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Dual graded graphs and Bratteli diagrams of towers of groups. (English) Zbl 1409.05212
Summary: 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.

##### MSC:
 05E10 Combinatorial aspects of representation theory 05C25 Graphs and abstract algebra (groups, rings, fields, etc.) 06A07 Combinatorics of partially ordered sets 06A11 Algebraic aspects of posets 20C30 Representations of finite symmetric groups
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