Quantum cluster variables via Serre polynomials

Abstract

Abstract. For skew-symmetric acyclic quantum cluster algebras, we express the quantum F -polynomials and the quantum cluster monomials in terms of Serre polynomials of quiver Grassmannians of rigid modules. As byproducts, we obtain the existence of counting polynomials for these varieties and the positivity conjecture with respect to acyclic seeds. These results complete previous work by Caldero and Reineke and confirm a recent conjecture by Rupel.

Quantum unipotent subgroup and dual canonical basis

“…In fact, these results apply only to cluster algebras, not quantum ones except [49]. Thus we have much stronger positivity.…”

“…• cluster algebras of finite type [16], • cluster algebras with bipartite seeds [47], • cluster algebras coming from triangulated surfaces [45], • acyclic cluster algebras at the initial seed [49].…”

Bar-invariant bases of the quantum cluster algebra of type A 2 (2)

“…This conjecture has been proved for acyclic equally valued quivers in [15]. Naturally, one may hope to construct Z[q ±1/2 ]-bases of quantum cluster algebras.…”

On natural maps from strata of quiver Grassmannians to ordinary Grassmannians

“…Recently the study of quiver Grassmannians has been very active [because of its important role in Cluster algebra, and the surprising fact that every projective variety is a quiver Grassmannian] (for instance, see [3,4,5,6,7,11,14,19,20,21]). Let Q be a quiver on vertices {1, ..., n}.…”

“…A very interesting result of [18,19,10,14] shows that if M is an indecomposable rigid representation of an acyclic quiver Q, then the Euler characteristic of any quiver Grassmannian is nonnegative. Caldero and Zelevinsky [6] studied the geometry of quiver Grassmannians for the Kronecker quiver (the quiver with two vertices and two arrows from one vertex to the other) and obtained simple formulas for their Euler characteristics by examining natural stratification of quiver Grassmannians.…”