Möbius conjugation and convolution formulae

Abstract Let P be a locally finite poset with the interval space Int ( P ) , and R be a ring with identity. We shall introduce the Mobius conjugation μ ⁎ sending each function f : P → R to an incidence function μ ⁎ ( f ) : Int ( P ) → R such that μ ⁎ ( f g ) = μ ⁎ ( f ) ⁎ μ ⁎ ( g ) . Taking P to be the intersection poset of a hyperplane arrangement A , we shall obtain a convolution identity for the number r ( A ) of regions and the number b ( A ) of relatively bounded regions, and a reciprocity theorem of the characteristic polynomial χ ( A , t ) which gives a combinatorial interpretation of the values | χ ( 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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