The theories of character of symmetric group and of symmetric functions make use of the combinatorics of Young tableaux, such as the Robinson-Schensted algorithm, Schuetzenberger's "jeu de taquin", and evacuation.
In 1995 Poirier and Reutenauer 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 Malvenuto and Reutenauer in 1995.
In 2006 Aguiar and Sottile studied in more detail the so called Malvenuto-Reutenauer algebra of permutations : among other things, they introduce a new basis, by Moebius inversion in the poset of weak Bruhat order, that allows them to describe the primitive elements of the Hopf algebra of permutations.
Using this method, we determine the primitive elements of the Poirier-Reutenauer algebra of tableaux, using a partial order on tableaux defined by Taskin.
(Joint work with Christophe Reutenauer)