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arXiv · 1209.2769

Mobius Conjugation and Convolution Formulae

Suijie Wang
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Abstract

Let $P$ be a locally finite poset with the interval space $\Int(P)$, and $R$ a ring with identity. We shall introduce the Möbius conjugation $μ^\ast$ sending each function $f:P\to R$ to an incidence function $μ^\ast(f):\Int(P)\to R$ such that $μ^\ast(fg)=μ^\ast(f)\astμ^\ast(g)$. 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 $χ(\mathcal{A},t)$, which also leads to a combinatorial interpretation to the values $|χ(\mathcal{A},-q)|$ 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 $f,g$.

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Suijie Wang. 2012-09-13. Mobius Conjugation and Convolution Formulae. https://doi.org/10.1016/j.jctb.2015.05.006

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