$f$-vectors and $F$-invariant in generalized cluster algebras

Abstract

We establish certain fundamental properties of $f$-vectors and $F$-matrices for generalized cluster algebras, including the initial and final seed mutation formulas, the compatibility property and the symmetry property. Along the way, we also generalize the construction of $F$-invariant for generalized cluster algebras without assuming positivity and prove certain basic properties.

17 pages