arXiv math/98051281998-05-27math.CO

Orlik-Solomon代数与Tutte多项式

Orlik-Solomon algebras and Tutte polynomials

Carrie Eschenbrenner, Michael Falk

备注：: 12 pages, 2 figures

摘要：

拟阵$M$的$OS$代数$A$是与$M$的平坦格的Whitney同调相关的分次代数。如果$M$是$\C^r$中的超平面安排\A的基础拟阵，$A$同构于补$\C^r\setminus \bigcup \A.$的上同调代数很少有例子是已知的对安排与非同构拟阵，但同构$OS$代数。在所有已知的例子中，Tutte多项式是相同的，并且补是同伦等价的但不是同胚的。对任意给定的简单拟阵$M_0$，我们构造了一对无限拟阵族$M_n$和$M'_n$，$n\geq 1$，每个拟阵族都包含$M_0$作为子拟阵，其中对应的拟阵对具有同构的$OS$代数.如果种子拟阵$ M_0$是连通的，则$M_n$和$M'_n$具有不同的Tutte多项式。作为建设的后果，我们得到，对于任何$m$，$m$不同拟阵同构的$OS$代数。假设给定一对中心复超平面排列$\A_0$和$\A_1$。令$§$表示由$\C^1$中的超平面$\{0\}$组成的布置。我们定义了并行连接$P(\A_0,\A_1)$，一种实现底层拟阵的并行连接的安排，并证明了直和$\A_0 \oplus \A_1$和$§\oplus P(\A_0,\A_1)$具有同构补。

英文摘要：

The $OS$ algebra $A$ of a matroid $M$ is a graded algebra related to the Whitney homology of the lattice of flats of $M$. In case $M$ is the underlying matroid of a hyperplane arrangement \A in $\C^r$, $A$ is isomorphic to the cohomology algebra of the complement $\C^r\setminus \bigcup \A.$ Few examples are known of pairs of arrangements with non-isomorphic matroids but isomorphic $OS$ algebras. In all known examples, the Tutte polynomials are identical, and the complements are homotopy equivalent but not homeomorphic. We construct, for any given simple matroid $M_0$, a pair of infinite families of matroids $M_n$ and $M'_n$, $n\geq 1$, each containing $M_0$ as a submatroid, in which corresponding pairs have isomorphic $OS$ algebras. If the seed matroid $ M_0$ is connected, then $M_n$ and $M'_n$ have different Tutte polynomials. As a consequence of the construction, we obtain, for any $m$, $m$ different matroids with isomorphic $OS$ algebras. Suppose one is given a pair of central complex hyperplane arrangements $\A_0$ and $\A_1$. Let $§$ denote the arrangement consisting of the hyperplane $\{0\}$ in $\C^1$. We define the parallel connection $P(\A_0,\A_1)$, an arrangement realizing the parallel connection of the underlying matroids, and show that the direct sums $\A_0 \oplus \A_1$ and $§\oplus P(\A_0,\A_1)$ have diffeomorphic complements.

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