Combinatorial Topology and the Global Dimension of Algebras Arising in Combinatorics
Combinatorial Topology and the Global Dimension of Algebras Arising in Combinatorics
复制标题
组合拓扑和组合数学中出现的代数的全局维数
DOI：
10.4171/jems/579
复制
发表时间：
2012
期刊：
影响因子：
--
通讯作者：
B. Steinberg
中科院分区：
文献类型：
--
作者：
S. Margolis;Franco V. Saliola;B. Steinberg
在一篇极具影响力的论文中，Bidigare、Hanlon和Rockmore证明了许多流行的马尔可夫链是超平面排列面上的随机游动。他们对这些马尔可夫链的分析利用了脸部集合上的么半群结构。这一理论后来被Brown推广到一类更大的么半群，称为左正则带。在这两种情况下，这些么半群的表示理论发挥了突出的作用。特别地，它被用来计算马氏链的转移算子的谱，并证明了转移算子的可对角化。
本文建立了左正则带的代数不变量与组合不变量之间的紧密联系：证明了左正则带的代数的某些同调不变量与与左正则带自然相关的偏序集序复形的上同调重合。例如，我们证明了这些代数的整体维度由相关序复形的Leray数以上有界。相反，我们将一个左正则带关联到每个旗形复形，其代数的整体维度正好是旗形复形的勒雷数。
In a highly influential paper, Bidigare, Hanlon and Rockmore showed that a number of popular Markov chains are random walks on the faces of a hyperplane arrangement. Their analysis of these Markov chains took advantage of the monoid structure on the set of faces. This theory was later extended by Brown to a larger class of monoids called left regular bands. In both cases, the representation theory of these monoids played a prominent role. In particular, it was used to compute the spectrum of the transition operators of the Markov chains and to prove diagonalizability of the transition operators.
In this paper, we establish a close connection between algebraic and combinatorial invariants of a left regular band: we show that certain homological invariants of the algebra of a left regular band coincide with the cohomology of order complexes of posets naturally associated to the left regular band. For instance, we show that the global dimension of these algebras is bounded above by the Leray number of the associated order complex. Conversely, we associate to every flag complex a left regular band whose algebra has global dimension precisely the Leray number of the flag complex.
影响因子：
0.8
作者：
Gray R
通讯作者：
Gray R