Dual graded graphs and Bratteli diagrams of towers of groups
The electronic journal of combinatorics, Tome 26 (2019) no. 1
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.
DOI :
10.37236/7790
Classification :
05E10, 05C25, 06A07, 06A11, 20C30
Mots-clés : dual graded graph, differential poset, tower of groups, Schensted correspondence, Bratteli diagram
Mots-clés : dual graded graph, differential poset, tower of groups, Schensted correspondence, Bratteli diagram
Affiliations des auteurs :
Christian Gaetz 1
@article{10_37236_7790,
author = {Christian Gaetz},
title = {Dual graded graphs and {Bratteli} diagrams of towers of groups},
journal = {The electronic journal of combinatorics},
year = {2019},
volume = {26},
number = {1},
doi = {10.37236/7790},
zbl = {1409.05212},
url = {http://geodesic.mathdoc.fr/articles/10.37236/7790/}
}
Christian Gaetz. Dual graded graphs and Bratteli diagrams of towers of groups. The electronic journal of combinatorics, Tome 26 (2019) no. 1. doi: 10.37236/7790
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