an:07140432
Zbl 1436.16018
Rupel, Dylan C.
Rank two non-commutative Laurent phenomenon and pseudo-positivity
EN
Algebr. Comb. 2, No. 6, 1239-1273 (2019).
00443059
2019
j
16G20 05E10 13F60
non-commutative cluster; Kontsevich automorphism; maximal Dyck path; quiver Grassmannian
Summary: 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 our work [``Greedy bases in rank 2 generalized cluster algebras'', Preprint, \url{arXiv:1309.2567}]) of the combinatorics of compatible pairs in a maximal Dyck path developed by \textit{K. Lee} et al. [Sel. Math., New Ser. 20, No. 1, 57--82 (2014; Zbl 1295.13031)].
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.
Zbl 1295.13031