MÖBIUS INVERSION AND THE LEFSCHETZ INVARIANTS OF SOME p-SUBGROUP COMPLEXES

ABSTRACT:

By generalising Möbius inversion to finite acyclic directed multigraphs, we show that the coefficients for the reduced Lefschetz invariant of a suitable G-poset of p-subgroups are, essentially, values of a Möbius function. This allows us to present the Knörr–Robinson reformulation of Alperin's conjecture as a case of Möbius inversion.