The character theory of symmetric groups, and the theory of symmetric functions, both make use of the combinatorics of Young tableaux, such as the Robinson–Schensted algorithm, Schützenberger’s “jeu de taquin”, and evacuation. In 1995 Poirier and the second author introduced some algebraic structures, different from the plactic monoid, which induce some products and coproducts of tableaux, with homomorphisms. Their starting point are the two dual Hopf algebras of permutations, introduced by the authors in 1995. In 2006 Aguiar and Sottile studied in more detail the Hopf algebra of permutations: among other things, they introduce a new basis, by Möbius inversion in the poset of weak order, that allows them to describe the primitive elements of the Hopf algebra of permutations. In the present Note, by a similar method, we determine the primitive elements of the Poirier–Reutenauer algebra of tableaux, using a partial order on tableaux defined by Taskin.

对称群的特征论和对称函数理论都利用了杨氏表的组合学，例如罗宾逊-申斯泰德算法、舒岑贝格的“jeu de taquin”, 和疏散。1995 年，Poirier 和第二作者介绍了一些代数结构，不同于 plac 幺半群，它产生了一些具有同态的 tableaux 的积和联积。他们的出发点是作者在 1995 年引入的两个对偶 Hopf 置换代数。 2006 年，Aguiar 和 Sottile 更详细地研究了置换的 Hopf 代数：除其他外，他们引入了一个新的基础，通过在偏序中的莫比乌斯反演弱阶，这使他们能够描述排列的 Hopf 代数的原始元素。在本笔记中，通过类似的方法，我们使用 Taskin 定义的 tableaux 上的偏序确定 tableaux 的 Poirier-Reutenauer 代数的原始元素。