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arXiv · 1707.02696

Rank Two Non-Commutative Laurent Phenomenon and Pseudo-Positivity

Dylan Rupel
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Abstract

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 of the combinatorics of compatible pairs in a maximal Dyck path developed by Lee, Li, and Zelevinsky. By specializing to quasi-commuting variables we obtain pseudo-positive expressions for rank 2 quantum generalized cluster variables. In the binomial case when 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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Dylan Rupel. 2017-07-10. Rank Two Non-Commutative Laurent Phenomenon and Pseudo-Positivity. https://arxiv.org/abs/1707.02696

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