Mobius Conjugation and Convolution Formulae

TL;DR

This paper introduces a M"{o}bius conjugation for functions on posets, leading to convolution identities and reciprocity theorems for hyperplane arrangements and Tutte polynomials, unifying various combinatorial results.

Contribution

It develops a new M"{o}bius conjugation framework for poset functions, deriving convolution identities and reciprocity theorems for arrangements and matroids.

Findings

Derived convolution identities for hyperplane arrangements.

Established reciprocity theorems for characteristic polynomials.

Unified known convolution identities for Tutte polynomials.

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 $f,g$.