论文标题
单词的包装单词的分解和自我二元性式对称函数
Decompositions of packed words and self duality of Word Quasisymmetric Functions
论文作者
论文摘要
通过Foissy的工作，单词quasisymmortric函数Hopf代数（WQSYM）的Bidendriform结构意味着它与双重偶性是同构的。但是，由于Vargas引起的唯一已知的显式同构不尊重双齿结构。这种结构完全由所谓的完全原始元素（两个半杂产元素消失）确定。在本文中，我们构建了两个新组合家族索引的基地，称为红色（双侧）和蓝色（原始的侧）双层森林，并用包装的单词进行两次射击。在这些基地中，原始元素由双层树和完全原始的元素索引，由一定的树木组成。我们小心地结合了红色和蓝色的森林，以获取双色森林。边缘的简单重新上色使我们能够获得WQSym的第一个显式双形自动形态。
By Foissy's work, the bidendriform structure of the Word Quasisymmetric Functions Hopf algebra (WQSym) implies that it is isomorphic to its dual. However, the only known explicit isomorphism due to Vargas does not respect the bidendriform structure. This structure is entirely determined by so-called totally primitive elements (elements such that the two half-coproducts vanish). In this paper, we construct two bases indexed by two new combinatorial families called red (dual side) and blue (primal side) biplane forests in bijection with packed words. In those bases, primitive elements are indexed by biplane trees and totally primitive elements by a certain subset of trees. We carefully combine red and blue forests to get bicolored forests. A simple recoloring of the edges allows us to obtain the first explicit bidendriform automorphism of WQSym.
