Partial categorification of Hopf algebras and representation theory of towers of \mathcalJ-trivial monoids
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014), DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014) (2014)
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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.
@article{DMTCS_2014_special_265_a63,
author = {Virmaux, Aladin},
title = {Partial categorification of {Hopf} algebras and representation theory of towers of {\mathcalJ-trivial} monoids},
journal = {Discrete mathematics & theoretical computer science},
year = {2014},
volume = {DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014)},
doi = {10.46298/dmtcs.2438},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2438/}
}
TY - JOUR AU - Virmaux, Aladin TI - Partial categorification of Hopf algebras and representation theory of towers of \mathcalJ-trivial monoids JO - Discrete mathematics & theoretical computer science PY - 2014 VL - DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014) UR - http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2438/ DO - 10.46298/dmtcs.2438 LA - en ID - DMTCS_2014_special_265_a63 ER -
%0 Journal Article %A Virmaux, Aladin %T Partial categorification of Hopf algebras and representation theory of towers of \mathcalJ-trivial monoids %J Discrete mathematics & theoretical computer science %D 2014 %V DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014) %U http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2438/ %R 10.46298/dmtcs.2438 %G en %F DMTCS_2014_special_265_a63
Virmaux, Aladin. Partial categorification of Hopf algebras and representation theory of towers of \mathcalJ-trivial monoids. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014), DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014) (2014). doi: 10.46298/dmtcs.2438
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