2017/07/10 by Rupel, Dylan
#13F60 #16G20 #Combinatorics (math.CO) #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.1707.02696
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.