Abstract
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.
Highlights
Differential posets are a class of partially ordered sets introduced by Stanley [13] which generalize many of the enumerative and combinatorial properties of Young’s lattice Y, the poset of integer partitions ordered by inclusion of Young diagrams
Dual graded graphs are a generalization of differential posets developed independently by Fomin [4, 5]
G is an r-dual tower of groups if and only if the Bratteli diagram of G is an r-dual graded graph P
Summary
Differential posets are a class of partially ordered sets introduced by Stanley [13] which generalize many of the enumerative and combinatorial properties of Young’s lattice Y , the poset of integer partitions ordered by inclusion of Young diagrams. Dual graded graphs are a generalization of differential posets developed independently by Fomin [4, 5]. In analogy with Hasse diagrams of partially ordered sets, we write x y if ρ(x) ρ(y) and there is a path from x to y in P taking only upward steps. What we have defined here are usually called self-dual graded graphs. Many of the combinatorial properties of Young’s lattice Y are shared by all dual graded graphs.
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