On Modular Homology in the Boolean Algebra, II

Abstract

LetRbe an associative ring with identity and Ω ann-element set. Fork≤nconsider theR-moduleMkwithk-element subsets of Ω as basis. Ther-step inclusion map∂r:Mk→Mk−ris the linear map defined on this basis through where the Γiare the (k−r)-element subsets of Δ. Form<rone obtains chains of inclusion maps which have interesting homological properties ifRhas characteristicp>0. V. B. Mnukhin and J. Siemons (J. Combin. Theory74, 1996 287–300;J. Algebra179, 1995, 191–199) introduced the notion ofp-homologyto examine such sequences whenr=1 and here we continue this investigation whenris a power ofp. We show that any section of M not containing certainmiddle termsisp-exact and we determine the homology modules for such middle terms. Numerous infinite families of irreducible modules for the symmetric groups arise in this fashion. Among these thesemi-simple inductive systemsdiscussed by A. Kleshchev (J. Algebra181, 1996, 584–592) appear and in the special casep=5 we obtain theFibonacci representationsof A. J. E. Ryba (J. Algebra170, 1994, 678–686). There are also applications to permutation groups of order co-prime top, resulting inEuler–Poincaréequations for the number of orbits on subsets of such groups.