Dual Graded Graphs and Bratteli Diagrams of Towers of Groups

An $r$-dual tower of groups is a nested sequence of finite groups, like the symmetric groups, whose Bratteli diagram forms an $r$-dual graded graph. Miller and Reiner introduced a special case of these towers in order to study the Smith forms of the up and down maps in a differential poset. Agarwal and the author have also used these towers to compute critical groups of representations of groups appearing in the tower. In this paper I prove that when $r=1$ or $r$ is prime, wreath products of a fixed group with the symmetric groups are the only $r$-dual tower of groups, and conjecture that this is the case for general values of $r$. This implies that these wreath products are the only groups for which one can define an analog of the Robinson-Schensted bijection in terms of a growth rule in a dual graded graph.

Dual filtered graphs

International audience We define a $K$ -theoretic analogue of Fomin’s dual graded graphs, which we call dual filtered graphs. The key formula in the definition is $DU - UD = D + I$. Our major examples are $K$ -theoretic analogues of Young’s lattice, the binary tree, and the graph determined by the Poirier-Reutenauer Hopf algebra. Most of our examples arise via two constructions, which we call the Pieri construction and the Möbius construction. The Pieri construction is closely related to the construction of dual graded graphs from a graded Hopf algebra, as described in Bergeron-Lam-Li, Nzeutchap, and Lam-Shimozono. The Möbius construction is more mysterious but also potentially more important, as it corresponds to natural insertion algorithms. Nous définissons un analogue $K$ -théorique aux graphes gradués en dualité de Fomin que nous appelons les graphes filtrés en dualité. La formule importante pour la définition est $DU - UD = D + I$. Nos principaux exemples sont un analogue $K$ -théorique aux graphe de Young, l’arbre binaire, et un graphe déterminé par l’algèbre de Hopf de Poirier-Reutenauer. La plupart de nos exemples surviennent de deux constructions que nous appelons la construction de Pieri et la construction de Möbius. La construction de Pieri est étroitement liée à la construction des graphes gradués en dualité d’une algèbre graduée de Hopf à la Bergeron-Lam-Li, Nzeutchap, et Lam-Shimozono. La construction de Möbius est plus mystérieuse, mais aussi peut-être plus importante car cette construction correspond aux algorithmes d’insertion naturelles.

Quantized Dual Graded Graphs

We study quantized dual graded graphs, which are graphs equipped with linear operators satisfying the relation $DU - qUD = rI$. We construct examples based upon: the Fibonacci differential poset, permutations, standard Young tableau, and plane binary trees.

A Combinatorial Model for $q$-Generalized Stirling and Bell Numbers

International audience We describe a combinatorial model for the $q$-analogs of the generalized Stirling numbers in terms of bugs and colonies. Using both algebraic and combinatorial methods, we derive explicit formulas, recursions and generating functions for these $q$-analogs. We give a weight preserving bijective correspondence between our combinatorial model and rook placements on Ferrer boards. We outline a direct application of our theory to the theory of dual graded graphs developed by Fomin. Lastly we define a natural $p,q$-analog of these generalized Stirling numbers.

Combinatorial Hopf Algebras and Towers of Algebras

International audience Bergeron and Li have introduced a set of axioms which guarantee that the Grothendieck groups of a tower of algebras $\bigoplus_{n \geq 0}A_n$ can be endowed with the structure of graded dual Hopf algebras. Hivert and Nzeutzhap, and independently Lam and Shimozono constructed dual graded graphs from primitive elements in Hopf algebras. In this paper we apply the composition of these constructions to towers of algebras. We show that if a tower $\bigoplus_{n \geq 0}A_n$ gives rise to graded dual Hopf algebras then we must have $\dim (A_n)=r^nn!$ where $r = \dim (A_1)$. Bergeron et Li ont donné un ensemble d'axiomes qui garanti que les groupes de Grothendieck d'une tour d'algèbres $\bigoplus_{n \geq 0}A_n$ peuvent être dotés d'une structure d'algèbres de Hopf graduées duales. Hivert et Nzeutzhap, et indépen\-damment Lam et Shimozono, ont construit des graphes gradués duals à partir d'éléments primitifs dans des algèbres de Hopf. Dans cet article, nous appliquons la composition de ces constructions aux tours des algèbres. Nous prouvons que si une tour $\bigoplus_{n \geq 0}A_n$ donne des algèbres de Hopf graduées duales, alors nous devons avoir $\dim (A_n)=r^nn!$ où $r = \dim (A_1)$.