Rank two non-commutative Laurent phenomenon and pseudo-positivity
Algebraic Combinatorics, Volume 2 (2019) no. 6, pp. 1239-1273.

We study polynomial generalizations of the Kontsevich automorphisms acting on the skew-field of formal rational expressions in two non-commuting variables. Our main result is the Laurentness and pseudo-positivity of iterations of these automorphisms. The resulting expressions are described combinatorially using a generalization (studied in [10]) of the combinatorics of compatible pairs in a maximal Dyck path developed by Lee, Li, and Zelevinsky in [8].

By specializing to quasi-commuting variables we obtain pseudo-positive expressions for rank 2 quantum generalized cluster variables. In the case that all internal exchange coefficients are zero, this quantum specialization provides a positive combinatorial construction of counting polynomials for Grassmannians of submodules in exceptional representations of valued quivers with two vertices.

Received: 2017-07-28
Revised: 2019-02-24
Accepted: 2019-04-22
Published online: 2019-12-04
DOI: https://doi.org/10.5802/alco.81
Classification: 13F60,  16G20
Keywords: non-commutative cluster, Kontsevich automorphism, maximal Dyck path, quiver Grassmannian
@article{ALCO_2019__2_6_1239_0,
     author = {Rupel, Dylan C.},
     title = {Rank two non-commutative Laurent phenomenon and pseudo-positivity},
     journal = {Algebraic Combinatorics},
     publisher = {MathOA foundation},
     volume = {2},
     number = {6},
     year = {2019},
     pages = {1239-1273},
     doi = {10.5802/alco.81},
     mrnumber = {4049845},
     zbl = {07140432},
     language = {en},
     url = {alco.centre-mersenne.org/item/ALCO_2019__2_6_1239_0/}
}
Rupel, Dylan C. Rank two non-commutative Laurent phenomenon and pseudo-positivity. Algebraic Combinatorics, Volume 2 (2019) no. 6, pp. 1239-1273. doi : 10.5802/alco.81. https://alco.centre-mersenne.org/item/ALCO_2019__2_6_1239_0/

[1] Bai, Liqian; Chen, Xueqing; Ding, Ming; Xu, Fan A quantum analogue of generalized cluster algebras (2016) (https://arxiv.org/abs/1610.09803) | Zbl 1408.16008

[2] Berenstein, Arkady; Retakh, Vladimir A Short Proof of Kontsevich Cluster Conjecture, C. R. Math. Acad. Sci. Paris, Volume 349 (2011) no. 3-4, pp. 119-122 | Article | MR 2769891 | Zbl 1266.16026

[3] Caldero, Philippe; Reineke, Markus On the quiver Grassmannian in the acyclic case, J. Pure Appl. Algebra, Volume 212 (2008) no. 11, pp. 2369-2380 | Article | MR 2440252 | Zbl 1153.14032

[4] Chekhov, Leonid; Shapiro, Michael Teichmüller Spaces of Riemann Surfaces with Orbifold Points of Arbitrary Order and Cluster Variables, Int. Math. Res. Not. (2014) no. 10, pp. 2746-2772 | Article | Zbl 1301.30042

[5] Di Francesco, Philippe; Kedem, Rinat Discrete Non-Commutative Integrability: Proof of a Conjecture of M. Kontsevich, Int. Math. Res. Not. (2010) no. 21, pp. 4042-4063 | MR 2738350 | Zbl 1276.16025

[6] Fomin, Sergey; Zelevinsky, Andrei Cluster Algebras I. Foundations, J. Amer. Math. Soc., Volume 15 (2002) no. 2, pp. 497-529 | Article | MR 1887642 | Zbl 1021.16017

[7] Kontsevich, Maxim Noncommutative identities (https://arxiv.org/abs/1109.2469)

[8] Lee, Kyungyong; Li, Li; Zelevinsky, Andrei Greedy elements in rank 2 cluster algebras, Selecta Math., Volume 20 (2014) no. 1, pp. 57-82 | MR 3147413 | Zbl 1295.13031

[9] Lee, Kyungyong; Schiffler, Ralf Proof of a Positivity Conjecture of M. Kontsevich on Non-Commutative Cluster Variables, Compos. Math., Volume 148 (2012) no. 6, pp. 1821-1832 | MR 2999306 | Zbl 1266.16027

[10] Rupel, Dylan Greedy bases in rank 2 generalized cluster algebras (https://arxiv.org/abs/1309.2567)

[11] Rupel, Dylan On a quantum analog of the Caldero–Chapoton formula, Int. Math. Res. Not. (2011) no. 14, pp. 3207-3236 | MR 2817677 | Zbl 1237.16013

[12] Rupel, Dylan Proof of the Kontsevich non-commutative cluster positivity conjecture, C. R. Math. Acad. Sci. Paris, Volume 350 (2012) no. 21-22, pp. 929-932 | Article | MR 2996767 | Zbl 1266.16028

[13] Rupel, Dylan Quantum cluster characters for valued quivers, Trans. Amer. Math. Soc., Volume 367 (2015) no. 10, pp. 7061-7102 | Article | MR 3378824 | Zbl 1371.16014

[14] Usnich, Alexandr Non-commutative Laurent phenomenon for two variables (https://arxiv.org/abs/1006.1211)

[15] Usnich, Alexandr Non-commutative cluster mutations, Dokl. Nats. Akad. Nauk Belarusi, Volume 53 (2009) no. 4, pp. 27-29 | MR 2606151

[16] Usnich, Alexandr Action of the Cremona group on a non-commutative ring, Adv. Math., Volume 228 (2011) no. 4, pp. 1863-1893 | Article | MR 2836108 | Zbl 1250.14008