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.

On the Rank Function of a Differential Poset

We study $r$-differential posets, a class of combinatorial objects introduced in 1988 by the first author, which gathers together a number of remarkable combinatorial and algebraic properties, and generalizes important examples of ranked posets, including the Young lattice. We first provide a simple bijection relating differential posets to a certain class of hypergraphs, including all finite projective planes, which are shown to be naturally embedded in the initial ranks of some differential poset. As a byproduct, we prove the existence, if and only if $r\geq 6$, of $r$-differential posets nonisomorphic in any two consecutive ranks but having the same rank function. We also show that the Interval Property, conjectured by the second author and collaborators for several sequences of interest in combinatorics and combinatorial algebra, in general fails for differential posets. In the second part, we prove that the rank function $p_n$ of any arbitrary $r$-differential poset has nonpolynomial growth; namely, $p_n\gg n^ae^{2\sqrt{rn}},$ a bound very close to the Hardy-Ramanujan asymptotic formula that holds in the special case of Young's lattice. We conclude by posing several open questions.

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.

The Induced Subgraph Order on Unlabelled Graphs

A differential poset is a partially ordered set with raising and lowering operators $U$ and $D$ which satisfy the commutation relation $DU-UD=rI$ for some constant $r$. This notion may be generalized to deal with the case in which there exist sequences of constants $\{q_n\}_{n\geq 0}$ and $\{r_n\}_{n\geq 0}$ such that for any poset element $x$ of rank $n$, $DU(x) = q_n UD(x) + r_nx$. Here, we introduce natural raising and lowering operators such that the set of unlabelled graphs, ordered by $G\leq H$ if and only if $G$ is isomorphic to an induced subgraph of $H$, is a generalized differential poset with $q_n=2$ and $r_n = 2^n$. This allows one to apply a number of enumerative results regarding walk enumeration to the poset of induced subgraphs.