We show that, for any cluster-tilted algebra of finite representation type over an algebraically closed field, the following three definitions of a maximal green sequence are equivalent: (1) the usual definition in terms of Fomin–Zelevinsky mutation of the extended exchange matrix, (2) a complete forward hom-orthogonal sequence of Schurian modules, (3) the sequence of wall crossings of a generic green path. Together with [24], this completes the foundational work needed to support the author’s work with P. J. Apruzzese [1], namely, to determine all lengths of all maximal green sequences for all quivers whose underlying graph is an oriented or unoriented cycle and to determine which are “linear”.
In an Appendix, written jointly with G. Todorov, we give a conjectural description of maximal green sequences of maximum length for any cluster-tilted algebra of finite representation type.
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.61
Keywords: c-vectors, forward hom-orthogonal sequences, Jacobian algebras, quivers with potential, cluster mutation, stability conditions, tilted algebras
Igusa, Kiyoshi 1
CC-BY 4.0
@article{ALCO_2019__2_5_753_0,
author = {Igusa, Kiyoshi},
title = {Maximal green sequences for cluster-tilted algebras of finite representation type},
journal = {Algebraic Combinatorics},
pages = {753--780},
publisher = {MathOA foundation},
volume = {2},
number = {5},
year = {2019},
doi = {10.5802/alco.61},
zbl = {1422.16010},
mrnumber = {4023565},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.61/}
}
TY - JOUR AU - Igusa, Kiyoshi TI - Maximal green sequences for cluster-tilted algebras of finite representation type JO - Algebraic Combinatorics PY - 2019 SP - 753 EP - 780 VL - 2 IS - 5 PB - MathOA foundation UR - https://alco.centre-mersenne.org/articles/10.5802/alco.61/ DO - 10.5802/alco.61 LA - en ID - ALCO_2019__2_5_753_0 ER -
%0 Journal Article %A Igusa, Kiyoshi %T Maximal green sequences for cluster-tilted algebras of finite representation type %J Algebraic Combinatorics %D 2019 %P 753-780 %V 2 %N 5 %I MathOA foundation %U https://alco.centre-mersenne.org/articles/10.5802/alco.61/ %R 10.5802/alco.61 %G en %F ALCO_2019__2_5_753_0
Igusa, Kiyoshi. Maximal green sequences for cluster-tilted algebras of finite representation type. Algebraic Combinatorics, Volume 2 (2019) no. 5, pp. 753-780. doi : 10.5802/alco.61. https://alco.centre-mersenne.org/articles/10.5802/alco.61/
[1] Stability conditions for affine type A (2018) (https://arxiv.org/abs/1804.09100)
[2] Tilted algebras of type , Comm in Algebra, Volume 19 (1982) no. 10, pp. 2121-2139 | DOI | Zbl
[3] A course on cluster-tilted algebras, Homological Methods, Representation Theory, and Cluster Algebras, Springer, 2018, pp. 127-176 | DOI | Zbl
[4] Cluster-tilted algebras as trivial extensions, Bull London Math Soc, Volume 40 (2008) no. 1, pp. 151-162 | DOI | MR
[5] Generalized tilted algebras of type , Comm in Algebra, Volume 20 (1981) no. 9, pp. 2101-2125 | DOI | MR | Zbl
[6] Elements of the representation theory of associative algebras. Vol. 1, London Mathematical Society Student Texts, 65, Cambridge University Press, Cambridge, 2006, x+458 pages (Techniques of representation theory) | MR | Zbl
[7] Homological and homotopical aspects of torsion theories, Memoirs of AMS, 883, American Mathematical Soc., 2007 | Zbl
[8] Stability conditions on triangulated categories, Ann. Math., Volume 166 (2007) no. 2, pp. 317-345 | DOI | MR | Zbl
[9] Semi-invariant pictures and two conjectures on maximal green sequences, J. Algebra, Volume 473 (2017), pp. 80-109 | DOI | MR | Zbl
[10] Stability conditions, tau-tilting theory and maximal green sequences (2017) (https://arxiv.org/abs/1805.04382)
[11] Mutation of cluster-tilting objects and potentials, Amer. J. Math., Volume 133 (2011) no. 4, pp. 835-887 | DOI | MR | Zbl
[12] Tilting theory and cluster combinatorics, Adv. Math., Volume 204 (2006) no. 2, pp. 572-618 | DOI | MR | Zbl
[13] Cluster-tilted algebras of finite representation type, J. Algebra, Volume 306 (2006) no. 2, pp. 412-431 | DOI | MR | Zbl
[14] Cluster-tilted algebras, Trans. Amer. Math. Soc., Volume 359 (2007) no. 1, p. 323-332 (electronic) | DOI | MR | Zbl
[15] Derived equivalence classification for cluster-tilted algebras of type , J. Algebra, Volume 319 (2008) no. 7, pp. 2723-2738 | DOI | MR | Zbl
[16] Auslander–Reiten sequences with few middle terms and applications to string algebras, Comm. Algebra, Volume 15 (1987), pp. 145-179 | DOI | MR | Zbl
[17] Quivers with relations arising from clusters ( case), Trans. Amer. Math. Soc., Volume 358 (2006) no. 3, pp. 1347-1364 | DOI | MR
[18] Minimal length maximal green sequences and triangulations of polygons, J. Algebraic Combin., Volume 44 (2016) no. 4, pp. 905-930 | DOI | MR | Zbl
[19] -tilting finite algebras, bricks and g-vectors, Int. Math. Res. Not. (2017), pp. 1-41 | Zbl
[20] Quivers with potentials and their representations I: Mutations, Selecta Math., Volume 14 (2008) no. 1, pp. 59-119 | DOI | MR | Zbl
[21] Cluster algebras. II: Finite type classification, Invent. Math., Volume 154 (2003), pp. 63-121 | DOI | MR | Zbl
[22] Cluster algebras. IV. Coefficients, Compos. Math., Volume 143 (2007) no. 1, pp. 112-164 | DOI | MR | Zbl
[23] Minimal length maximal green sequences, Adv. in Appl. Math., Volume 96 (2018), pp. 76-138 | DOI | MR | Zbl
[24] Linearity of stability conditions (2017) (https://arxiv.org/abs/1706.06986)
[25] From triangulated categories to abelian categories: cluster tilting in a general framework, Math. Z., Volume 258 (2008), pp. 143-160 | DOI | MR | Zbl
[26] Stability conditions and quantum dilogarithm identities for Dynkin quivers, Adv. Math., Volume 269 (2015), pp. 229-264 | MR | Zbl
[27] The Harder–Narasimhan system in quantum groups and cohomology of quiver moduli, Invent. Math., Volume 152 (2003) no. 2, pp. 349-368 | DOI | MR | Zbl
Cited by Sources: