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Large-vertex merge (tau_u) and small source/sink reflection (sigma_u) from Domokos (doi:10.4171/JCA/97), with tau_sigma_reduce and is_taus_minimal. Formulas use QuiverTools' King sign, so the tau case split is mirrored.
Check tau/sigma against the paper's Section 9 pair and reductions to P^2 and the 3-Kronecker 6-fold, sigma involution, and tau-sigma minimality.
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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 drivertau_sigma_reduce, and a boundedis_taus_minimalcheck (Definition 2.3). Each reduction is available both on(Q, d, theta)triples and on aQuiverModuliSpace.Notes
theta_QuiverTools = -theta_paper). The formulas are written in QuiverTools' sign; since they are linear intheta, the only visible consequence is that thetaucase split onsign(theta(u))is mirrored. This is documented in the module header and thetau_reductiondocstring.tau_sigma_reduceis a terminating greedy descent (weaker than full Definition 2.3 minimality).is_taus_minimalis a bounded breadth-first search: afalseanswer is always a genuine witness, whiletrueis "not disproved within the search bound" (sigma-orbits can be infinite, as the paper's Section 9 example shows).Tests
Doctests plus a
@testsetcovering:taureductions of1 → 2 ⇉⇉⇉ 3to the 3-Kronecker quiver:[1,3,1] → P^2and[2,9,3] →the 6-foldM(K_3, [2,3]), with matching Hodge diamonds (Theorem 2.5);(Q, [2,1,3]): no large vertex, small source/sink, greedy-reduce fixpoint, dimension preserved undersigma;sigmais an involution;[1,1] →a single loop, and the correspondingis_taus_minimaltransitions.Full test suite is green.