. Among these the semi-simple inductive systems discussed by A. Kleshchev (J. Algebra 181, 1996, 584-592) appear and in the special ca se p = 5 we obtain the Fibonacci representations of A. J. E. Ryba (J. Algebra 170, 1994, 678-686). There are also applications to permutatio n groups of order co-prime to p, resulting in Euler-Poincare' equation s for the number of orbits on subsets of such groups. (C) 1998 Academi c Press.