Simple reflections on the dimension vectors of a quiver #
For a vertex i of a finite quiver Q, the simple reflection sᵢ is the reflection of the
dimension-vector lattice Q → ℤ that negates the simple dimension vector αᵢ = Pi.single i 1
and fixes the hyperplane orthogonal to it for the polarized Tits form. It is the numerical
shadow of the Bernstein-Gelfand-Ponomarev reflection functor at i, and the generator of the
Weyl group action under which the roots of the Tits form are stable.
The reflection is built from Mathlib's Module.preReflection, taking the linear form to be the
polarized Tits form paired with αᵢ. Following that naming, TauCeti.vertexPreReflection is
available with no hypothesis on i, while the reflection identities need q(αᵢ) = 1,
equivalently that i carries no loop; TauCeti.vertexReflection is the automorphism obtained
at such a loopless vertex. In an acyclic quiver every vertex is loopless, by
TauCeti.Quiver.IsAcyclic.isEmpty_hom_self.
The reflection of the quiver itself, which reverses the arrows at i, is
TauCeti.Quiver.Reflect in TauCeti.RepresentationTheory.Quiver.Reflection.Basic;
TauCeti.RepresentationTheory.Quiver.Reflection.EulerForm relates the two.
Main definitions and results #
TauCeti.vertexPreReflection: the mapsᵢ, as aℤ-linear endomorphism of the dimension-vector lattice, defined at every vertex.TauCeti.vertexReflection: the same map at a loopless vertex, where it is a genuine reflection, packaged as a linear automorphism.TauCeti.titsForm_vertexPreReflection: the simple reflection at a loopless vertex preserves the Tits form;TauCeti.titsPolarForm_vertexPreReflectionis the bilinear counterpart.TauCeti.bijOn_vertexPreReflection: consequently the simple reflection permutes each level set of the Tits form, in particular the rootsq(d) = 1.
References #
This implements the “simple reflection at a vertex” target of Layer 4 of
TauCetiRoadmap/RepresentationTheory/QuiverRepresentations/README.md, whose Layer 5 root-system
bridge consumes the symmetrized Gram matrix TauCeti.titsPolarForm_single_single. See
Derksen--Weyman, An Introduction to Quiver Representations.
The simple reflection sᵢ at a vertex i, acting on dimension vectors by
d ↦ d - ⟨αᵢ, d⟩ αᵢ for the polarized Tits form and the simple dimension vector
αᵢ = Pi.single i 1.
No hypothesis on i is imposed here, following Module.preReflection; the reflection identities
hold at a loopless vertex, where ⟨αᵢ, αᵢ⟩ = 2, and there vertexReflection packages this map
as an automorphism.
Equations
- TauCeti.vertexPreReflection Q i = Module.preReflection (Pi.single i 1) ((TauCeti.titsPolarForm Q) (Pi.single i 1))
Instances For
The defining formula for the simple reflection at a vertex.
Away from i, the simple reflection at i leaves a dimension vector unchanged.
The coordinate of the simple reflection at the reflected vertex.
At a loopless vertex the reflected coordinate is the classical formula
-dᵢ + ∑_{v ≠ i} (#(i ⟶ v) + #(v ⟶ i)) d_v, the sum running over the neighbours of i.
The simple reflection at a loopless vertex negates the corresponding simple dimension vector.
The simple reflection at i adds a multiple of αᵢ to any other simple dimension vector, the
multiple being the number of arrows joining the two vertices in either direction.
The simple reflection fixes every dimension vector orthogonal to αᵢ for the polarized Tits
form.
The simple reflection at a loopless vertex is an involution.
The simple reflection at a loopless vertex, as a linear automorphism of the dimension-vector lattice.
Equations
Instances For
The simple reflection at a loopless vertex is its own inverse.
Invariance of the Tits form #
The simple reflection at a loopless vertex preserves the Tits form.
The simple reflection at a loopless vertex preserves the polarized Tits form.
The simple reflection at a loopless vertex permutes every level set of the Tits form; taking
the level 1 this says that it permutes the roots of Q.