feat: add covering quiver support

.JuliaFormatter.jl

@@ -4,6 +4,7 @@ files = [
   "src/Bundles.jl",
   "src/Chow.jl",
   "src/Constructors.jl",
+  "src/CoveringQuiver.jl",
   "src/Hodge.jl",
   "src/Misc.jl",
   "src/Moduli.jl",
@@ -17,6 +18,7 @@ files = [
   "ext/QuiverToolsOscarExt.jl",
   "docs/docs.jl",
   "benchmark/benchmarks.jl",
+  "test/covering_quiver.jl",
   "test/runtests.jl",
   ".JuliaFormatter.jl",
 ]

docs/docs.jl

@@ -29,6 +29,7 @@ makedocs(;
       "Constructors" => "methods/constructors.md",
       "Quiver moduli" => "methods/quiver-moduli.md",
       "Representation theory" => "methods/representation-theory.md",
+      "Covering quiver" => "methods/covering-quiver.md",
       "Teleman quantization" => "methods/teleman-quantization.md",
       "Chow rings" => "methods/chow-rings.md",
       "Walls and chamber decompositions" => "methods/walls-and-chambers.md",

docs/src/methods/covering-quiver.md

@@ -0,0 +1,116 @@
+# Covering quiver
+
+For a finite quiver ``Q`` with arrow set ``Q_1 = \{a_1, \ldots, a_m\}``,
+the *covering quiver* ``Q(w)`` is the (infinite) quiver
+
+```math
+\begin{aligned}
+Q(w)_0 &= Q_0 \times \mathbb{Z}^m,\\
+Q(w)_1 &= \bigl\{ (s(a_k), \xi) \to (t(a_k), \xi + e_k)
+  \;\bigm|\; a_k \in Q_1,\ \xi \in \mathbb{Z}^m \bigr\},
+\end{aligned}
+```
+
+where ``(e_k)_{k=1}^m`` is the standard basis of ``\mathbb{Z}^m``. The projection
+``(i, \xi) \mapsto i`` is the universal abelian cover of ``Q`` with respect to the
+free abelian group on ``Q_1``. Coordinates use the order returned by
+[`arrows`](@ref), so a character vector and an arrow index always refer to
+the same ordering.
+
+A *compatible dimension vector* for ``d \in \mathbb{N}^{Q_0}`` is a function
+``\beta\colon Q_0 \times \mathbb{Z}^m \to \mathbb{N}`` with finite support such that
+``\sum_\xi \beta(i, \xi) = d_i`` for every ``i \in Q_0``. The group ``\mathbb{Z}^m``
+acts on compatible dimension vectors by
+``s_\chi(\beta)(i, \xi) = \beta(i, \xi + \chi)``.
+
+Over an algebraically closed field, every connected component of the
+natural-torus fixed locus of the stable moduli space ``M^\theta(Q, d)`` is of
+the form
+``F_\beta \cong M^{\hat\theta}(Q(w), \beta)``, where
+``\hat\theta_{i, \xi} = \theta_i``, for a shift-equivalence class of compatible
+``\beta``. A class contributes only when this lifted stable moduli space is
+nonempty. This is the distinction between the candidates returned by
+[`compatible_dimension_vectors`](@ref) and the actual components returned by
+[`torus_fixed_components`](@ref); see
+[[Theorem 3.1, Boos--Franzen](https://doi.org/10.1112/blms.12649)] and
+[[Theorem 3.8, Weist](https://doi.org/10.1090/S1088-4165-2013-00436-3)].
+
+For a semistable moduli space, `torus_fixed_components` requires the stable and
+semistable loci to agree. Constructing `M` with `condition="stable"` explicitly
+requests the fixed locus of the stable locus. Candidate enumeration is
+combinatorial and can grow quickly with ``d`` and ``|Q_1|``; it is intended for
+small and medium dimension vectors.
+
+At a point of a stable fixed component, the dimension of the ``\chi``-weight
+space of the tangent space is given by
+[[Theorem 6.1, Boos--Franzen](https://doi.org/10.1112/blms.12649)]:
+
+```math
+\dim (T_{[M]} \mathcal M)_\chi
+  = \delta_{\chi, 0} - \langle \beta, s_{-\chi}\beta\rangle_{Q(w)}.
+```
+
+Accordingly, [`weight_space_dimension`](@ref) and
+[`tangent_weight_multiplicities`](@ref) accept the ambient moduli space and
+verify that ``\beta`` has a nonempty stable lift. The latter includes the zero
+character when the fixed component itself has positive dimension.
+
+## Typical workflow
+
+First enumerate candidates, then filter them using stability:
+
+```julia
+Q = kronecker_quiver(3)
+M = QuiverModuliSpace(Q, [2, 3])
+
+candidates = compatible_dimension_vectors(Q, M.d)
+components = torus_fixed_components(M)
+
+length(candidates)  # 55 connected-support shift classes
+length(components)  # 13 nonempty stable fixed components
+
+component = first(components)
+tangent_weight_multiplicities(M, component.beta)
+```
+
+## Data structure
+
+```@docs
+CoveringDimVector
+```
+
+## Operations on covering dimension vectors
+
+```@docs
+shift_beta
+covering_euler_form
+extract_finite_subquiver
+compatible_dimension_vectors
+```
+
+## Fixed components
+
+```@docs
+torus_fixed_components
+```
+
+## Tangent weight spaces at fixed points
+
+```@docs
+weight_space_dimension
+tangent_weight_multiplicities
+```
+
+## References
+
+- M. Boos and H. Franzen, *Weight spaces and attracting sets for torus actions
+  on quiver moduli*, Bulletin of the London Mathematical Society **54** (2022),
+  1658--1682. [doi:10.1112/blms.12649](https://doi.org/10.1112/blms.12649),
+  [arXiv:2002.12049](https://arxiv.org/abs/2002.12049).
+- A. D. King, *Moduli of representations of finite-dimensional algebras*,
+  Quarterly Journal of Mathematics **45** (1994), 515--530.
+  [doi:10.1093/qmath/45.4.515](https://doi.org/10.1093/qmath/45.4.515).
+- T. Weist, *Localization in quiver moduli spaces*, Representation Theory
+  **17** (2013), 382--425.
+  [doi:10.1090/S1088-4165-2013-00436-3](https://doi.org/10.1090/S1088-4165-2013-00436-3),
+  [arXiv:0903.5442](https://arxiv.org/abs/0903.5442).

docs/src/tutorial.md

@@ -61,8 +61,10 @@ julia> is_coprime(d, θ)
 true
 ```
 
-Here, `is_coprime()` checks if ``d`` is θ-coprime, i.e., if none of the
-proper subdimension vectors ``0 \neq d' \nleq d`` satisfies ``\theta \cdot d' = 0``.
+Here, `is_coprime()` checks if ``d`` is θ-coprime, i.e., if no proper nonzero
+subdimension vector has the same slope as ``d``. When ``\theta \cdot d=0``,
+this is equivalent to requiring ``\theta \cdot d' \neq 0`` for every proper
+nonzero subdimension vector ``d'``.
 
 The bilinear Euler form relative to a quiver Q of any two vectors
 in ``\mathbb{Z}^{Q_0}`` can be computed: