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.
Highlights
Since Frobenius it has been known that the self-dual Hopf algebra of symmetric functions encodes the representation theory of the tower of symmetric groups Sym through the Frobenius character map
In [KT97], Krob and Thibon discovered that the same construction for the tower of Hecke algebras at q 0 gives rise to the pair of dual Hopf algebras Non Commutative Symmetric Functions (NCSF) and QSym
A natural but long running open question is that of categorification: Problem 1.1 Which combinatorial Hopf algebras can be recovered as Grothendieck groups of some tower of algebras?
Summary
Since Frobenius it has been known that the self-dual Hopf algebra of symmetric functions encodes the representation theory of the tower of symmetric groups Sym through the Frobenius character map. In [KT97], Krob and Thibon discovered that the same construction for the tower of Hecke algebras at q 0 gives rise to the pair of dual Hopf algebras NCSF and QSym (since the algebras are not semisimple, one needs to consider both the categories of simple and projective modules, which gives two p q p q Grothendieck rings G0 A and K0 A ). We obtain two copies of NCSF on different bases, including the well known ribbon basis RI
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