On the partial categorification of some Hopf algebras using the representation theory of towers of J -trivial monoids and semilattices
Résumé
This paper considers the representation theory of towers of algebras of J-trivial monoids. Using a very general lemma on induction, we derive a combinatorial description of the algebra and coalgebra structure on the Grothendieck rings G0 and K0. We then apply our theory to some examples. We first retrieve the classical Krob-Thibon's categorification of the pair of Hopf algebras QSym/NCSF as representation theory of the tower of 0-Hecke algebras. Considering the towers of semilattices given by the permutohedron, associahedron, and Boolean lattices, we categorify the algebra and the coalgebra structure of the Hopf algebras FQSym, PBT, and NCSF respectively. Lastly we completely describe the representation theory of the tower of the monoids of Non Decreasing Parking Functions. Résumé. Cet article traite de la théorie des représentations des tours d'alg ebres de mono¨des J-triviaux. Nous introduisons un lemme général d'induction, duquel nous déduisons une description combinatoire des alg ebres et cog ebres des groupes de Grothendieck G0 et K0.
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