ALGEBRAIC COMBINATORICS

no. 4
Towards a uniform subword complex description of acyclic finite type cluster algebras
Algebraic Combinatorics, Volume 1 (2018) no. 4, pp. 545-572.

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 c- and g-vectors, and by providing a conjectured description of the Newton polytopes of the F-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 F-polynomials or of the cluster variables, respectively. We prove this conjectured description to hold in type A and in all types of rank at most 8 including all exceptional types, leaving types B, C, and D conjectural.

Received: 2017-12-20
Accepted: 2018-06-12
Published online: 2018-09-10
DOI: https://doi.org/10.5802/alco.25
Keywords: cluster algebra, F-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},
     publisher = {MathOA foundation},
     volume = {1},
     number = {4},
     year = {2018},
     pages = {545-572},
     doi = {10.5802/alco.25},
     mrnumber = {3875076},
     zbl = {06963904},
     language = {en},
     url = {alco.centre-mersenne.org/item/ALCO_2018__1_4_545_0/}
}
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/item/ALCO_2018__1_4_545_0/

[1] Borovik, Alexandre V.; Gelfand, Israil M.; White, Neil Coxeter matroids, Progress in Mathematics, Volume 216, Birkhäuser, 2003 | MR 1989953 | Zbl 1050.52005

[2] Brodsky, Sarah B.; Ceballos, Cesar; Labbé, Jean-Philippe Cluster algebras of type D 4 , tropical planes, and the positive tropical Grassmannian, Beitr. Algebra Geom., Volume 58 (2017) no. 1, pp. 25-46 | Article | MR 3607668 | Zbl 06695676

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

[4] Ceballos, Cesar; Pilaud, Vincent Denominator vectors and compatibility degrees in cluster algebras of finite type, Trans. Am. Math. Soc., Volume 367 (2015) no. 2, pp. 1421-1439 | Article | MR 3280049 | Zbl 1350.13020

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

[6] Fomin, Sergey; Zelevinsky, Andrei Cluster algebras I: foundations, J. Am. Math. Soc., Volume 15 (2002) no. 2, pp. 497-529 | Article | MR 1887642 | Zbl 1021.16017

[7] 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

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

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

[10] Humphreys, James E. Reflection groups and Coxeter groups Volume 29, Cambridge University Press, 1990, xii+204 pages | MR 1066460 | Zbl 0725.20028

[11] 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

[12] Knutson, Allen; Miller, Ezra Gröbner geometry of Schubert polynomials, Ann. Math., Volume 161 (2005) no. 3, pp. 1245-1318 | Article | Zbl 1089.14007

[13] Lange, Carsten Minkowski decomposition of associahedra and related combinatorics, Discrete Comput. Geom., Volume 50 (2013) no. 4, pp. 903-939 | Article | MR 3138141 | Zbl 1283.52014

[14] Lange, Carsten; Pilaud, Vincent Associahedra via spines, Combinatorica, Volume 38 (2018) no. 2, pp. 443-486 | Article | MR 3800847 | Zbl 06909521

[15] Loday, Jean-Louis Realization of the Stasheff polytope, Arch. Math., Volume 83 (2004) no. 3, pp. 267-278 | MR 2108555 | Zbl 1059.52017

[16] Musiker, Gregg; Schiffler, Ralf Cluster expansion formulas and perfect matchings, J. Algebr. Comb., Volume 32 (2010) no. 2, pp. 187-209 | Article | MR 2661414 | Zbl 1246.13035

[17] Musiker, Gregg; Schiffler, Ralf; Williams, Lauren Positivity for cluster algebras from surfaces, Adv. Math., Volume 227 (2011) no. 6, pp. 2241-2308 | Article | MR 2807089 | Zbl 1331.13017

[18] Nakanishi, Tomoki; Zelevinsky, Andrei On tropical dualities in cluster algebras, Algebraic groups and quantum groups (Nagoya, 2010) (Contemporary Mathematics) Volume 565, American Mathematical Society, 2012, pp. 217-226 | Article | MR 2932428 | Zbl 1317.13054

[19] 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 06436290

[20] Pilaud, Vincent; Stump, Christian Vertex barycenter of generalized associahedra, Proc. Am. Math. Soc., Volume 153 (2015) no. 6, pp. 2623-2636 | Article | MR 3326042 | Zbl 1316.52022

[21] Postnikov, Alexander Permutahedra, associahedra, and beyond, Int. Math. Res. Not., Volume 2009 (2009) no. 6, pp. 1026-1106 | Article | Zbl 1162.52007

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

[23] Reading, Nathan; Speyer, David Combinatorial frameworks for cluster algebras, Int. Math. Res. Not., Volume 2016 (2016) no. 1, pp. 109-173 | Article | MR 3514060 | Zbl 1330.05167

[24] Reading, Nathan; Speyer, David Cambrian frameworks for cluster algebras of affine type, Trans. Am. Math. Soc., Volume 370 (2018) no. 2, pp. 1429-1468 | Article | MR 3729507 | Zbl 06814531

[25] Schiffler, Ralf A cluster expansion formula (A n case), Electron. J. Comb., Volume 15 (2008), 9 pages | MR 2398856 | Zbl 1184.13064

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

[27] Tran, Thao Quantum F-polynomials in the theory of cluster algebras (2010), 99 pages (Ph. D. Thesis) | MR 2941308

[28] Yang, Shih-Wei; Zelevinsky, Andrei Cluster algebras of finite type via Coxeter elements and principal minors, Transform. Groups, Volume 13 (2008) no. 3-4, pp. 855-895 | Article | MR 2452619 | Zbl 1177.16010