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.

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Accepted:
Published online:
DOI: 10.5802/alco.81
Classification: 13F60,  16G20
Keywords: non-commutative cluster, Kontsevich automorphism, maximal Dyck path, quiver Grassmannian
Rupel, Dylan C. 1

1 Michigan State University Department of Mathematics 619 Red Cedar Road C212 Wells Hall East Lansing, MI 48824, USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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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/

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