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Partial categorification of Hopf algebras and representation theory of towers of \mathcalJ-trivial monoids

Abstract : 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.
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Aladin Virmaux. Partial categorification of Hopf algebras and representation theory of towers of \mathcalJ-trivial monoids. 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014), 2014, Chicago, United States. pp.741-752, ⟨10.46298/dmtcs.2438⟩. ⟨hal-01207568⟩

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