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feat: Domokos reduction for quiver moduli#34

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domokos
Open

feat: Domokos reduction for quiver moduli#34
pbelmans wants to merge 11 commits into
bocklandtfrom
domokos

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@pbelmans pbelmans commented Jul 9, 2026

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Implements the two reduction operations of Domokos, Quiver moduli spaces of a given dimension (doi:10.4171/JCA/97), which transform a quiver-dimension vector pair while preserving its moduli space up to isomorphism (Theorem 2.5):

  • tau_reduction (τ_u): merge and delete a large vertex (Definition 2.1, Lemma 3.1);
  • sigma_reduction (σ_u): reflect a small source or sink (Definition 2.2, Lemma 3.3).

Plus the supporting predicates is_large, is_small_source, is_small_sink, a greedy driver tau_sigma_reduce, and a bounded is_taus_minimal check (Definition 2.3). Each reduction is available both on (Q, d, theta) triples and on a QuiverModuliSpace.

Notes

  • Sign convention. The paper's stability is the opposite King sign to QuiverTools (theta_QuiverTools = -theta_paper). The formulas are written in QuiverTools' sign; since they are linear in theta, the only visible consequence is that the tau case split on sign(theta(u)) is mirrored. This is documented in the module header and the tau_reduction docstring.
  • Minimality. tau_sigma_reduce is a terminating greedy descent (weaker than full Definition 2.3 minimality). is_taus_minimal is a bounded breadth-first search: a false answer is always a genuine witness, while true is "not disproved within the search bound" (sigma-orbits can be infinite, as the paper's Section 9 example shows).

Tests

Doctests plus a @testset covering:

  • tau reductions of 1 → 2 ⇉⇉⇉ 3 to the 3-Kronecker quiver: [1,3,1] → P^2 and [2,9,3] → the 6-fold M(K_3, [2,3]), with matching Hodge diamonds (Theorem 2.5);
  • the Section 9 pair (Q, [2,1,3]): no large vertex, small source/sink, greedy-reduce fixpoint, dimension preserved under sigma;
  • sigma is an involution;
  • the 2-cycle [1,1] → a single loop, and the corresponding is_taus_minimal transitions.

Full test suite is green.

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