Partial F-invariants and cluster categorifications

Abstract

The F-invariant in cluster algebras is a combinatorial invariant that unifies the E-invariant from additive categorification and the d-invariant from monoidal categorification. In this paper, we study its refinement, the partial F-invariant, and establish its mutation formula under changes of the initial seed. As an application, we prove a conjecture of Reading, which asserts that the non-compatible cluster variables can be separated by sign-coherence of g-vectors upon varying the initial seed. We further show that, for cluster monomials, the partial F-invariants coincide with both the partial E-invariants for reachable decorated representations of quivers with potentials and the pole orders of normalized R-matrices (or partial d-invariants) for finite-dimensional reachable simple modules over quantum affine algebras. As consequences, we obtain a combinatorial formula for the pole orders for reachable simple modules in terms of q-characters; we verify the conjectural explicit formula for the pole orders between Kirillov--Reshetikhin modules.

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BibTeX

@misc{cao2026partial,
  title = {{Partial $F$-invariants and cluster categorifications}},
  author = {Peigen Cao and Ryo Fujita and Kota Murakami},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.08781},
  eprint = {2609.08781},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.08781},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Partial F-invariants and cluster categorifications
AU  - Peigen Cao
AU  - Ryo Fujita
AU  - Kota Murakami
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2609.08781
UR  - https://arxiv.org/abs/2609.08781
AB  - The F-invariant in cluster algebras is a combinatorial invariant that unifies the E-invariant from additive categorification and the d-invariant from monoidal categorification. In this paper, we study its refinement, the partial F-invariant, and establish its mutation formula under changes of the initial seed. As an application, we prove a conjecture of Reading, which asserts that the non-compatible cluster variables can be separated by sign-coherence of g-vectors upon varying the initial seed. We further show that, for cluster monomials, the partial F-invariants coincide with both the partial E-invariants for reachable decorated representations of quivers with potentials and the pole orders of normalized R-matrices (or partial d-invariants) for finite-dimensional reachable simple modules over quantum affine algebras. As consequences, we obtain a combinatorial formula for the pole orders for reachable simple modules in terms of q-characters; we verify the conjectural explicit formula for the pole orders between Kirillov--Reshetikhin modules.
ER  -

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