Dual graded graphs and Bratteli diagrams of towers of groups
Christian Gaetz

TL;DR
This paper characterizes specific towers of groups with dual graded graph structures, proving that for certain parameters, only wreath products of a fixed group with symmetric groups form such towers, impacting combinatorial bijections.
Contribution
It proves that for r=1 or prime, wreath products are the only r-dual towers of groups, and conjectures this for all r.
Findings
Wreath products are the only r-dual towers for r=1 or prime.
Implication that only these groups admit an analog of the Robinson-Schensted bijection.
Conjecture that this uniqueness extends to all r.
Abstract
An -dual tower of groups is a nested sequence of finite groups, like the symmetric groups, whose Bratteli diagram forms an -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 is one or prime, wreath products of a fixed group with the symmetric groups are the only -dual tower of groups, and conjecture that this is the case for general values of . 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.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
