Dylan Rupel

On a Quantum Analog of the Caldero–Chapoton Formula

Let Q be any valued quiver without oriented cycles. We study connections between the category of valued representations of Q and expansions of cluster variables in terms of an initial cluster in quantum cluster algebras. We show that an analog of the Caldero–Chapoton formula holds for the specializations , where is a finite field, in all quantum cluster algebras of finite type, including nonsimply laced types, and for any cluster variable in an almost acyclic cluster.

Proof of the Kontsevich non-commutative cluster positivity conjecture

Abstract We extend the Lee–Schiffler Dyck path model to give a proof of the Kontsevich non-commutative cluster positivity conjecture with unequal parameters.

Quantum Cluster Characters

Let

Greedy bases in rank 2 quantum cluster algebras

\FF

String diagrams for traced and compact categories are oriented 1-cobordisms

be a finite field and

The existence of greedy bases in rank 2 quantum cluster algebras

(Q,\bfd)

The operad of temporal wiring diagrams: formalizing a graphical language for discrete-time processes.

an acyclic valued quiver with associated exchange matrix

Quantum cluster characters of Hall algebras

\tilde{B}

Greedy bases in rank 2 generalized cluster algebras

. We follow Huberys approach \cite{hub1} to prove our main conjecture of \cite{rupel}: the quantum cluster character gives a bijection from the isoclasses of indecomposable rigid valued representations of

The Feigin Tetrahedron

Q