1Operations Research and Information Engineering, Cornell University 2Department of Mathematics, Cornell University 3Department of Mathematical Sciences, DePaul University
The electronic journal of combinatorics, Tome 28 (2021) no. 1
We study the connection between triangulations of a type $A$ root polytope and the resonance arrangement, a hyperplane arrangement that shows up in a surprising number of contexts. Despite an elementary definition for the resonance arrangement, the number of resonance chambers has only been computed up to the $n=8$ dimensional case. We focus on data structures for labeling chambers, such as sign vectors and sets of alternating trees, with an aim at better understanding the structure of the resonance arrangement, and, in particular, enumerating its chambers. Along the way, we make connections with similar (and similarly difficult) enumeration questions. With the root polytope viewpoint, we relate resonance chambers to the chambers of polynomiality of the Kostant partition function. With the hyperplane viewpoint, we clarify the connections between resonance chambers and threshold functions. In particular, we show that the base-2 logarithm of the number of resonance chambers is asymptotically $n^2$.
Samuel C. Gutekunst
1
;
Karola Mészáros
2
;
T. Kyle Petersen
3
1
Operations Research and Information Engineering, Cornell University
2
Department of Mathematics, Cornell University
3
Department of Mathematical Sciences, DePaul University
@article{10_37236_8759,
author = {Samuel C. Gutekunst and Karola M\'esz\'aros and T. Kyle Petersen},
title = {Root cones and the resonance arrangement},
journal = {The electronic journal of combinatorics},
year = {2021},
volume = {28},
number = {1},
doi = {10.37236/8759},
zbl = {1456.52028},
url = {http://geodesic.mathdoc.fr/articles/10.37236/8759/}
}
TY - JOUR
AU - Samuel C. Gutekunst
AU - Karola Mészáros
AU - T. Kyle Petersen
TI - Root cones and the resonance arrangement
JO - The electronic journal of combinatorics
PY - 2021
VL - 28
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/8759/
DO - 10.37236/8759
ID - 10_37236_8759
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%0 Journal Article
%A Samuel C. Gutekunst
%A Karola Mészáros
%A T. Kyle Petersen
%T Root cones and the resonance arrangement
%J The electronic journal of combinatorics
%D 2021
%V 28
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/8759/
%R 10.37236/8759
%F 10_37236_8759
Samuel C. Gutekunst; Karola Mészáros; T. Kyle Petersen. Root cones and the resonance arrangement. The electronic journal of combinatorics, Tome 28 (2021) no. 1. doi: 10.37236/8759
