It has been established in recent years how to approach acyclic cluster algebras of finite type using subword complexes. We continue this study by uniformly describing the - and -vectors, and by providing a conjectured description of the Newton polytopes of the -polynomials. We moreover show that this conjectured description would imply that finite type cluster complexes are realized by the duals of the Minkowski sums of the Newton polytopes of either the -polynomials or of the cluster variables, respectively. We prove this conjectured description to hold in type and in all types of rank at most including all exceptional types, leaving types , , and conjectural.
Accepted:
Published online:
DOI: https://doi.org/10.5802/alco.25
Keywords: cluster algebra, -polynomial, subword complexes
@article{ALCO_2018__1_4_545_0,
author = {Brodsky, Sarah B. and Stump, Christian},
title = {Towards a uniform subword complex description of acyclic finite type cluster algebras},
journal = {Algebraic Combinatorics},
pages = {545--572},
publisher = {MathOA foundation},
volume = {1},
number = {4},
year = {2018},
doi = {10.5802/alco.25},
mrnumber = {3875076},
zbl = {06963904},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.25/}
}
Brodsky, Sarah B.; Stump, Christian. Towards a uniform subword complex description of acyclic finite type cluster algebras. Algebraic Combinatorics, Volume 1 (2018) no. 4, pp. 545-572. doi : 10.5802/alco.25. https://alco.centre-mersenne.org/articles/10.5802/alco.25/
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