Broken circuit complexes: Factorizations and generalizations
Broken circuit complexes: Factorizations and generalizations
复制标题
断路复合体：因式分解和概括
DOI：
10.1016/0095-8956(91)90008-8
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发表时间：
1991
期刊：
影响因子：
--
通讯作者：
G. Ziegler
中科院分区：
文献类型：
--
作者：
A. Björner;G. Ziegler
基于矩阵的特征多项式何时分解的问题，我们研究了断路复合体和根复合体（一类更一般的复合体）的联合分解。复合体的这种因式分解不仅可以引申出特征多项式的因式分解，而且可以引申出矩阵的ork - solomon代数的因式分解。当且仅当矩阵是超可解的，则矩阵的断路复形分解为零维子复形的多重连接。本案例的其他几个特征被导出。证明了一个矩阵是否可超解，可以从它的三元电路的知识和它的秩来判断。此外，一个超可解的矩阵可以由它的点和线的重合来重建。介绍了根络合物的一类，并证明了断路络合物的许多基本理论是可推广的。然而，对于非超可解的拟阵，根复合体的完全因子分解也是可能的，仍然会引起特征多项式的因子分解。
Motivated by the question of when the characteristic polynomial of a matroid factorizes, we study join-factorizations of broken circuit complexes and rooted complexes (a more general class of complexes). Such factorizations of complexes induce factorizations not only of the characteristic polynomial but also of the Orlik-Solomon algebra of the matroid. The broken circuit complex of a matroid factors into a multiple join of zero-dimensional subcomplexes for some linear order of the ground set if and only if the matroid is supersolvable. Several other characterizations of this case are derived. It is shown that whether a matroid is supersolvable can be determined from the knowledge of its 3-element circuits and its rank alone. Also, a supersolvable matroid can be reconstructed from the incidences of its points and lines. The class of rooted complexes is introduced, and much of the basic theory for broken circuit complexes is shown to generalize. Complete factorization of rooted complexes is, however, possible also for non-supersolvable matroids, still inducing factorization of the characteristic polynomial.