Aladin Virmaux
-
Partial categorification of Hopf algebras and representation theory of towers of \mathcalJ-trivial 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 \mathcalJ-trivial monoids
Authors: Aladin Virmaux 1
NULL
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 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.