Mobius Conjugation and Convolution Formulae

Abstract

Let $P$ be a locally finite poset with the interval space $\Int(P)$, and $R$ a ring with identity. We shall introduce the M\"{o}bius conjugation $\mu^\ast$ sending each function $f:P\to R$ to an incidence function $\mu^\ast(f):\Int(P)\to R$ such that $\mu^\ast(fg)=\mu^\ast(f)\ast\mu^\ast(g)$. Taking $P$ to be the intersection poset of a hyperplane arrangement $\mathcal{A}$, we shall obtain a convolution identity for the number $r(\mathcal{A})$ of regions and the number $b(\mathcal{A})$ of relatively bounded regions, and a reciprocity theorem of the characteristic polynomial $\chi(\mathcal{A},t)$, which also leads to a combinatorial interpretation to the values $|\chi(\mathcal{A},-q)|$ for large primes $q$. Moreover, all known convolution identities on Tutte polynomials of matroids will be direct consequences after specializing the poset $P$ and functions $f,g$.