Browse issues
no. 4
Universal quivers
Fomin, Sergey1; Igusa, Kiyoshi2; Lee, Kyungyong3
1 Department of Mathematics University of Michigan Ann Arbor MI 48109, USA
2 Department of Mathematics Brandeis University Waltham MA 02454, USA
3 Department of Mathematics University of Alabama Tuscaloosa AL 35487, USA;
Algebraic Combinatorics, Volume 4 (2021) no. 4, pp. 683-702.

We show that for any positive integer n, there exists a quiver Q with O(n 2 ) vertices and O(n 2 ) edges such that any quiver on n vertices is a full subquiver of a quiver mutation equivalent to Q. We generalize this statement to skew-symmetrizable matrices, and obtain other related results.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.175
Classification: 13F60, 05C25, 05E16, 16G20, 18G80
Keywords: Quiver mutation, universal quiver, cluster algebra.

Fomin, Sergey 1; Igusa, Kiyoshi 2; Lee, Kyungyong 3

1 Department of Mathematics University of Michigan Ann Arbor MI 48109, USA
2 Department of Mathematics Brandeis University Waltham MA 02454, USA
3 Department of Mathematics University of Alabama Tuscaloosa AL 35487, USA;
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{ALCO_2021__4_4_683_0,
     author = {Fomin, Sergey and Igusa, Kiyoshi and Lee, Kyungyong},
     title = {Universal quivers},
     journal = {Algebraic Combinatorics},
     pages = {683--702},
     publisher = {MathOA foundation},
     volume = {4},
     number = {4},
     year = {2021},
     doi = {10.5802/alco.175},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.175/}
}
TY  - JOUR
AU  - Fomin, Sergey
AU  - Igusa, Kiyoshi
AU  - Lee, Kyungyong
TI  - Universal quivers
JO  - Algebraic Combinatorics
PY  - 2021
SP  - 683
EP  - 702
VL  - 4
IS  - 4
PB  - MathOA foundation
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.175/
DO  - 10.5802/alco.175
LA  - en
ID  - ALCO_2021__4_4_683_0
ER  - 
%0 Journal Article
%A Fomin, Sergey
%A Igusa, Kiyoshi
%A Lee, Kyungyong
%T Universal quivers
%J Algebraic Combinatorics
%D 2021
%P 683-702
%V 4
%N 4
%I MathOA foundation
%U https://alco.centre-mersenne.org/articles/10.5802/alco.175/
%R 10.5802/alco.175
%G en
%F ALCO_2021__4_4_683_0
Fomin, Sergey; Igusa, Kiyoshi; Lee, Kyungyong. Universal quivers. Algebraic Combinatorics, Volume 4 (2021) no. 4, pp. 683-702. doi : 10.5802/alco.175. https://alco.centre-mersenne.org/articles/10.5802/alco.175/

[1] Amiot, Claire Cluster categories for algebras of global dimension 2 and quivers with potential, Ann. Inst. Fourier (Grenoble), Volume 59 (2009) no. 6, pp. 2525-2590 | DOI | Numdam | MR | Zbl

[2] Buan, Aslak Bakke; Iyama, Osamu; Reiten, Idun; Scott, Jeanne Cluster structures for 2-Calabi–Yau categories and unipotent groups, Compos. Math., Volume 145 (2009) no. 4, pp. 1035-1079 | DOI | MR | Zbl

[3] Buan, Aslak Bakke; Marsh, Bethany R.; Reiten, Idun Cluster mutation via quiver representations, Comment. Math. Helv., Volume 83 (2008) no. 1, pp. 143-177 | DOI | MR | Zbl

[4] Derksen, Harm; Weyman, Jerzy; Zelevinsky, Andrei Quivers with potentials and their representations II: Applications to cluster algebras, J. Amer. Math. Soc., Volume 23 (2010) no. 3, pp. 749-790 | DOI | MR | Zbl

[5] Fomin, Sergey; Pylyavskyy, Pavlo; Shustin, Eugenii; Thurston, Dylan Morsifications and mutations (2020) (https://arxiv.org/abs/1711.10598)

[6] Fomin, Sergey; Williams, Lauren; Zelevinsky, Andrei Introduction to cluster algebras, Chapters 1–5 (https://arxiv.org/abs/1608.05735, https://arxiv.org/abs/1707.07190)

[7] Fomin, Sergey; Zelevinsky, Andrei Cluster algebras. I. Foundations, J. Amer. Math. Soc., Volume 15 (2002) no. 2, pp. 497-529 | DOI | MR | Zbl

[8] Fomin, Sergey; Zelevinsky, Andrei Cluster algebras. II. Finite type classification, Invent. Math., Volume 154 (2003) no. 1, pp. 63-121 | DOI | MR | Zbl

[9] Fomin, Sergey; Zelevinsky, Andrei Cluster algebras. IV. Coefficients, Compos. Math., Volume 143 (2007) no. 1, pp. 112-164 | DOI | MR | Zbl

[10] Ford, Nicolas; Serhiyenko, Khrystyna Green-to-red sequences for positroids, J. Combin. Theory Ser. A, Volume 159 (2018), pp. 164-182 | DOI | MR | Zbl

[11] Galashin, Pavel; Lam, Thomas Positroid varieties and cluster algebras (https://arxiv.org/abs/1906.03501)

[12] Gross, Mark; Hacking, Paul; Keel, Sean; Kontsevich, Maxim Canonical bases for cluster algebras, J. Amer. Math. Soc., Volume 31 (2018) no. 2, pp. 497-608 | DOI | MR | Zbl

[13] Henrich, Thilo Mutation classes of diagrams via infinite graphs, Math. Nachr., Volume 284 (2011) no. 17-18, pp. 2184-2205 | DOI | MR | Zbl

[14] Hernandez, David; Leclerc, Bernard Cluster algebras and quantum affine algebras, Duke Math. J., Volume 154 (2010) no. 2, pp. 265-341 | DOI | MR | Zbl

[15] Keller, Bernhard Quiver mutation and combinatorial DT-invariants (https://arxiv.org/abs/1709.03143)

[16] Keller, Bernhard Quiver mutation in JavaScript and Java (software freely available at https://webusers.imj-prg.fr/~bernhard.keller/quivermutation/)

[17] Keller, Bernhard Cluster algebras, quiver representations and triangulated categories, Triangulated categories (London Math. Soc. Lecture Note Ser.), Volume 375, Cambridge Univ. Press, Cambridge, 2010, pp. 76-160 | DOI | MR | Zbl

[18] Keller, Bernhard On cluster theory and quantum dilogarithm identities, Representations of algebras and related topics (EMS Ser. Congr. Rep.), Eur. Math. Soc., Zürich, 2011, pp. 85-116 | DOI | Zbl

[19] Lee, Kyungyong; Schiffler, Ralf Positivity for cluster algebras, Ann. of Math. (2), Volume 182 (2015) no. 1, pp. 73-125 | DOI | MR | Zbl

[20] Marsh, Bethany R. Lecture notes on cluster algebras, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2013, ii+117 pages | MR | Zbl

[21] Marsh, Bethany R.; Scott, Jeanne Twists of Plücker coordinates as dimer partition functions, Comm. Math. Phys., Volume 341 (2016) no. 3, pp. 821-884 | DOI | Zbl

[22] Muller, Greg The existence of a maximal green sequence is not invariant under quiver mutation, Electron. J. Combin., Volume 23 (2016) no. 2, Paper no. Paper 2.47, 23 pages | MR | Zbl

[23] Musiker, Gregg; Stump, Christian A compendium on the cluster algebra and quiver package in Sage, Sém. Lothar. Combin., Volume 65 (2010/12), Paper no. Art. B65d, 67 pages | MR | Zbl

[24] Nagao, Kentaro Donaldson–Thomas theory and cluster algebras, Duke Math. J., Volume 162 (2013) no. 7, pp. 1313-1367 | DOI | MR | Zbl

[25] Nakanishi, Tomoki; Zelevinsky, Andrei On tropical dualities in cluster algebras, Algebraic groups and quantum groups (Contemp. Math.), Volume 565, Amer. Math. Soc., 2012, pp. 217-226 | DOI | MR | Zbl

[26] Plamondon, Pierre-Guy Cluster characters for cluster categories with infinite-dimensional morphism spaces, Adv. Math., Volume 227 (2011) no. 1, pp. 1-39 | DOI | MR | Zbl

[27] Postnikov, Alexander Total positivity, Grassmannians, and networks (https://arxiv.org/abs/math/0609764)

[28] Székely, László A. Turán’s brick factory problem: the status of the conjectures of Zarankiewicz and Hill, Graph theory—favorite conjectures and open problems. 1, Springer, Cham, 2016, pp. 211-230 | Zbl

Cited by Sources: