Orlik-Solomon Algebras and Tutte Polynomials
Orlik-Solomon Algebras and Tutte Polynomials
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Orlik-Solomon 代数和 Tutte 多项式
DOI：
10.1023/a:1018735815621
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发表时间：
1998
期刊：
影响因子：
--
通讯作者：
M. Falk
中科院分区：
文献类型：
--
作者：
Carrie Eschenbrenner;M. Falk
拟阵M的OS代数A是与M的平面格的Whitney同调有关的分次代数。如果M是超平面排列的下层拟阵Ainℂr，Ai同构于补ℂr∖∪A的上同调代数，则很少有非同构拟阵但OS代数同构的排列对的例子。在所有已知的例子中，图特多项式是恒等的，补是同伦等价的，但不是同胚的.对于任意给定的简单拟阵M0，我们构造了一对无穷族拟阵Mn和Mn‘，n≥1，每个都包含M0作为子拟阵，其中相应的对具有同构的OSalgebras.如果种子拟阵M_0是连通的，则M_n和M_n‘具有不同的图特多项式。作为构造的结果，我们得到了对任意m，m个具有同构OS代数的不同拟阵。假设有一对中心复超平面排列A0和A1。设S表示∪1中由超平面{0}组成的排列，我们定义了并行连接P(A0，A1)，这是一种实现底层拟阵的并行连接的排列，并证明了直和A0⊕A1和S⊕P(A0，A1)有异同胚补.
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 arrangementAin ℂr,Ais isomorphic to the cohomology algebra of the complement ℂr∖∪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 matroidM0, a pair of infinite families of matroidsMnandMn′,n≥ 1, each containingM0as a submatroid, in which corresponding pairs have isomorphicOSalgebras. If the seed matroidM0is connected, thenMnandMn′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 arrangementsA0andA1. LetSdenote the arrangement consisting of the hyperplane {0} in ∪1. We define the parallel connectionP(A0,A1), an arrangement realizing the parallel connection of the underlying matroids, and show that the direct sumsA0⊕A1andS⊕P(A0,A1) have diffeomorphic complements.