Conjugation of reddening sequences and conjugation difference
Siyang Liu, Jie Pan
TL;DR
This work addresses how reddening and maximal green sequences behave under seed mutations in totally sign-skew-symmetric cluster algebras. It develops a purely combinatorial framework by formulating the Conjugation Lemma, proving a Rotation Lemma, and introducing the conjugation difference to quantify changes in red mutations, all without recourse to categorification. The results extend key properties—such as mutation-invariance of reddening sequence existence and Target before Source phenomena—to a broad class beyond skew-symmetric cases, including acyclic and general totally sign-skew-symmetric matrices. Overall, the paper provides a unified, mutation-aware approach to analyzing reddening/maximal green sequences in a wide family of cluster algebras, with implications for combinatorial and computational aspects of mutation dynamics.
Abstract
We describe the conjugation of the reddening sequence according to the formula of $c$-vectors with respect to changing the initial seed. As applications, we extend the Rotation Lemma, the Target before Source Theorem, and the mutation invariant property of the existence of reddening sequences to totally sign-skew-symmetric cluster algebras. Furthermore, this also leads to the construction of conjugation difference which characterizes the number of red mutations a maximal green sequence should admit in any matrix pattern with the initial seed changed via mutations.