Initial-seed recursions and dualities for d-vectors

TL;DR

This paper introduces a new initial-seed-mutation formula for d-vectors in cluster algebras, explores dualities, and proves its validity in certain cases, including surface cluster algebras.

Contribution

It presents a novel recursion for d-vectors, duality reformulations, and proves the recursion in surface cluster algebras, advancing understanding of cluster algebra structures.

Findings

The initial-seed-mutation recursion holds in some cluster algebras but not all.

Duality formulas relate d-vectors to g-vectors and c-vectors.

The conjecture is proven for surface cluster algebras.

Abstract

We present an initial-seed-mutation formula for d-vectors of cluster variables in a cluster algebra. We also give two rephrasings of this recursion: one as a duality formula for d-vectors in the style of the g-vectors/c-vectors dualities of Nakanishi and Zelevinsky, and one as a formula expressing the highest powers in the Laurent expansion of a cluster variable in terms of the d-vectors of any cluster containing it. We prove that the initial-seed-mutation recursion holds in a varied collection of cluster algebras, but not in general. We conjecture further that the formula holds for source-sink moves on the initial seed in an arbitrary cluster algebra, and we prove this conjecture in the case of surfaces.