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Mutation Of Matrices Over Group Rings

Dani Kaufman, Carmen Alves Sabin

TL;DR

This work extends mutation theory to σ-skew-symmetric matrices over group rings Z[G] by framing them as folded quivers under a G-action and showing their mutations align with folding when diagonal entries vanish. It introduces generalized mutations via Green-to-Red sequences to handle nonzero diagonal entries, and develops canonical unfoldings that connect folded matrices to familiar mutation classes, including A-type and Grassmannian cluster algebras, for cyclic groups G ≅ Z/nZ with n = 3,4. The paper provides explicit constructions, formulas, and examples (3-cycle, 4-cycle, double 3-cycle) that illustrate how folding, weaving, and unfolding interrelate and how generalized mutations can be realized via framed quivers. These results broaden mutation theory to noncommutative settings with symmetry, enabling structured analysis of cluster algebras arising from quivers with group actions and their canonical unfoldings. The practical impact lies in a unified framework for folding and mutating group-ring valued matrices, with concrete links to well-studied Grassmannian cluster algebras and DT-type transformations.

Abstract

We give a precise definition of mutation of skew symmetrizable matrices over group rings and relate it to folding and mutation of quivers with symmetries. These matrices can have non-zero diagonal entries and we explain a mutation rule in some of these cases as well. This new rule comes from a notion of a generalized mutation of an entire quiver or sub-quiver.

Mutation Of Matrices Over Group Rings

TL;DR

This work extends mutation theory to σ-skew-symmetric matrices over group rings Z[G] by framing them as folded quivers under a G-action and showing their mutations align with folding when diagonal entries vanish. It introduces generalized mutations via Green-to-Red sequences to handle nonzero diagonal entries, and develops canonical unfoldings that connect folded matrices to familiar mutation classes, including A-type and Grassmannian cluster algebras, for cyclic groups G ≅ Z/nZ with n = 3,4. The paper provides explicit constructions, formulas, and examples (3-cycle, 4-cycle, double 3-cycle) that illustrate how folding, weaving, and unfolding interrelate and how generalized mutations can be realized via framed quivers. These results broaden mutation theory to noncommutative settings with symmetry, enabling structured analysis of cluster algebras arising from quivers with group actions and their canonical unfoldings. The practical impact lies in a unified framework for folding and mutating group-ring valued matrices, with concrete links to well-studied Grassmannian cluster algebras and DT-type transformations.

Abstract

We give a precise definition of mutation of skew symmetrizable matrices over group rings and relate it to folding and mutation of quivers with symmetries. These matrices can have non-zero diagonal entries and we explain a mutation rule in some of these cases as well. This new rule comes from a notion of a generalized mutation of an entire quiver or sub-quiver.
Paper Structure (14 sections, 6 theorems, 23 equations, 6 figures)

This paper contains 14 sections, 6 theorems, 23 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Basic Definitions
  3. Exchange matrices over group rings
  4. Symmetric Folding
  5. Folding by the action of a group
  6. Weaving
  7. Unfolding
  8. Examples
  9. Exchange Matrices and Mutations over Cyclic Groups
  10. The green to red mutation sequence
  11. Cyclic Groups
  12. The 3-Cycle
  13. The 4-Cycle
  14. The Double 3-Cycle

Key Result

Theorem 3.4

The matrix $B_G^Q(x_1,\dots,x_m)$ is a skew symmetrizable matrix over the group ring $\mathbb{Z}[G]$.

Figures (6)

  • Figure 1: Folding Q into 3 folding sets, namely {1}, {2,4} and {3,5}.
  • Figure 2: Folding the same quiver with respect to two different groups.
  • Figure 3: Combinations of arrows of Q with two folding sets of 3 nodes with assigned elements of $G=\mathbb{Z}/3\mathbb{Z} = \{ 1, \omega, \omega^2 \}$.
  • Figure 4: 4-cycle returns to itself after performing the generalized mutation.
  • Figure 5: Unfolding the double 3-cycle
  • ...and 1 more figures

Theorems & Definitions (40)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8: Mutation of Exchange Matrices
  • Definition 2.9
  • Definition 2.10
  • ...and 30 more