Aladin Virmaux

Partial categorification of Hopf algebras and representation theory of towers of \mathcalJtrivial monoids
dmtcs:2438 
Discrete Mathematics & Theoretical Computer Science,
January 1, 2014,
DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014)

https://doi.org/10.46298/dmtcs.2438
Partial categorification of Hopf algebras and representation theory of towers of \mathcalJtrivial monoids
Authors: Aladin Virmaux ^{1}
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Aladin Virmaux
1 Laboratoire de Recherche en Informatique
This paper considers the representation theory of towers of algebras of $\mathcal{J} trivial$ monoids. Using a very general lemma on induction, we derive a combinatorial description of the algebra and coalgebra structure on the Grothendieck rings $G_0$ and $K_0$. We then apply our theory to some examples. We first retrieve the classical KrobThibon's categorification of the pair of Hopf algebras QSym$/NCSF$ as representation theory of the tower of 0Hecke 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.