Mobius共轭和卷积公式
Mobius Conjugation and Convolution Formulae
摘要：
设$P$是区间空间$\Int(P)$的局部有限偏序集，$R$是有单位元的环.我们将介绍莫比乌斯共轭$μ^\ast$发送每个功能$f:P\to R$的关联功能$μ^\ast(f):\Int(P)\to R$，使$μ^\ast(fg)=μ^\ast(f)\astμ^\ast(g)$。设$P$是超平面排列$\mathcal{A}$的交偏序集，我们将得到区域数$r(\mathcal{A})$与相对有界区域数$b(\mathcal{A})$的卷积恒等式，以及特征多项式$χ(\mathcal{A},t)$的互易定理，这也导致对大素数$q$的值$|χ(\mathcal{A},-q)|$的组合解释.此外，所有已知的Tutte多项式的拟阵的卷积恒等式将是专门化偏序集$P$和函数$f,g$后的直接结果。
英文摘要：
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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