We give an explicit subword complex description of the generators of the type cone of the -vector fan of a finite type cluster algebra with acyclic initial seed. This yields in particular a description of the Newton polytopes of the -polynomials in terms of subword complexes as conjectured by S. Brodsky and the third author. We then show that the cluster complex is combinatorially isomorphic to the totally positive part of the tropicalization of the cluster variety as conjectured by D. Speyer and L. Williams.
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Keywords: Cluster algebras, F-polynomials, subword complex, type cone.
@article{ALCO_2021__4_5_757_0, author = {Jahn, Dennis and L\"owe, Robert and Stump, Christian}, title = {Minkowski decompositions for generalized associahedra of acyclic type}, journal = {Algebraic Combinatorics}, pages = {757--775}, publisher = {MathOA foundation}, volume = {4}, number = {5}, year = {2021}, doi = {10.5802/alco.177}, language = {en}, url = {http://www.numdam.org/articles/10.5802/alco.177/} }
TY - JOUR AU - Jahn, Dennis AU - Löwe, Robert AU - Stump, Christian TI - Minkowski decompositions for generalized associahedra of acyclic type JO - Algebraic Combinatorics PY - 2021 SP - 757 EP - 775 VL - 4 IS - 5 PB - MathOA foundation UR - http://www.numdam.org/articles/10.5802/alco.177/ DO - 10.5802/alco.177 LA - en ID - ALCO_2021__4_5_757_0 ER -
%0 Journal Article %A Jahn, Dennis %A Löwe, Robert %A Stump, Christian %T Minkowski decompositions for generalized associahedra of acyclic type %J Algebraic Combinatorics %D 2021 %P 757-775 %V 4 %N 5 %I MathOA foundation %U http://www.numdam.org/articles/10.5802/alco.177/ %R 10.5802/alco.177 %G en %F ALCO_2021__4_5_757_0
Jahn, Dennis; Löwe, Robert; Stump, Christian. Minkowski decompositions for generalized associahedra of acyclic type. Algebraic Combinatorics, Volume 4 (2021) no. 5, pp. 757-775. doi : 10.5802/alco.177. http://www.numdam.org/articles/10.5802/alco.177/
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