Minkowski decompositions for generalized associahedra of acyclic type
Algebraic Combinatorics, Volume 4 (2021) no. 5, pp. 757-775.

We give an explicit subword complex description of the generators of the type cone of the g-vector fan of a finite type cluster algebra with acyclic initial seed. This yields in particular a description of the Newton polytopes of the F-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.

Received:
Revised:
Accepted:
Published online:
DOI: https://doi.org/10.5802/alco.177
Classification: 13F60,  20F55,  52B05,  05E45
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 = {https://alco.centre-mersenne.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
DA  - 2021///
SP  - 757
EP  - 775
VL  - 4
IS  - 5
PB  - MathOA foundation
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.177/
UR  - https://doi.org/10.5802/alco.177
DO  - 10.5802/alco.177
LA  - en
ID  - ALCO_2021__4_5_757_0
ER  - 
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. https://alco.centre-mersenne.org/articles/10.5802/alco.177/

[1] Arkani-Hamed, Nima; He, Song; Lam, Thomas Cluster configuration spaces of finite type (2020) (Preprint, available at arxiv.org/abs/2005.11419)

[2] Bazier-Matte, Véronique; Douville, Guillaume; Mousavand, Kaveh; Thomas, Hugh; Yıldırım, Emine ABHY Associahedra and Newton polytopes of F-polynomials for finite type cluster algebras (2018) (Preprint, available at arxiv.org/abs/1808.09986)

[3] Brodsky, Sarah B.; Stump, Christian Towards a uniform subword complex description of acyclic finite type cluster algebras, Algebr. Comb., Volume 1 (2018) no. 4, pp. 545-572 | Article | MR 3875076 | Zbl 1423.13118

[4] Ceballos, Cesar; Labbé, Jean-Philippe; Stump, Christian Subword complexes, cluster complexes, and generalized multi-associahedra, J. Algebraic Combin., Volume 39 (2014) no. 1, pp. 17-51 | Article | MR 3144391 | Zbl 1286.05180

[5] Chapoton, Frédéric; Fomin, Sergey; Zelevinsky, Andrei Polytopal realizations of generalized associahedra, Canad. Math. Bull., Volume 45 (2002) no. 4, pp. 537-566 | Article | MR 1941227 | Zbl 1018.52007

[6] Demonet, Laurent Mutations of group species with potentials and their representations. Applications to cluster algebras (2010) (Preprint, available at arxiv.org/abs/1003.5078)

[7] Derksen, Harm; Weyman, Jerzy; Zelevinsky, Andrei Quivers with potentials and their representations II: Applications to cluster algebras, J. Amer. Math. Soc., Volume 23 (2010) no. 3, pp. 749-790 | Article | MR 2629987 | Zbl 1208.16017

[8] Fomin, Sergey; Zelevinsky, Andrei Cluster algebras. II. Finite type classification, Invent. Math., Volume 154 (2003) no. 1, pp. 63-121 | Article | MR 2004457 | Zbl 1054.17024

[9] Fomin, Sergey; Zelevinsky, Andrei Y-systems and generalized associahedra, Ann. of Math. (2), Volume 158 (2003) no. 3, pp. 977-1018 | Article | MR 2031858 | Zbl 1057.52003

[10] Fomin, Sergey; Zelevinsky, Andrei Cluster algebras. IV. Coefficients, Compos. Math., Volume 143 (2007) no. 1, pp. 112-164 | Article | MR 2295199 | Zbl 1127.16023

[11] Hohlweg, Christophe; Lange, Carsten E. M. C.; Thomas, Hugh Permutahedra and generalized associahedra, Adv. Math., Volume 226 (2011) no. 1, pp. 608-640 | Article | MR 2735770 | Zbl 1233.20035

[12] Hohlweg, Christophe; Pilaud, Vincent; Stella, Salvatore Polytopal realizations of finite type g-vector fans, Adv. Math., Volume 328 (2018), pp. 713-749 | Article | MR 3771140 | Zbl 1382.05075

[13] Knutson, Allen; Miller, Ezra Subword complexes in Coxeter groups, Adv. Math., Volume 184 (2004) no. 1, pp. 161-176 | Article | MR 2047852 | Zbl 1069.20026

[14] Lampe, Philipp On the approximate periodicity of sequences attached to non-crystallographic root systems, Exp. Math., Volume 27 (2018) no. 3, pp. 265-271 | Article | MR 3857662 | Zbl 1423.13126

[15] McMullen, Peter Representations of polytopes and polyhedral sets, Geometriae Dedicata, Volume 2 (1973), pp. 83-99 | Article | MR 326574 | Zbl 0273.52006

[16] Padrol, Arnau; Palu, Yann; Pilaud, Vincent; Plamondon, Pierre-Guy Associahedra for finite type cluster algebras and minimal relations between g-vectors (2019) (Preprint, available at arxiv.org/abs/1906.06861)

[17] Pilaud, Vincent; Stump, Christian Brick polytopes of spherical subword complexes and generalized associahedra, Adv. Math., Volume 276 (2015), pp. 1-61 | Article | MR 3327085 | Zbl 1405.05196

[18] Reading, Nathan Sortable elements and Cambrian lattices, Algebra Universalis, Volume 56 (2007) no. 3-4, pp. 411-437 | Article | MR 2318219 | Zbl 1184.20038

[19] Speyer, David; Williams, Lauren The tropical totally positive Grassmannian, J. Algebraic Combin., Volume 22 (2005) no. 2, pp. 189-210 | Article | MR 2164397 | Zbl 1094.14048

Cited by Sources: