论文标题
在物种中的Cohen-Macaulay Hopf中
On Cohen-Macaulay Hopf monoids in species
论文作者
论文摘要
我们研究了物种类别中的Cohen-Macaulay Hopf Monoids。目的是将从拓扑组合学的技术应用于组合HOPF代数产生的多项式不变剂的研究。鉴于由线性化的hopf monoid引起的多项式不变，我们表明在某些条件下，它是相对简单复合物的希尔伯特多项式。如果Hopf Monoid是Cohen-Macaulay，我们为相应的相对简单复合物提供了必要的条件，以相对cohen-Macaulay，这意味着多项式具有非负$ H $ -vector。我们将结果应用于无环混合图的弱和强色多项式，以及双poset的多项式阶。
We study Cohen-Macaulay Hopf monoids in the category of species. The goal is to apply techniques from topological combinatorics to the study of polynomial invariants arising from combinatorial Hopf algebras. Given a polynomial invariant arising from a linearized Hopf monoid, we show that under certain conditions it is the Hilbert polynomial of a relative simplicial complex. If the Hopf monoid is Cohen-Macaulay, we give necessary and sufficient conditions for the corresponding relative simplicial complex to be relatively Cohen-Macaulay, which implies that the polynomial has a nonnegative $h$-vector. We apply our results to the weak and strong chromatic polynomials of acyclic mixed graphs, and the order polynomial of a double poset.
