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

International audience 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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Convolution Powers of the Identity

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2365 ◽

2013 ◽

Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽

Author(s):

Marcelo Aguiar ◽

Aaron Lauve

Keyword(s):

Lie Algebra ◽

Characteristic Polynomial ◽

Hopf Algebras ◽

Homogeneous Component ◽

Nous Obtenons ◽

Combinatorial Hopf Algebras ◽

Combinatorial Description ◽

Free Generators ◽

International Audience ◽

Convolution Powers

International audience We study convolution powers $\mathtt{id}^{\ast n}$ of the identity of graded connected Hopf algebras $H$. (The antipode corresponds to $n=-1$.) The chief result is a complete description of the characteristic polynomial - both eigenvalues and multiplicity - for the action of the operator $\mathtt{id}^{\ast n}$ on each homogeneous component $H_m$. The multiplicities are independent of $n$. This follows from considering the action of the (higher) Eulerian idempotents on a certain Lie algebra $\mathfrak{g}$ associated to $H$. In case $H$ is cofree, we give an alternative (explicit and combinatorial) description in terms of palindromic words in free generators of $\mathfrak{g}$. We obtain identities involving partitions and compositions by specializing $H$ to some familiar combinatorial Hopf algebras. Nous étudions les puissances de convolution $\mathtt{id}^{\ast n}$ de l’identité d’une algèbre de Hopf graduée et connexe $H$ quelconque. (L’antipode correspond à $n=-1$.) Le résultat principal est une description complète du polynôme caractéristique (des valeurs propres et de leurs multiplicités) de l’opérateur $\mathtt{id}^{\ast n}$ agissant sur chaque composante homogène $H_m$. Les multiplicités sont indépendants de $n$. Ceci résulte de l’examen de l’action des idempotents eulériens (supérieures) sur une algèbre de Lie $\mathfrak{g}$ associée à $H$. Dans le cas où $H$ est colibre, nous donnons une description alternative (explicite et combinatoire) en termes de mots palindromes dans les générateurs libres de $\mathfrak{g}$. Nous obtenons des identités impliquant des partitions et compositions en choisissant comme $H$ certaines algèbres de Hopf combinatoires connues.

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Extending the parking space

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2325 ◽

2013 ◽

Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽

Author(s):

Andrew Berget ◽

Brendon Rhoades

Keyword(s):

Representation Theory ◽

Symmetric Group ◽

Algebraic Combinatorics ◽

Parking Space ◽

Parking Functions ◽

Dyck Paths ◽

International Audience ◽

Special Case

International audience The action of the symmetric group $S_n$ on the set $\mathrm{Park}_n$ of parking functions of size $n$ has received a great deal of attention in algebraic combinatorics. We prove that the action of $S_n$ on $\mathrm{Park}_n$ extends to an action of $S_{n+1}$. More precisely, we construct a graded $S_{n+1}$-module $V_n$ such that the restriction of $V_n$ to $S_n$ is isomorphic to $\mathrm{Park}_n$. We describe the $S_n$-Frobenius characters of the module $V_n$ in all degrees and describe the $S_{n+1}$-Frobenius characters of $V_n$ in extreme degrees. We give a bivariate generalization $V_n^{(\ell, m)}$ of our module $V_n$ whose representation theory is governed by a bivariate generalization of Dyck paths. A Fuss generalization of our results is a special case of this bivariate generalization. L’action du groupe symétrique $S_n$ sur l’ensemble $\mathrm{Park}_n$ des fonctions de stationnement de longueur $n$ a reçu beaucoup d’attention dans la combinatoire algébrique. Nous démontrons que l’action de $S_n$ sur $\mathrm{Park}_n$ s’étend à une action de $S_{n+1}$. Plus précisément, nous construisons un gradué $S_{n+1}$-module $V_n$ telles que la restriction de $S_n$ est isomorphe à $\mathrm{Park}_n$. Nous décrivons la $S_n$-Frobenius caractères des modules $V_n$ à tous les degrés et décrivent le $S_{n+1}$-Frobenius caractères de $V_n$ en degrés extrêmes. Nous donnons une généralisation bivariée $V_n^{(\ell, m)}$ de notre module $V_n$ dont la représentation théorie est régie par une généralisation bivariée des chemins de Dyck. Une généralisation Fuss de nos résultats est un cas particulier de cette généralisation bivariée.

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The coalgebra structure of Hopf algebras

Journal of Algebra ◽

10.1016/0021-8693(78)90152-7 ◽

1978 ◽

Vol 50 (2) ◽

pp. 245-264 ◽

Cited By ~ 2

Author(s):

Kenneth Newman

Keyword(s):

Hopf Algebras ◽

Coalgebra Structure

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On Several types of universal invariants of framed links and 3-manifolds derived from Hopf algebras

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004106009674 ◽

2007 ◽

Vol 142 (1) ◽

pp. 73-92

Author(s):

JIANJUN PAUL TIAN

Keyword(s):

Representation Theory ◽

Hopf Algebras ◽

Comparison Study ◽

Comprehensive Comparison ◽

The Difference

AbstractWithout using representations of quasitriangular ribbon Hopf algebras, Hennings, Kauffman and Radford, Ohtsuki and the author gave different methods to construct invariants of links and 3-manifolds respectively. To understand these different methods and the resultant invariants, we made a comprehensive comparison study in this paper. We show the relations among the universal invariants of framed links defined by these different authors. We also figured out the relations among the resultant invariants of 3-manifolds defined by these different authors. Ignoring the difference by scalar constants and the inversion of Hopf algebras, we found that the invariants of 3-manifolds obtained by these authors are the same or equivalent to each other. Therefore, in a sense, there is only one way to construct invariants of 3-manifolds without using representation theory of Hopf algebras.

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Multi-cluster complexes

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.3014 ◽

2012 ◽

Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽

Author(s):

Cesar Ceballos ◽

Jean-Philippe Labbé ◽

Christian Stump

Keyword(s):

Coxeter Groups ◽

Type A ◽

Simplicial Complexes ◽

Cluster Complex ◽

Geometric Properties ◽

Cluster Complexes ◽

Combinatorial Description ◽

Compatibility Relation ◽

Positive Roots ◽

International Audience

International audience We present a family of simplicial complexes called \emphmulti-cluster complexes. These complexes generalize the concept of cluster complexes, and extend the notion of multi-associahedra of types ${A}$ and ${B}$ to general finite Coxeter groups. We study combinatorial and geometric properties of these objects and, in particular, provide a simple combinatorial description of the compatibility relation among the set of almost positive roots in the cluster complex. Nous présentons une famille de complexes simpliciaux appelés \emphcomplexes des multi-amas. Ces complexes généralisent le concept de complexes des amas et étendent la notion de multi-associaèdre de type ${A}$ et ${B}$ aux groupes de Coxeter finis. Nous étudions des propriétés combinatoires et géométriques de ces objets et, en particulier nous fournissons une description combinatoire simple de la relation de compatibilité sur l'ensemble des racines presque positives du complexe des amas.

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