论文标题
图案HOPF代数
Pattern Hopf algebras
论文作者
论文摘要
这是对组合结构中模式HOPF代数的研究。我们通过将物种的代数框架调整为组合术中的子结构的研究来介绍组合前膜的概念。之后，我们考虑计算对象模式数量的函数，并赋予这些函数的线性跨度，并使用乘积和共同点。通过这种方式，任何接受下部结构概念的组合对象家族都会产生HOPF代数，并且该关联起作用。例如，瓦尔加斯（Vargas）在2014年研究的置换和对称函数的霍普夫代数为对称函数所研究的hopf代数是这种结构的特殊情况。特定模式的Hopf代数家族引起了人们的关注，这些家族是由合并的前照示例引起的。这包括图表，posets和广义定居者上的预示。在这里，我们证明了对应于交换性预示的所有模式HOPF代数都是免费的。我们还研究了针对明显排列的非交换性预性，即具有明显元素的排列。这些对象具有称为通货膨胀的固有产品，这是由排列的分解定理动机的操作。在本文中，我们发现对标记排列的新分解定理，并用它们证明这是免费的hopf代数的另一个示例。
This is a study on pattern Hopf algebras in combinatorial structures. We introduce the notion of combinatorial presheaf, by adapting the algebraic framework of species to the study of substructures in combinatorics. Afterwards, we consider functions that count the number of patterns of objects and endow the linear span of these functions with a product and a coproduct. In this way, any well behaved family of combinatorial objects that admits a notion of substructure generates a Hopf algebra, and this association is functorial. For example, the Hopf algebra on permutations studied by Vargas in 2014 and the Hopf algebra on symmetric functions are particular cases of this construction. A specific family of pattern Hopf algebras is of interest, the ones arising from commutative combinatorial presheaves. This includes the presheaves on graphs, posets and generalized permutahedra. Here, we show that all the pattern Hopf algebras corresponding to commutative presheaves are free. We also study a non-commutative presheaf on marked permutations, i.e. permutations with a marked element. These objects have an inherent product called inflation, which is an operation motivated by factorization theorems of permutations. In this paper we find new factorization theorems on marked permutations, and use them to show that this is another example of a pattern Hopf algebra that is free.
