Polynomial criteria for quiver mutation-acyclicity
Can a fixed finite set of polynomial tests on a quiver’s arrow counts decide whether mutations can remove all directed cycles? A quiver is a finite directed graph that allows repeated arrows but no loops or opposite-arrow pairs. Mutation is a local rule that changes arrows at one vertex. A quiver is mutation-acyclic if some finite sequence of mutations reaches a graph with no directed cycle.
Why it matters and what counts as progress
Why it matters: Quiver mutation is part of cluster algebra theory. A criterion for reaching an acyclic quiver connects an infinite search over mutations with the structure of representations and cluster algebras. Limits on polynomial criteria show which classification strategies cannot work.
What counts as progress: Give a decision algorithm under explicit vertex and weight assumptions, find exact criteria for a restricted class, or prove that a proposed type of criterion is impossible. Separate bounds in arrow multiplicities from bounds in binary input length; finite searches alone do not settle unbounded families.
How the problem developed
- 2008 · Rank-three algorithm [1]
I. Assem, M. Blais, T. Brüstle and A. Samson Gave a descent algorithm for mutation classes on three vertices. The later paper contrasts this solved rank-three problem with higher-rank classification.
- 2011 · Polynomial criterion [1]
A. Beineke, T. Brüstle and L. Hille Characterized mutation-cyclic oriented triangles through their arrow multiplicities and Markov constant. This supplies the rank-three polynomial-sign criterion that motivates the higher-rank question.
Kymani T. K. Armstrong-Williams, Edward Hirst, Blake Jackson and Kyu-Hwan Lee with GPT-Astra Presented the model-discovered alternating quiver family and a proof that fixed finite polynomial-sign tests cannot decide mutation-acyclicity for connected integral quivers on any vertices. Human generalization and exposition retain separate credit.
Selected milestones, as described and cited in the primary paper. They do not constitute a complete history of cluster algebras or mutation invariants.
AI contribution
Mutation-acyclicity in rank four is decidable but not semialgebraic
For every , no fixed finite Boolean combination of polynomial equations and strict polynomial inequalities with real coefficients in the arrow weights decides mutation-acyclicity of all integral n-vertex quivers, even restricted to connected quivers. The rank-four family with weights is mutation-acyclic exactly when or t is even; attaching a path extends the obstruction to higher ranks.
Sources
Publication history
Mutation-acyclicity in rank four is decidable but not semialgebraic
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