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.
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.81
Keywords: non-commutative cluster, Kontsevich automorphism, maximal Dyck path, quiver Grassmannian
Rupel, Dylan C. 1
@article{ALCO_2019__2_6_1239_0, author = {Rupel, Dylan C.}, title = {Rank two non-commutative {Laurent} phenomenon and pseudo-positivity}, journal = {Algebraic Combinatorics}, pages = {1239--1273}, publisher = {MathOA foundation}, volume = {2}, number = {6}, year = {2019}, doi = {10.5802/alco.81}, zbl = {07140432}, mrnumber = {4049845}, language = {en}, url = {https://alco.centre-mersenne.org/articles/10.5802/alco.81/} }
TY - JOUR AU - Rupel, Dylan C. TI - Rank two non-commutative Laurent phenomenon and pseudo-positivity JO - Algebraic Combinatorics PY - 2019 SP - 1239 EP - 1273 VL - 2 IS - 6 PB - MathOA foundation UR - https://alco.centre-mersenne.org/articles/10.5802/alco.81/ DO - 10.5802/alco.81 LA - en ID - ALCO_2019__2_6_1239_0 ER -
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/articles/10.5802/alco.81/
[1] A quantum analogue of generalized cluster algebras (2016) (https://arxiv.org/abs/1610.09803) | Zbl
[2] A Short Proof of Kontsevich Cluster Conjecture, C. R. Math. Acad. Sci. Paris, Volume 349 (2011) no. 3-4, pp. 119-122 | DOI | MR | Zbl
[3] On the quiver Grassmannian in the acyclic case, J. Pure Appl. Algebra, Volume 212 (2008) no. 11, pp. 2369-2380 | DOI | MR | Zbl
[4] Teichmüller Spaces of Riemann Surfaces with Orbifold Points of Arbitrary Order and Cluster Variables, Int. Math. Res. Not. (2014) no. 10, pp. 2746-2772 | DOI | Zbl
[5] Discrete Non-Commutative Integrability: Proof of a Conjecture of M. Kontsevich, Int. Math. Res. Not. (2010) no. 21, pp. 4042-4063 | MR | Zbl
[6] Cluster Algebras I. Foundations, J. Amer. Math. Soc., Volume 15 (2002) no. 2, pp. 497-529 | DOI | MR | Zbl
[7] Noncommutative identities (https://arxiv.org/abs/1109.2469)
[8] Greedy elements in rank 2 cluster algebras, Selecta Math., Volume 20 (2014) no. 1, pp. 57-82 | MR | Zbl
[9] Proof of a Positivity Conjecture of M. Kontsevich on Non-Commutative Cluster Variables, Compos. Math., Volume 148 (2012) no. 6, pp. 1821-1832 | MR | Zbl
[10] Greedy bases in rank 2 generalized cluster algebras (https://arxiv.org/abs/1309.2567)
[11] On a quantum analog of the Caldero–Chapoton formula, Int. Math. Res. Not. (2011) no. 14, pp. 3207-3236 | MR | Zbl
[12] Proof of the Kontsevich non-commutative cluster positivity conjecture, C. R. Math. Acad. Sci. Paris, Volume 350 (2012) no. 21-22, pp. 929-932 | DOI | MR | Zbl
[13] Quantum cluster characters for valued quivers, Trans. Amer. Math. Soc., Volume 367 (2015) no. 10, pp. 7061-7102 | DOI | MR | Zbl
[14] Non-commutative Laurent phenomenon for two variables (https://arxiv.org/abs/1006.1211)
[15] Non-commutative cluster mutations, Dokl. Nats. Akad. Nauk Belarusi, Volume 53 (2009) no. 4, pp. 27-29 | MR
[16] Action of the Cremona group on a non-commutative ring, Adv. Math., Volume 228 (2011) no. 4, pp. 1863-1893 | DOI | MR | Zbl
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