Exchange matrices of I-boxes
Masaki Kashiwara, Myungho Kim
TL;DR
The paper provides an explicit combinatorial description of the exchange matrix for maximal commuting families of i-boxes, connecting i-box envelopes and effective ends to seed mutations in cluster algebras. It defines the exchange matrix tilde B(F) from horizontal/vertical relations among i-boxes and proves, in two monoidal categorifications (C_w and C_g^{[a,b]}), that the pair (F, tilde B(F)) forms a seed whose mutations correspond to short exact sequences, i.e., T-systems. Through a detailed case analysis of vertical arrows and a general mutation framework, it establishes a concrete bridge between i-box combinatorics and the cluster structure in Grothendieck rings, including explicit examples. The results illuminate how determinantial and root modules realize cluster variables and mutations, offering a practical pathway to compute seeds from combinatorial data and advancing monoidal categorification in non-symmetric settings. Overall, the work provides both a rigorous theoretical foundation and tangible tools for constructing and mutating seeds in quantum cluster algebras via i-box combinatorics.
Abstract
Admissible chains of i-boxes are important combinatorial tools in the monoidal categorification of cluster algebras, as they provide seeds of the cluster algebra. In this paper, we explore the properties of maximal commuting families of i-boxes in a more general setting, and define a certain matrix associated with such a family, which we call the exchange matrix. It turns out that, when considering the cluster algebra structure on the Grothendieck rings, this matrix is indeed the exchange matrix of the seed associated with the family, both in certain categories of modules over quantum affine algebras and over quiver Hecke algebras. We prove this by constructing explicit short exact sequences that represent the mutation relations.