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: 2018-06-12
Published online: 2018-09-10
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/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] Coxeter matroids, Progress in Mathematics, Volume 216, Birkhäuser, 2003 | MR 1989953 | Zbl 1050.52005
[2] Cluster algebras of type , tropical planes, and the positive tropical Grassmannian, Beitr. Algebra Geom., Volume 58 (2017) no. 1, pp. 25-46 | Article | MR 3607668 | Zbl 06695676
[3] 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] 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] Polytopal realizations of generalized associahedra, Can. Math. Bull., Volume 45 (2002) no. 4, pp. 537-566 | Article | MR 1941227 | Zbl 1018.52007
[6] Cluster algebras I: foundations, J. Am. Math. Soc., Volume 15 (2002) no. 2, pp. 497-529 | Article | MR 1887642 | Zbl 1021.16017
[7] Cluster algebras II: finite type classification, Invent. Math., Volume 154 (2003) no. 1, pp. 63-121 | Article | MR 2004457 | Zbl 1054.17024
[8] Cluster algebras IV: coefficients, Compos. Math., Volume 143 (2007) no. 1, pp. 112-164 | Article | MR 2295199 | Zbl 1127.16023
[9] Permutahedra and generalized associahedra, Adv. Math., Volume 226 (2011) no. 1, pp. 608-640 | Article | MR 2735770 | Zbl 1233.20035
[10] Reflection groups and Coxeter groups Volume 29, Cambridge University Press, 1990, xii+204 pages | MR 1066460 | Zbl 0725.20028
[11] Subword complexes in Coxeter groups, Adv. Math., Volume 184 (2004) no. 1, pp. 161-176 | Article | MR 2047852 | Zbl 1069.20026
[12] Gröbner geometry of Schubert polynomials, Ann. Math., Volume 161 (2005) no. 3, pp. 1245-1318 | Article | Zbl 1089.14007
[13] 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] Associahedra via spines, Combinatorica, Volume 38 (2018) no. 2, pp. 443-486 | Article | MR 3800847 | Zbl 06909521
[15] Realization of the Stasheff polytope, Arch. Math., Volume 83 (2004) no. 3, pp. 267-278 | MR 2108555 | Zbl 1059.52017
[16] Cluster expansion formulas and perfect matchings, J. Algebr. Comb., Volume 32 (2010) no. 2, pp. 187-209 | Article | MR 2661414 | Zbl 1246.13035
[17] Positivity for cluster algebras from surfaces, Adv. Math., Volume 227 (2011) no. 6, pp. 2241-2308 | Article | MR 2807089 | Zbl 1331.13017
[18] 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] Brick polytopes of spherical subword complexes and generalized associahedra, Adv. Math., Volume 276 (2015), pp. 1-61 | Article | MR 3327085 | Zbl 06436290
[20] Vertex barycenter of generalized associahedra, Proc. Am. Math. Soc., Volume 153 (2015) no. 6, pp. 2623-2636 | Article | MR 3326042 | Zbl 1316.52022
[21] Permutahedra, associahedra, and beyond, Int. Math. Res. Not., Volume 2009 (2009) no. 6, pp. 1026-1106 | Article | Zbl 1162.52007
[22] Sortable elements and Cambrian lattices, Algebra Univers., Volume 56 (2007) no. 3-4, pp. 411-437 | Article | MR 2318219 | Zbl 1184.20038
[23] Combinatorial frameworks for cluster algebras, Int. Math. Res. Not., Volume 2016 (2016) no. 1, pp. 109-173 | Article | MR 3514060 | Zbl 1330.05167
[24] 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] A cluster expansion formula ( case), Electron. J. Comb., Volume 15 (2008), 9 pages | MR 2398856 | Zbl 1184.13064
[26] The tropical totally positive Grassmannian, J. Algebr. Comb., Volume 22 (2005) no. 2, pp. 189-210 | Article | MR 2164397 | Zbl 1094.14048
[27] Quantum F-polynomials in the theory of cluster algebras (2010), 99 pages (Ph. D. Thesis) | MR 2941308
[28] 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