F-invariant and E-invariant
Peigen Cao
TL;DR
This paper addresses how to obtain mutation-invariant, coordinate-free invariants in cluster algebras and how these invariants relate to representation-theoretic data. It provides a new mutation-invariance proof for the $F$-invariant $(u\mid\mid u')_F$ in the spirit of Derksen–Weyman–Zelevinsky, and shows that this invariant matches the $E$-invariant on cluster monomials, connecting combinatorial and categorical perspectives. A key contribution is proving Reading's conjecture on the separation of non-compatible cluster variables via sign-coherence of $g$-vectors, with implications for when products of cluster variables remain clusters. The results extend to skew-symmetrizable cases and offer a unified framework linking $g$-vectors, $F$-polynomials, and decorated representations in cluster algebras.
Abstract
$F$-invariant for a pair of good elements (e.g. cluster monomials) in cluster algebras is introduced by the author in a previous work. A key feature of $F$-invariant is that it is a coordinate-free invariant, that is, it is mutation invariant under the initial seed mutations. $E$-invariant for a pair of decorated representations of quivers with potentials is introduced by Derksen, Weyman and Zelevinsky, which is also a coordinate-free invariant. The strategies used to show the mutation-invariance of $F$-invariant and $E$-invariant are totally different. In this paper, we give a new proof of the mutation-invariance of $F$-invariant following the strategy used by Derksen, Weyman and Zelevinsky. As a result, we prove that $F$-invariant coincides with $E$-invariant on cluster monomials. We also give a proof of Reading's conjecture, which says that the non-compatible cluster variables in cluster algebras can be separated by the sign-coherence of $g$-vectors.