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Towards a uniform subword complex description of acyclic finite type cluster algebras. (English) Zbl 1423.13118

Summary: 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.

MSC:

13F60 Cluster algebras
05E15 Combinatorial aspects of groups and algebras (MSC2010)
20F55 Reflection and Coxeter groups (group-theoretic aspects)
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[1] Borovik, Alexandre V.; Gelfand, Israil M.; White, Neil, Coxeter matroids, 216, (2003), Birkhäuser · 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., 58, 1, 25-46, (2017) · Zbl 1401.13063 · doi:10.1007/s13366-016-0316-4
[3] Ceballos, Cesar; Labbé, Jean-Philippe; Stump, Christian, Subword complexes, cluster complexes, and generalized multi-associahedra, J. Algebr. Comb., 39, 1, 17-51, (2014) · Zbl 1286.05180
[4] Ceballos, Cesar; Pilaud, Vincent, Denominator vectors and compatibility degrees in cluster algebras of finite type, Trans. Am. Math. Soc., 367, 2, 1421-1439, (2015) · Zbl 1350.13020
[5] Chapoton, Frédéric; Fomin, Sergey; Zelevinsky, Andrei, Polytopal realizations of generalized associahedra, Can. Math. Bull., 45, 4, 537-566, (2002) · Zbl 1018.52007 · doi:10.4153/CMB-2002-054-1
[6] Fomin, Sergey; Zelevinsky, Andrei, Cluster algebras I: foundations, J. Am. Math. Soc., 15, 2, 497-529, (2002) · Zbl 1021.16017 · doi:10.1090/S0894-0347-01-00385-X
[7] Fomin, Sergey; Zelevinsky, Andrei, Cluster algebras II: finite type classification, Invent. Math., 154, 1, 63-121, (2003) · Zbl 1054.17024 · doi:10.1007/s00222-003-0302-y
[8] Fomin, Sergey; Zelevinsky, Andrei, Cluster algebras IV: coefficients, Compos. Math., 143, 1, 112-164, (2007) · Zbl 1127.16023
[9] Hohlweg, Christophe; Lange, Carsten; Thomas, Hugh, Permutahedra and generalized associahedra, Adv. Math., 226, 1, 608-640, (2011) · Zbl 1233.20035 · doi:10.1016/j.aim.2010.07.005
[10] Humphreys, James E., Reflection groups and Coxeter groups, 29, xii+204 pp., (1990), Cambridge University Press · Zbl 0725.20028
[11] Knutson, Allen; Miller, Ezra, Subword complexes in Coxeter groups, Adv. Math., 184, 1, 161-176, (2004) · Zbl 1069.20026 · doi:10.1016/S0001-8708(03)00142-7
[12] Knutson, Allen; Miller, Ezra, Gröbner geometry of Schubert polynomials, Ann. Math., 161, 3, 1245-1318, (2005) · Zbl 1089.14007 · doi:10.4007/annals.2005.161.1245
[13] Lange, Carsten, Minkowski decomposition of associahedra and related combinatorics, Discrete Comput. Geom., 50, 4, 903-939, (2013) · Zbl 1283.52014 · doi:10.1007/s00454-013-9546-5
[14] Lange, Carsten; Pilaud, Vincent, Associahedra via spines, Combinatorica, 38, 2, 443-486, (2018) · Zbl 1413.52017 · doi:10.1007/s00493-015-3248-y
[15] Loday, Jean-Louis, Realization of the stasheff polytope, Arch. Math., 83, 3, 267-278, (2004) · Zbl 1059.52017
[16] Musiker, Gregg; Schiffler, Ralf, Cluster expansion formulas and perfect matchings, J. Algebr. Comb., 32, 2, 187-209, (2010) · Zbl 1246.13035 · doi:10.1007/s10801-009-0210-3
[17] Musiker, Gregg; Schiffler, Ralf; Williams, Lauren, Positivity for cluster algebras from surfaces, Adv. Math., 227, 6, 2241-2308, (2011) · Zbl 1331.13017 · doi:10.1016/j.aim.2011.04.018
[18] Nakanishi, Tomoki; Zelevinsky, Andrei, Algebraic groups and quantum groups (Nagoya, 2010), 565, On tropical dualities in cluster algebras, 217-226, (2012), American Mathematical Society · Zbl 1317.13054
[19] Pilaud, Vincent; Stump, Christian, Brick polytopes of spherical subword complexes and generalized associahedra, Adv. Math., 276, 1-61, (2015) · Zbl 1405.05196 · doi:10.1016/j.aim.2015.02.012
[20] Pilaud, Vincent; Stump, Christian, Vertex barycenter of generalized associahedra, Proc. Am. Math. Soc., 153, 6, 2623-2636, (2015) · Zbl 1316.52022
[21] Postnikov, Alexander, Permutahedra, associahedra, and beyond, Int. Math. Res. Not., 2009, 6, 1026-1106, (2009) · Zbl 1162.52007 · doi:10.1093/imrn/rnn153
[22] Reading, Nathan, Sortable elements and Cambrian lattices, Algebra Univers., 56, 3-4, 411-437, (2007) · Zbl 1184.20038 · doi:10.1007/s00012-007-2009-1
[23] Reading, Nathan; Speyer, David, Combinatorial frameworks for cluster algebras, Int. Math. Res. Not., 2016, 1, 109-173, (2016) · Zbl 1330.05167 · doi:10.1093/imrn/rnv101
[24] Reading, Nathan; Speyer, David, Cambrian frameworks for cluster algebras of affine type, Trans. Am. Math. Soc., 370, 2, 1429-1468, (2018) · Zbl 1423.13131
[25] Schiffler, Ralf, A cluster expansion formula (\(A_n\) case), Electron. J. Comb., 15, (2008) · Zbl 1184.13064
[26] Speyer, David; Williams, Lauren, The tropical totally positive Grassmannian, J. Algebr. Comb., 22, 2, 189-210, (2005) · Zbl 1094.14048 · doi:10.1007/s10801-005-2513-3
[27] Tran, Thao, Quantum F-polynomials in the theory of cluster algebras, 99 pp., (2010)
[28] Yang, Shih-Wei; Zelevinsky, Andrei, Cluster algebras of finite type via Coxeter elements and principal minors, Transform. Groups, 13, 3-4, 855-895, (2008) · Zbl 1177.16010 · doi:10.1007/s00031-008-9025-x
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