# ALGEBRAIC COMBINATORICS

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/

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