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.

Keywords

• algebraic combinatorics
• combinatorial representation theory
• combinatorial hopf algebras
• lattices
• categorification
• [info.info-dm] computer science [cs]/discrete mathematics [cs.dm]
• [math.math-co] mathematics [math]/combinatorics [math.co]