# American Institute of Mathematical Sciences

doi: 10.3934/era.2021025

## Picture groups and maximal green sequences

 1 Brandeis University, MS050, 415 South St, Waltham, MA 02454-9110, USA 2 Northeastern University, 360 Huntington Ave, Boston, MA 02115, USA

* Corresponding author: Kiyoshi Igusa

Received  July 2020 Revised  February 2021 Published  March 2021

Fund Project: The first author is supported by the Simons Foundation

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. Electronic Research Archive, doi: 10.3934/era.2021025
##### References:
 [1] T. Adachi, O. Iyama and I. Reiten, $\tau$-tilting theory, Compos. Math., 150 (2014), 415-452.  doi: 10.1112/S0010437X13007422.  Google Scholar [2] C. Amiot, Cluster categories for algebras of global dimension 2 and quivers with potential, Ann. Inst. Fourier (Grenoble), 59 (2009), 2525-2590.  doi: 10.5802/aif.2499.  Google Scholar [3] T. Bridgeland, Stability conditions on triangulated categories, Ann. of Math., 166 (2007), 317-345.  doi: 10.4007/annals.2007.166.317.  Google Scholar [4] T. Brüstle, G. Dupont and M. Pérotin, On maximal green sequences, Int. Math. Res. Not. IMRN, 2014 (2014), 4547-4586.  doi: 10.1093/imrn/rnt075.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] A. B. Buan, R. Marsh, M. Reineke, I. Reiten and G. Todorov, Tilting theory and cluster combinatorics, Adv. Math., 204 (2006), 572-618.  doi: 10.1016/j.aim.2005.06.003.  Google Scholar [9] W. Crawley-Boevey, Exceptional sequences of representations of quivers, Representations of Algebras, (Ottawa, ON, 1992) 14 (1993), 117-124.  Google Scholar [10] H. Derksen and J. Weyman, Semi-invariants of quivers and saturation for Littlewood-Richardson coefficients, J. Amer. Math. Soc., 13 (2000), 467-479.  doi: 10.1090/S0894-0347-00-00331-3.  Google Scholar [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.  Google Scholar [12] H. Derksen, J. Weyman and A. Zelevinsky, Quivers with potentials and their representations II: Applications to cluster algebras, J. Amer. Math. Soc., 23 (2010), 749-790.  doi: 10.1090/S0894-0347-10-00662-4.  Google Scholar [13] S. Fomin and A. Zelevinsky, Cluster algebras I: Foundations, J. Amer. Math. Soc., 15 (2002), 497-529.  doi: 10.1090/S0894-0347-01-00385-X.  Google Scholar [14] S. Fomin and A. Zelevinsky, Cluster algebras II: Finite type classification, Invent. Math., 154 (2003), 63-121.  doi: 10.1007/s00222-003-0302-y.  Google Scholar [15] S. M. Gersten, $K\sb{3}$ of a ring is $H\sb{3}$ of the Steinberg group, Proc. Amer. Math. Soc., 37 (1973), 366-368.  doi: 10.2307/2039440.  Google Scholar [16] A. Hatcher and J. Wagoner, Pseudo-Isotopies of Compact Manifolds, Société mathématique de France, 1973.  Google Scholar [17] K. Igusa, The $Wh_3(\pi)$ Obstruction for Pseudoisotopy, PhD thesis, Princeton University, 1979.  Google Scholar [18] ▬▬▬▬▬, The Borel regulator map on pictures, I: A dilogarithm formula, K-Theory, 7, (1993). Google Scholar [19] ▬▬▬▬▬, The category of noncrossing partitions, preprint, arXiv: 1411.0196. Google Scholar [20] ▬▬▬▬▬, Linearity of stability conditions, Communications in Algebra, (2020), 1-26. Google Scholar [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.  Google Scholar [22] K. Igusa and J. Klein, The Borel regulator map on pictures. II. An example from Morse theory, $K$-Theory, 7 (1993), 225-267.  doi: 10.1007/BF00961065.  Google Scholar [23] K. Igusa and K. E. Orr, Links, pictures and the homology of nilpotent groups, Topology, 40 (2001), 1125-1166.  doi: 10.1016/S0040-9383(00)00002-1.  Google Scholar [24] K. Igusa, K. Orr, G. Todorov and J. Weyman, Cluster complexes via semi-invariants, Compos. Math., 145 (2009), 1001-1034.  doi: 10.1112/S0010437X09004151.  Google Scholar [25] ▬▬▬▬▬, Modulated semi-invariants, preprint, arXiv: 1507.03051. Google Scholar [26] K. Igusa and G. Todorov, Signed exceptional sequences and the cluster morphism category, preprint, arXiv: 1706.02041. Google Scholar [27] K. Igusa, G. Todorov and J. Weyman, Picture groups of finite type and cohomology in type $A_n$, preprint, arXiv: 1609.02636. Google Scholar [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.  Google Scholar [29] A. D. King, Moduli of representations of finite dimensional algebras, Quart. J. Math. Oxford Ser., 45 (1994), 515-530.  doi: 10.1093/qmath/45.4.515.  Google Scholar [30] M. Kontsevich and Y. Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, preprint, arXiv: 0811.2435, 2008. Google Scholar [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.  Google Scholar [32] J. -L. Loday, Homotopical syzygies, Contemporary Math., 265 (2000), 99-127. doi: 10.1090/conm/265/04245.  Google Scholar [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.  Google Scholar [34] R. Peiffer, Über Identitäten zwischen Relationen, Math. Ann., 121 (1949/1950), 67-99. doi: 10.1007/BF01329617.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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##### References:
 [1] T. Adachi, O. Iyama and I. Reiten, $\tau$-tilting theory, Compos. Math., 150 (2014), 415-452.  doi: 10.1112/S0010437X13007422.  Google Scholar [2] C. Amiot, Cluster categories for algebras of global dimension 2 and quivers with potential, Ann. Inst. Fourier (Grenoble), 59 (2009), 2525-2590.  doi: 10.5802/aif.2499.  Google Scholar [3] T. Bridgeland, Stability conditions on triangulated categories, Ann. of Math., 166 (2007), 317-345.  doi: 10.4007/annals.2007.166.317.  Google Scholar [4] T. Brüstle, G. Dupont and M. Pérotin, On maximal green sequences, Int. Math. Res. Not. IMRN, 2014 (2014), 4547-4586.  doi: 10.1093/imrn/rnt075.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] A. B. Buan, R. Marsh, M. Reineke, I. Reiten and G. Todorov, Tilting theory and cluster combinatorics, Adv. Math., 204 (2006), 572-618.  doi: 10.1016/j.aim.2005.06.003.  Google Scholar [9] W. Crawley-Boevey, Exceptional sequences of representations of quivers, Representations of Algebras, (Ottawa, ON, 1992) 14 (1993), 117-124.  Google Scholar [10] H. Derksen and J. Weyman, Semi-invariants of quivers and saturation for Littlewood-Richardson coefficients, J. Amer. Math. Soc., 13 (2000), 467-479.  doi: 10.1090/S0894-0347-00-00331-3.  Google Scholar [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.  Google Scholar [12] H. Derksen, J. Weyman and A. Zelevinsky, Quivers with potentials and their representations II: Applications to cluster algebras, J. Amer. Math. Soc., 23 (2010), 749-790.  doi: 10.1090/S0894-0347-10-00662-4.  Google Scholar [13] S. Fomin and A. Zelevinsky, Cluster algebras I: Foundations, J. Amer. Math. Soc., 15 (2002), 497-529.  doi: 10.1090/S0894-0347-01-00385-X.  Google Scholar [14] S. Fomin and A. Zelevinsky, Cluster algebras II: Finite type classification, Invent. Math., 154 (2003), 63-121.  doi: 10.1007/s00222-003-0302-y.  Google Scholar [15] S. M. Gersten, $K\sb{3}$ of a ring is $H\sb{3}$ of the Steinberg group, Proc. Amer. Math. Soc., 37 (1973), 366-368.  doi: 10.2307/2039440.  Google Scholar [16] A. Hatcher and J. Wagoner, Pseudo-Isotopies of Compact Manifolds, Société mathématique de France, 1973.  Google Scholar [17] K. Igusa, The $Wh_3(\pi)$ Obstruction for Pseudoisotopy, PhD thesis, Princeton University, 1979.  Google Scholar [18] ▬▬▬▬▬, The Borel regulator map on pictures, I: A dilogarithm formula, K-Theory, 7, (1993). Google Scholar [19] ▬▬▬▬▬, The category of noncrossing partitions, preprint, arXiv: 1411.0196. Google Scholar [20] ▬▬▬▬▬, Linearity of stability conditions, Communications in Algebra, (2020), 1-26. Google Scholar [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.  Google Scholar [22] K. Igusa and J. Klein, The Borel regulator map on pictures. II. An example from Morse theory, $K$-Theory, 7 (1993), 225-267.  doi: 10.1007/BF00961065.  Google Scholar [23] K. Igusa and K. E. Orr, Links, pictures and the homology of nilpotent groups, Topology, 40 (2001), 1125-1166.  doi: 10.1016/S0040-9383(00)00002-1.  Google Scholar [24] K. Igusa, K. Orr, G. Todorov and J. Weyman, Cluster complexes via semi-invariants, Compos. Math., 145 (2009), 1001-1034.  doi: 10.1112/S0010437X09004151.  Google Scholar [25] ▬▬▬▬▬, Modulated semi-invariants, preprint, arXiv: 1507.03051. Google Scholar [26] K. Igusa and G. Todorov, Signed exceptional sequences and the cluster morphism category, preprint, arXiv: 1706.02041. Google Scholar [27] K. Igusa, G. Todorov and J. Weyman, Picture groups of finite type and cohomology in type $A_n$, preprint, arXiv: 1609.02636. Google Scholar [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.  Google Scholar [29] A. D. King, Moduli of representations of finite dimensional algebras, Quart. J. Math. Oxford Ser., 45 (1994), 515-530.  doi: 10.1093/qmath/45.4.515.  Google Scholar [30] M. Kontsevich and Y. Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, preprint, arXiv: 0811.2435, 2008. Google Scholar [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.  Google Scholar [32] J. -L. Loday, Homotopical syzygies, Contemporary Math., 265 (2000), 99-127. doi: 10.1090/conm/265/04245.  Google Scholar [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.  Google Scholar [34] R. Peiffer, Über Identitäten zwischen Relationen, Math. Ann., 121 (1949/1950), 67-99. doi: 10.1007/BF01329617.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
On the left is the semi-invariant picture $L(\mathcal{S}_0)$ for the admissible subsequences $\mathcal{S}_0 = (\alpha_1,\alpha_2,\alpha_4)$ of $\mathcal{S}$ from Example 1.13. $L(\mathcal{S}_0)$ is a subset of $S^2\subset\mathbb{R}^3$. Thus, e.g., $D(\alpha_i)$ are actually coordinate hyperplanes. The $\mathcal{S}_0$-compartments are the components of the complement of $L(\mathcal{S}_0)$. For example, $\mathcal{U}_{++0} = \mathcal{U}_{++}$ is the region on the positive side of the two hyperplanes $D(\alpha_1),D(\alpha_2)$. $\mathcal{U}_{-++}$ is the set of point in $\mathcal{U}_{-+}$ on the positive side of $D(\alpha_4)$. On the right, the wall $D(\alpha_3)$ cuts all five $\mathcal{S}_0$-compartments in half giving the semi-invariant picture for $\mathcal{S} = (\alpha_1,\alpha_2,\alpha_4,\alpha_3)$ with ten compartments
Semi-invariant picture $L(\mathcal{S}')$ for the weakly admissible sequence $\mathcal{S}' = (\alpha_1,\alpha_4,\alpha_3)$ from Example 1.13. The solid green arrow indicates an $\mathcal{S}'$-green path giving the maximal $\mathcal{S}'$-green sequence $\mathcal{U}_{---},\mathcal{U}_{+0-},\mathcal{U}_{+0+}$. Note that the dashed green arrow indicates another $\mathcal{S}'$-green path giving the maximal $\mathcal{S}'$-green sequence $\mathcal{U}_{---}, \mathcal{U}_{-+-}$, $\mathcal{U}_{+0-}, \mathcal{U}_{+0+}$. So, "maximal" is a misnomer when $\mathcal{S}'$ is only weakly admissible. Also, $L(\mathcal{S}')$ is not a "planar picture" for $G(\mathcal{S}')$ as defined in Section 3 since $\mathcal{S}'$ is not admissible
A typical intersection of two walls $D(\alpha_1)$ and $D(\alpha_2)$ producing walls $D(\beta_i)$. In this drawing there is only $\beta = \alpha_1+\alpha_2$. The green path $\gamma$ crosses $D(\alpha_1),D(\alpha_2)$ and $\gamma'$ crosses $D(\alpha_2),D(\beta),D(\alpha_1)$. The homotopy $h:\gamma\simeq\gamma'$ passes through $x_0$
The green path $\gamma_1$ is in Class 1 since it is disjoint from $D(\beta_m)$. The green path $\gamma_2$ is in Class 2 and passes through three $\mathcal{S}_0$-compartments $\mathcal{U}_{\epsilon(p)},\mathcal{U}_{\epsilon(r)},\mathcal{U}_{\epsilon(q)}$ in $\mathcal{V}_0 = int(\mathcal{V}(\beta_m)\backslash\mathcal{W}(\beta_m))$. Each of these is divided into two $\mathcal{S}$-compartments by the wall $D(\beta_m)$ and $\gamma_2$ passes through four of these $\mathcal{S}$-compartments in $\mathcal{V}_0$. $D(\beta_m)$ is the part of the hyperplane $H(\beta_m)$ inside the oval region $\mathcal{V}(\beta_m)$ and outside of $\mathcal{W}(\beta_m)$
The cone of $E_r$ in $D^2$ is the part inside the circle $S^1$. The asterisks $\ast$ indicates the position of the basepoint $1\in S^1$. The labels are drawn on the negative side of each edge
and $r_2$ are relations (or inverse relations). $L_1$ is the "standard partial picture" for $q(L_1) = (ab,r_1)(c,r_2)\in Q(G)$. On the right is $L_2$, a deformation of $L_1$ with $\partial L_2 = cc^{-1}\partial L_1$. $q(L_2) = (c,r_2)(cr_2^{-1}c^{-1}ab,r_1)$ since the vertex for $r_2$ is on the left and $cr_2^{-1}c^{-1}ab$ is given by reading the labels on the dotted path $\ell_1'$. Then $q(L_1) = q(L_2)$ by (4).">Figure 6.  On the left, $L_1$ is a partial picture with $\partial L_1 = abr_1b^{-1}a^{-1}cr_2c^{-1}$ where $r_1 = x^{-1}y^{-1}z$ and $r_2$ are relations (or inverse relations). $L_1$ is the "standard partial picture" for $q(L_1) = (ab,r_1)(c,r_2)\in Q(G)$. On the right is $L_2$, a deformation of $L_1$ with $\partial L_2 = cc^{-1}\partial L_1$. $q(L_2) = (c,r_2)(cr_2^{-1}c^{-1}ab,r_1)$ since the vertex for $r_2$ is on the left and $cr_2^{-1}c^{-1}ab$ is given by reading the labels on the dotted path $\ell_1'$. Then $q(L_1) = q(L_2)$ by (4).
Dotted lines $\ell_1,\ell_2$ are given by definition of $q(L_2)$. Take dashed lines parallel to $\ell_1,\ell_2$ and connected with small semicircles below vertices $v_1,v_2$. Push the dashed line down to the $x$-axis. This gives an admissible deformation of $L_2$ (on the left) to $L_{q(L_2)}$ (on the night). The dotted lines $\ell_1,\ell_2$ cross the same edges in both partial pictures
The X-letters $a,b$ have edge sets which are smooth at the vertex. The basepoint direction is on the negative side of both X-edges $E(a),E(b)$
The atom $A_\mathcal{A}(\alpha, \beta, \omega)$. There are three circles labeled $\alpha, \beta, \omega$. There is only one vertex (black dot) outside the brown circle labeled $\omega$. There is only one vertex inside the $\alpha$ circle. The faint gray line is deleted since, in this example, its label is not in the set $\mathcal{S}$
Illustrating proof of Atomic Deformation Theorem 3.18: $\Sigma'$ (in red) is on the negative side of an innermost $E(\omega)$ curve $\Sigma$ (in blue). The picture $L = L_0\cup L_1$, on the left, is deformation equivalent to the disjoint union of two pictures: $L" = L_0\cup \rho(L_0')$, in the middle, and $L' = \rho(L_0)\cup L_1$ on the right. The $E(\omega)$ component $\Sigma$ lies either in $L'$ or $L"$. (Here it is in $L"$ in the middle.) In either case, it can be removed by the Sliding Lemma 3.17
Illustrating proof of Sliding Lemma 3.17: $\Sigma$ in blue is a disjoint union of $E(\omega)$ closed curves which encloses a region $\overline U = \Sigma\cup U$. All Y-edges for vertices on $\Sigma$ lie outside $U$. The atom $\mathcal{A}(\alpha,\beta,\omega)$ in the proof has already been added on the left. The new region $U'$ is the complement of the new $\omega$ oval in $U$. The vertex $v$ has been cancelled with the vertex in the atom on the right
(Proof of Lemma E) The partial picture $L$ for $G(\mathcal{S}_-(\lambda))$ is divided into two parts $L = L_0\cup L_1$ by $\Sigma'$. Applying $\rho:G(\mathcal{S}_-(\lambda))\to G(\mathcal{R}_-(\lambda))$ to $L_0$ eliminates $x(\lambda)$ from the word $w_0 = w_1x(\lambda)x_2$ but does not eliminage $x(\beta_m)$. Then $w_1\rho(w_2)$ commutes with $x(\beta_m)$ contradicting the minimality of $w_0$
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