arXiv subjects
Publications and source records attributed to Dylan Rupel.
We identify a quantum lift of the greedy basis for rank 2 coefficient-free cluster algebras. Our main result is that our construction does not depend on the choice of initial cluster, that it builds all cluster monomials, and that it produces bar-invariant elements. We also present several conjectures related to this quantum greedy basis and the triangular basis of Berenstein and Zelevinsky.
In this note we extend the notion of greedy bases developed by Lee, Li, and Zelevinsky to rank two generalized cluster algebras, i.e. binomial exchange relations are replaced by polynomial exchange relations. In the process we give a combinatorial construction in terms of a refined notion of compatible pairs on a maximal Dyck path.
We investigate the hierarchical structure of processes using the mathematical theory of operads. Information or material enters a given process as a stream of inputs, and the process converts it to a stream of outputs. Output streams can then be supplied to other processes in an organized manner, and the resulting system of interconnected processes can itself be considered a macro process. To model the inherent structure in this kind of system, we define an operad $\mathcal{W}$ of black boxes and directed wiring diagrams, and we define a $\mathcal{W}$-algebra $\mathcal{P}$ of processes (which we call propagators, after Radul and Sussman). Previous operadic models of wiring diagrams use undirected wires without length, useful for modeling static systems of constraints, whereas we use directed wires with length, useful for modeling dynamic flows of information. We give multiple examples throughout to ground the ideas.
Let $\FF$ be a finite field and $(Q,\bfd)$ an acyclic valued quiver with associated exchange matrix $\tilde{B}$. We follow Hubery's 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 $Q$ to the set of non-initial quantum cluster variables for the quantum cluster algebra $\cA_{|\FF|}(\tilde{B},Λ)$. As a corollary we find that, for any rigid valued representation $V$ of $Q$, all Grassmannians of subrepresentations $Gr_\bfe^V$ have counting polynomials.
We extend the Lee-Schiffler Dyck path model to give a proof of the Kontsevich non-commutative cluster positivity conjecture with unequal parameters.
In this paper we give a direct proof of the positivity conjecture for adapted quantum cluster variables. Moreover, our process allows one to explicitly compute formulas for all adapted cluster monomials and certain ordered products of adapted cluster monomials. In particular, we describe all cluster monomials in cluster algebras and quantum cluster algebras of rank 2. One may obtain similar formulas for all finite type cluster monomials. The above results are achieved by computing explicit set-theoretic decompositions of Grassmannians of subrepresentations in adapted valued quiver representations into a disjoint union of products of standard vector space Grassmannians. We actually prove a more general result which should be of independent interest: we compute these decompositions for arbitrary flags of subrepresentations in adapted valued quiver representations. This implies the existence of counting polynomials for the number of points in these sets over different finite fields. Using this we extend the results of \cite{rupel} to quantum cluster algebras $\Acal_q(Q,\bfd)$, where $q$ is an indeterminate.
Let $Q$ be any invertible valued quiver without oriented cycles. We study connections between the category of valued representations of $Q$ and expansions of cluster variables in terms of the initial cluster in quantum cluster algebras. We show that an analogue of the Caldero-Chapoton formula holds for all quantum cluster algebras of finite type and for any cluster variable in an almost acyclic cluster.