F-invariant in cluster algebras
Peigen Cao
TL;DR
This paper introduces two invariants, the tropical invariant $\langle\cdot,\cdot\rangle_{\rm trop}$ and the $F$-invariant $(\cdot\mid\mid\cdot)_F$, for good elements in $\mathsf{\Lambda}$-upper cluster algebras and shows they encode key compatibility data across seeds. It furnishes explicit formulas expressing these invariants in terms of extended $g$-vectors and $F$-polynomials, and proves that the vanishing of the $F$-invariant characterizes when the product of two cluster monomials remains a cluster monomial. The work connects these invariants to categorical invariants in both monoidal and additive cluster categorifications, establishing precise equalities with $\Lambda$-invariants, $\mathfrak{d}$-invariants, and $E$-invariants for cluster monomials, and highlights a unified perspective across settings. By introducing dominant sets, the authors prove the oriented exchange graphs are acyclic and thereby obtain a natural partial order on seeds, with the graphs coinciding with Hasse quivers of seed posets. Overall, the paper provides a cohesive framework linking combinatorial invariants of cluster algebras with categorical invariants, with concrete computational formulas and implications for the structure of exchange graphs.
Abstract
We consider skew-symmetrizable (upper) cluster algebras with a compatible Poisson structure, called $\mathsfΛ$-(upper) cluster algebras. For any two good elements (e.g., cluster monomials) in a $\mathsfΛ$-upper cluster algebra, we introduce two invariants, called tropical invariant and $F$-invariant. We prove that (i) the product of two cluster monomials is still a cluster monomial if and only if their $F$-invariant is zero; (ii) if two cluster variables are log-canonical, then they are contained in the same cluster; and (iii) the notion of $F$-invariant for a pair of cluster monomials can be defined for any (upper) cluster algebra, regardless of whether it is a $\mathsfΛ$-(upper) cluster algebra. When restricting to cluster monomials, we prove that the tropical invariant and $F$-invariant respectively coincide with the $Λ$-invariant and twice $\mathfrak{d}$-invariant in the monoidal cluster categorification using various monoidal subcategories of finite-dimensional modules over quantum affine algebras and quiver Hecke algebras; and we prove that the $F$-invariant coincides with the $E$-invariant in the additive cluster categorification using the theory of quivers with potentials. Inspired by $F$-invariant, we introduce the dominant sets for seeds of cluster algebras as a replacement of torsion classes for $τ$-tilting pairs in $τ$-tilting theory. With the help of the dominant sets, we prove that the oriented exchange graphs of cluster algebras are acyclic. In particular, this implies that green mutations induce a partial order on the set of seeds (up to seed equivalence) of cluster algebras. We prove that the oriented exchange graphs of cluster algebras coincide with the Hasse quivers of the above posets of seeds.