### Electronic Research Archive

2021, Issue 5: 3031-3068. doi: 10.3934/era.2021025
Special Issues

# Picture groups and maximal green sequences

• Received: 01 July 2020 Revised: 01 February 2021 Published: 16 March 2021
• Primary: 16G20; Secondary: 20F55

• We show that picture groups are directly related to maximal green sequences for valued Dynkin quivers of finite type. Namely, there is a bijection between maximal green sequences and positive expressions (words in the generators without inverses) for the Coxeter element of the picture group. We actually prove the theorem for the more general set up of finite "vertically and laterally ordered" sets of positive real Schur roots for any hereditary algebra (not necessarily of finite type).

Furthermore, we show that every picture for such a set of positive roots is a linear combination of "atoms" and we give a precise description of atoms as special semi-invariant pictures.

Citation: Kiyoshi Igusa, Gordana Todorov. Picture groups and maximal green sequences[J]. Electronic Research Archive, 2021, 29(5): 3031-3068. doi: 10.3934/era.2021025

### Related Papers:

• We show that picture groups are directly related to maximal green sequences for valued Dynkin quivers of finite type. Namely, there is a bijection between maximal green sequences and positive expressions (words in the generators without inverses) for the Coxeter element of the picture group. We actually prove the theorem for the more general set up of finite "vertically and laterally ordered" sets of positive real Schur roots for any hereditary algebra (not necessarily of finite type).

Furthermore, we show that every picture for such a set of positive roots is a linear combination of "atoms" and we give a precise description of atoms as special semi-invariant pictures.

 [1] \begin{document}$\tau$\end{document}-tilting theory. Compos. Math. (2014) 150: 415-452. [2] Cluster categories for algebras of global dimension 2 and quivers with potential. Ann. Inst. Fourier (Grenoble) (2009) 59: 2525-2590. [3] Stability conditions on triangulated categories. Ann. of Math. (2007) 166: 317-345. [4] On maximal green sequences. Int. Math. Res. Not. IMRN (2014) 2014: 4547-4586. [5] T. Brüstle, S. Hermes, K. Igusa and G. Todorov, Semi-invariant pictures and two conjectures on maximal green sequences, J. Algebra, 473 (2017), 80-109. doi: 10.1016/j. jalgebra. 2016.10.025 [6] T. Brüstle, D. Smith and H. Treffinger, Wall and chamber structure for finite-dimensional algebras, Adv. Math., 354 (2019), 106746, 31 pp. doi: 10.1016/j. aim. 2019.106746 [7] A. B. Buan, R. J. Marsh and I. Reiten, Cluster-tilted algebras, Trans. Amer. Math. Soc., 359 (2007), 323-332. doi: 10.1090/S0002-9947-06-03879-7 [8] Tilting theory and cluster combinatorics. Adv. Math. (2006) 204: 572-618. [9] W. Crawley-Boevey, Exceptional sequences of representations of quivers, Representations of Algebras, (Ottawa, ON, 1992) 14 (1993), 117-124. [10] Semi-invariants of quivers and saturation for Littlewood-Richardson coefficients. J. Amer. Math. Soc. (2000) 13: 467-479. [11] H. Derksen, J. Weyman and A. Zelevinsky, Quivers with potentials and their representations I: Mutations, Selecta Math., 14 (2008), 59-119. doi: 10.1007/s00029-008-0057-9 [12] Quivers with potentials and their representations II: Applications to cluster algebras. J. Amer. Math. Soc. (2010) 23: 749-790. [13] Cluster algebras I: Foundations. J. Amer. Math. Soc. (2002) 15: 497-529. [14] Cluster algebras II: Finite type classification. Invent. Math. (2003) 154: 63-121. [15] \begin{document}$K\sb{3}$\end{document} of a ring is \begin{document}$H\sb{3}$\end{document} of the Steinberg group. Proc. Amer. Math. Soc. (1973) 37: 366-368. [16] A. Hatcher and J. Wagoner, Pseudo-Isotopies of Compact Manifolds, Société mathématique de France, 1973. [17] K. Igusa, The $Wh_3(\pi)$ Obstruction for Pseudoisotopy, PhD thesis, Princeton University, 1979. [18] ▬▬▬▬▬, The Borel regulator map on pictures, I: A dilogarithm formula, K-Theory, 7, (1993). [19] ▬▬▬▬▬, The category of noncrossing partitions, preprint, arXiv: 1411.0196. [20] ▬▬▬▬▬, Linearity of stability conditions, Communications in Algebra, (2020), 1-26. [21] K. Igusa, Maximal green sequences for cluster-tilted algebras of finite representation type, Algebr. Comb., 2 (2019), 753-780. doi: 10.5802/alco. 61 [22] The Borel regulator map on pictures. II. An example from Morse theory. \begin{document}$K$\end{document}-Theory (1993) 7: 225-267. [23] Links, pictures and the homology of nilpotent groups. Topology (2001) 40: 1125-1166. [24] Cluster complexes via semi-invariants. Compos. Math. (2009) 145: 1001-1034. [25] ▬▬▬▬▬, Modulated semi-invariants, preprint, arXiv: 1507.03051. [26] K. Igusa and G. Todorov, Signed exceptional sequences and the cluster morphism category, preprint, arXiv: 1706.02041. [27] K. Igusa, G. Todorov and J. Weyman, Picture groups of finite type and cohomology in type $A_n$, preprint, arXiv: 1609.02636. [28] B. Keller, On cluster theory and quantum dilogarithm identities, in Representations of Algebras and Related Topics, pages 85-116. European Mathematical Society Zürich, 2011. doi: 10.4171/101-1/3 [29] Moduli of representations of finite dimensional algebras. Quart. J. Math. Oxford Ser. (1994) 45: 515-530. [30] M. Kontsevich and Y. Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, preprint, arXiv: 0811.2435, 2008. [31] M. Kontsevich and Y. Soibelman, Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants, Commun. Number Theory Phys., 5 (2011), 231-352. doi: 10.4310/CNTP. 2011. v5. n2. a1 [32] J. -L. Loday, Homotopical syzygies, Contemporary Math., 265 (2000), 99-127. doi: 10.1090/conm/265/04245 [33] R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory, Springer-Verlag, Berlin-New York, 1977, reprinted in "Classics in Mathematics" series, Springer-Verlag, Berlin, 2001. doi: 10.1007/978-3-642-61896-3 [34] R. Peiffer, Über Identitäten zwischen Relationen, Math. Ann., 121 (1949/1950), 67-99. doi: 10.1007/BF01329617 [35] M. Reineke, The Harder-Narasimhan system in quantum groups and cohomology of quiver moduli, Invent. Math., 152 (2003), 349-368. doi: 10.1007/s00222-002-0273-4 [36] C. M. Ringel, The braid group action on the set of exceptional sequences of a hereditary Artin algebra, Contemp. Math., 171 (1994), 339-352. [37] J. B. Wagoner, A picture description of the boundary map in algebraic $K$-theory, Algebraic $K$-Theory, Lecture Notes in Math., Springer, Berlin, Heidelberg, 966 (1982), 362-389.
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