This paper completes the project of constructing combinatorial models (called frameworks) for the exchange graph and $\mathbf{g}$-vector fan associated to any exchange matrix $B$ whose Cartan companion is of finite or affine type, using the combinatorics and geometry of Coxeter-sortable elements and Cambrian lattices/fans. Specifically, we construct a framework in the unique non-acyclic affine case, the cyclically oriented $n$-cycle. In the acyclic affine case, a framework was constructed by combining a copy of the Cambrian fan for $B$ with an antipodal copy of the Cambrian fan for $-B$. In this paper, we extend this "doubled Cambrian fan'' construction to the oriented $n$-cycle, using a more general notion of sortable elements for quivers with cycles.
@article{10_37236_5387,
author = {Nathan Reading and David E Speyer},
title = {A {Cambrian} framework for the oriented cycle},
journal = {The electronic journal of combinatorics},
year = {2015},
volume = {22},
number = {4},
doi = {10.37236/5387},
zbl = {1329.05166},
url = {http://geodesic.mathdoc.fr/articles/10.37236/5387/}
}
TY - JOUR
AU - Nathan Reading
AU - David E Speyer
TI - A Cambrian framework for the oriented cycle
JO - The electronic journal of combinatorics
PY - 2015
VL - 22
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/5387/
DO - 10.37236/5387
ID - 10_37236_5387
ER -
%0 Journal Article
%A Nathan Reading
%A David E Speyer
%T A Cambrian framework for the oriented cycle
%J The electronic journal of combinatorics
%D 2015
%V 22
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/5387/
%R 10.37236/5387
%F 10_37236_5387
Nathan Reading; David E Speyer. A Cambrian framework for the oriented cycle. The electronic journal of combinatorics, Tome 22 (2015) no. 4. doi: 10.37236/5387
