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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DOI : 10.5802/alco.81
Mots clés : 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},
pages = {1239--1273},
publisher = {MathOA foundation},
volume = {2},
number = {6},
year = {2019},
doi = {10.5802/alco.81},
mrnumber = {4049845},
zbl = {07140432},
language = {en},
url = {http://www.numdam.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 - http://www.numdam.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, Tome 2 (2019) no. 6, pp. 1239-1273. doi : 10.5802/alco.81. http://www.numdam.org/articles/10.5802/alco.81/
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