AI 中文总结
研究在置换上定义耦合积和抽取余积，构建霍普夫代数$\mathbb{KS}$，并在特征零情况下证明其与特定余交换霍普夫代数同构，耦合积和抽取余积是对称函数相关积和余积的置换类似物。
AI 中文摘要
我们在置换上定义了一个耦合积和一个抽取余积，并表明它们定义了一个分次、连通、余交换、自由的霍普夫代数$\mathbb{KS}$。在特征零的情况下，这意味着$\mathbb{KS}$同构于与置换上的马尔韦努托 - 罗伊滕瑙尔霍普夫代数的余根基滤过的对偶相关的某个余交换霍普夫代数。耦合积和抽取余积是对称函数单项式基上的积和余积的置换类似物。
英文摘要
We define a coupling product and a draw coproduct on permutations and show that they define a graded, connected, cocommutative, free Hopf algebra $\mathbb{KS}$. In characteristic zero, this implies that $\mathbb{KS}$ is isomorphic to a certain cocommutative Hopf algebra associated with the dual of the coradical filtration of the Malvenuto--Reutenauer Hopf algebra on permutations. The coupling product and draw coproduct are permutation analogues of the product and coproduct on the monomial basis of symmetric functions; therefore, we can say that our presentation is a monomial basis for $\mathbb{KS}$.
Comments14 pages
