Browse issues
no. 1
Multiplicative Inequalities in Cluster Algebras of Finite Type
Gekhtman, Michael1; Greenberg, Zachary2; Soskin, Daniel3
1The University of Notre Dame, 255 Hurley Bldg, 46556 Notre Dame, Indiana, USA
2Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, 04103 Leipzig, Germany
3Institute for Advanced Study, 1 Einstein Drive, 08540 Princeton, New Jersey, USA
Algebraic Combinatorics, Volume 9 (2026) no. 1, pp. 51-74

Generalizing the notion of a multiplicative inequality among minors of a totally positive matrix, we describe, over full rank cluster algebras of finite type, the cone of Laurent monomials in cluster variables that are bounded as real-valued functions on the positive locus of the cluster variety. We prove that the extreme rays of this cone are the $u$-variables of the cluster algebra. Using this description, we prove that all bounded ratios are bounded by 1 and give a sufficient condition for all such ratios to be subtraction free. This allows us to show in $\mathrm{Gr}(2,n)$, $\mathrm{Gr}(3,6)$, $\mathrm{Gr}(3,7)$, and $\mathrm{Gr}(3,8)$ that every bounded Laurent monomial in Plücker coordinates factors into a positive integer combination of so-called primitive ratios. In $\mathrm{Gr}(4,8)$ this factorization does not exists, but we provide the full list of extreme rays of the cone of bounded Laurent monomials in Plücker coordinates.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.463
Classification: 13F60, 17B05, 14M15
Keywords: Cluster algebras, total positivity, root systems, Grassmannians

Gekhtman, Michael  1 ; Greenberg, Zachary  2 ; Soskin, Daniel  3

1 The University of Notre Dame, 255 Hurley Bldg, 46556 Notre Dame, Indiana, USA
2 Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, 04103 Leipzig, Germany
3 Institute for Advanced Study, 1 Einstein Drive, 08540 Princeton, New Jersey, USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{ALCO_2026__9_1_51_0,
     author = {Gekhtman, Michael and Greenberg, Zachary and Soskin, Daniel},
     title = {Multiplicative {Inequalities} in {Cluster} {Algebras} of {Finite} {Type}},
     journal = {Algebraic Combinatorics},
     pages = {51--74},
     year = {2026},
     publisher = {The Combinatorics Consortium},
     volume = {9},
     number = {1},
     doi = {10.5802/alco.463},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.463/}
}
TY  - JOUR
AU  - Gekhtman, Michael
AU  - Greenberg, Zachary
AU  - Soskin, Daniel
TI  - Multiplicative Inequalities in Cluster Algebras of Finite Type
JO  - Algebraic Combinatorics
PY  - 2026
SP  - 51
EP  - 74
VL  - 9
IS  - 1
PB  - The Combinatorics Consortium
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.463/
DO  - 10.5802/alco.463
LA  - en
ID  - ALCO_2026__9_1_51_0
ER  - 
%0 Journal Article
%A Gekhtman, Michael
%A Greenberg, Zachary
%A Soskin, Daniel
%T Multiplicative Inequalities in Cluster Algebras of Finite Type
%J Algebraic Combinatorics
%D 2026
%P 51-74
%V 9
%N 1
%I The Combinatorics Consortium
%U https://alco.centre-mersenne.org/articles/10.5802/alco.463/
%R 10.5802/alco.463
%G en
%F ALCO_2026__9_1_51_0
Gekhtman, Michael; Greenberg, Zachary; Soskin, Daniel. Multiplicative Inequalities in Cluster Algebras of Finite Type. Algebraic Combinatorics, Volume 9 (2026) no. 1, pp. 51-74. doi: 10.5802/alco.463

[1] Arkani-Hamed, Nima; He, Song; Lam, Thomas Cluster configuration spaces of finite type, SIGMA Symmetry Integrability Geom. Methods Appl., Volume 17 (2021), Paper no. 092, 41 pages | DOI | MR | Zbl

[2] Boocher, Adam; Froehle, Bradley On generators of bounded ratios of minors for totally positive matrices, Linear Algebra Appl., Volume 428 (2008) no. 7, pp. 1664-1684 | DOI | MR | Zbl

[3] Bruns, W.; Ichim, B.; Söger, C.; Ohe, U. von der Normaliz. Algorithms for rational cones and affine monoids https://www.normaliz.uni-osnabrueck.de

[4] Ceballos, Cesar; Pilaud, Vincent Denominator vectors and compatibility degrees in cluster algebras of finite type, Trans. Amer. Math. Soc., Volume 367 (2015) no. 2, pp. 1421-1439 | DOI | MR | Zbl

[5] Chang, Wen; Duan, Bing; Fraser, Chris; Li, Jian-Rong Quantum affine algebras and Grassmannians, Math. Z., Volume 296 (2020) no. 3-4, pp. 1539-1583 | DOI | MR | Zbl

[6] Fallat, Shaun M.; Gekhtman, Michael I.; Johnson, Charles R. Multiplicative principal-minor inequalities for totally nonnegative matrices, Adv. in Appl. Math., Volume 30 (2003) no. 3, pp. 442-470 | DOI | MR | Zbl

[7] Fallat, Shaun M.; Johnson, Charles R. Determinantal inequalities: ancient history and recent advances, Algebra and its applications (Athens, OH, 1999) (Contemp. Math.), Volume 259, Amer. Math. Soc., Providence, RI (2000), pp. 199-212 | DOI | MR | Zbl

[8] Fischer, E. Über den Hadamardschen Determinantensatz, Arch. Math. (Basel), Volume 13 (1908), pp. 32-40 | Zbl

[9] Fock, Vladimir V.; Goncharov, Alexander B. Cluster ensembles, quantization and the dilogarithm, Ann. Sci. Éc. Norm. Supér. (4), Volume 42 (2009) no. 6, pp. 865-930 | DOI | MR | Zbl | Numdam

[10] Fomin, Sergey; Williams, Lauren; Zelevinsky, Andrei Introduction to Cluster Algebras. Chapters 1-3 (2021) | arXiv

[11] Fomin, Sergey; Williams, Lauren; Zelevinsky, Andrei Introduction to Cluster Algebras. Chapters 6 (2021) | arXiv

[12] Fomin, Sergey; Zelevinsky, Andrei Double Bruhat cells and total positivity, J. Amer. Math. Soc., Volume 12 (1999) no. 2, pp. 335-380 | DOI | MR | Zbl

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

[14] Fomin, Sergey; Zelevinsky, Andrei Y-systems and generalized associahedra, Ann. of Math. (2), Volume 158 (2003) no. 3, pp. 977-1018 | DOI | MR | Zbl

[15] Gekhtman, Michael; Shapiro, Michael; Vainshtein, Alek Cluster algebras and Poisson geometry, Mathematical Surveys and Monographs, 167, American Mathematical Society, Providence, RI, 2010, xvi+246 pages | DOI | MR | Zbl

[16] Gekhtman, Michael; Shapiro, Michael; Vainshtein, Alek Cluster algebras and Poisson geometry, Mathematical Surveys and Monographs, 167, American Mathematical Society, Providence, RI, 2010, xvi+246 pages | DOI | MR | Zbl

[17] Hadamard, Jacques Resolution d’une question relative aux determinants, Bull. des sciences math., Volume 2 (1893), pp. 240-246 | Zbl

[18] 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 | MR | Zbl | DOI

[19] Koteljanskiĭ, D. M. The theory of nonnegative and oscillating matrices, Amer. Math. Soc. Transl. (2), Volume 27 (1963), pp. 1-8 | DOI | MR | Zbl

[20] Kotelyanskiĭ, D. M. On the theory of nonnegative and oscillating matrices, Ukrain. Mat. Žurnal, Volume 2 (1950) no. 2, pp. 94-101 | MR | Zbl

[21] Lam, Thomas; Speyer, David E. Cohomology of cluster varieties I: Locally acyclic case, Algebra Number Theory, Volume 16 (2022) no. 1, pp. 179-230 | DOI | MR | Zbl

[22] Lusztig, George Total positivity and canonical bases, Algebraic groups and Lie groups (Austral. Math. Soc. Lect. Ser), Volume 9, Cambridge Univ. Press, 1997, pp. 281-295 | MR | Zbl

[23] Postnikov, Alexander Total positivity, Grassmannians, and networks (2006) | arXiv | Zbl

[24] Scott, Jeanne S. Grassmannians and cluster algebras, Proc. London Math. Soc. (3), Volume 92 (2006) no. 2, pp. 345-380 | DOI | MR | Zbl

[25] Skandera, Mark Inequalities in products of minors of totally nonnegative matrices, J. Algebraic Combin., Volume 20 (2004) no. 2, pp. 195-211 | DOI | MR | Zbl

[26] Soskin, Daniel; Gekhtman, Michael On bounded ratios of minors of totally positive matrices, Linear Algebra Appl., Volume 715 (2025), pp. 46-67 | DOI | MR | Zbl

[27] Zickert, Christian K. Fock-Goncharov coordinates for rank two Lie groups, Math. Z., Volume 294 (2020) no. 1-2, pp. 251-286 | DOI | MR | Zbl

Cited by Sources: