The extended Dynkin quiver E₆~ has infinite representation type #
The quiver TauCeti.Quiver.AffineE6, whose underlying graph is the extended Dynkin diagram E₆~,
has infinitely many finite-dimensional indecomposable representations over every field
(TauCeti.not_isFiniteRepType_affineE6).
The proof reduces to the loop quiver •↺, which is already known to have infinite representation
type (TauCeti.not_isFiniteRepType_oneLoop). A representation of •↺ is a vector space V with
an endomorphism f, and TauCeti.affineE6LoopRep turns it into the representation of E₆~ with
V × V × V at the centre, V × V at each inner vertex and V at each outer vertex. All its
arrows are injective, so it is a configuration of three flags Lᵢ ⊆ Pᵢ of V × V × V:
L₀ = V × 0 × 0insideP₀ = V × V × 0;L₁ = 0 × 0 × VinsideP₁ = 0 × V × V;L₂ = {(x, x + f x, f x)}insideP₂ = {(a, a + c, c)}.
This construction is fully faithful (TauCeti.fullyFaithfulAffineE6LoopFunctor). A morphism
between two such configurations preserves the first two flags, so its centre component acts on
the three coordinates separately; preserving P₂ forces the three coordinate actions to agree,
and preserving L₂ forces the common one to commute with the endomorphisms. The construction
therefore preserves indecomposability and reflects isomorphism, and it preserves
finite-dimensionality, so it embeds the finite-dimensional indecomposables of •↺, up to
isomorphism, in those of E₆~. No algebraic closedness and no infinitude of the field is needed:
the nilpotent Jordan blocks already give infinitely many indecomposables of •↺ over every field.
Main definitions #
TauCeti.affineE6LoopRep: the representation ofE₆~attached to a representation of•↺.TauCeti.affineE6LoopFunctor: the same construction, as a functor.
Main results #
TauCeti.fullyFaithfulAffineE6LoopFunctor: the functor is fully faithful.TauCeti.IsFiniteRepType.oneLoop_of_affineE6: finite representation type ofE₆~would force it on•↺.TauCeti.not_isFiniteRepType_affineE6:E₆~has infinite representation type over every field.
Implementation notes #
TauCeti.affineE6LoopRep carries @[expose] for the reason recorded in
TauCeti.RepresentationTheory.Quiver.OneLoop.FiniteRepType: a functor built by
CategoryTheory.Paths.lift reveals its vertex spaces only through its definition, and the
components of the functor on morphisms are typed by them.
In the proof of fullness the components of a morphism are recorded as linear maps between
products of the vertex spaces, so that its six naturality squares, read on elements through
CategoryTheory.NatTrans.naturality_apply, are equations between pairs and triples which rw
and simp can take apart coordinatewise.
References #
- I. Assem, D. Simson, A. Skowroński, Elements of the Representation Theory of Associative Algebras, Volume I, Chapter VII.
- H. Derksen, J. Weyman, An Introduction to Quiver Representations, Chapter 4.
A vector space with an endomorphism, as a representation of E₆~. For a representation
M of •↺, with vertex space V on which the loop acts by f, this is the representation of
TauCeti.Quiver.AffineE6 with V × V × V at the centre, V × V at each inner vertex and V at
each outer vertex. The outer arrows are x ↦ (x, 0), x ↦ (0, x) and x ↦ (x, f x), and the
inner arrows are (a, b) ↦ (a, b, 0), (b, c) ↦ (0, b, c) and (a, c) ↦ (a, a + c, c).
Instances For
The centre of TauCeti.affineE6LoopRep M carries the cube of the vertex space of M.
Each inner vertex of TauCeti.affineE6LoopRep M carries the square of the vertex space of
M.
Each outer vertex of TauCeti.affineE6LoopRep M carries the vertex space of M.
The first outer arrow is the inclusion of the first coordinate axis.
The second outer arrow is the inclusion of the second coordinate axis.
The third outer arrow is the graph of the endomorphism by which the loop acts on M.
The first inner arrow is (a, b) ↦ (a, b, 0).
The second inner arrow is (b, c) ↦ (0, b, c).
The third inner arrow is (a, c) ↦ (a, a + c, c).
A finite-dimensional representation of •↺ gives a finite-dimensional representation of
E₆~: the vertex spaces of TauCeti.affineE6LoopRep M are powers of the vertex space of M.
The functor #
A vector space with an endomorphism, as a representation of E₆~, as a functor: a
morphism with component ψ acts by ψ at every outer vertex, by ψ × ψ at the inner vertices
and by ψ × ψ × ψ at the centre.
Instances For
The functor TauCeti.affineE6LoopFunctor sends a representation to its E₆~ representation.
At the centre, TauCeti.affineE6LoopFunctor maps a morphism by the triple product of its
component at the loop vertex, transported to the functor's vertex spaces.
At each inner vertex, TauCeti.affineE6LoopFunctor maps a morphism by the double product of
its component at the loop vertex, transported to the functor's vertex spaces.
At each outer vertex, TauCeti.affineE6LoopFunctor maps a morphism by its component at the
loop vertex, transported to the functor's vertex spaces.
Full faithfulness #
The six naturality squares of g, read on elements. Each is
CategoryTheory.NatTrans.naturality_apply along one arrow, whose action is the one recorded in
the docstring of TauCeti.affineE6LoopRep.
The functor TauCeti.affineE6LoopFunctor is fully faithful. A morphism between images is
determined by its component ψ at the first outer vertex: by naturality along the six arrows,
every other component is ψ, ψ × ψ or ψ × ψ × ψ, and ψ commutes with the loop.
Instances For
The functor TauCeti.affineE6LoopFunctor reflects and preserves isomorphism.
The image of an indecomposable representation of •↺ is indecomposable: the functor is
fully faithful, so it matches the idempotent endomorphisms of M with those of its image.
Infinite representation type #
Finite representation type of E₆~ would force it on the loop quiver: the functor
TauCeti.affineE6LoopFunctor carries the finite-dimensional indecomposables of •↺ to
finite-dimensional indecomposables of TauCeti.Quiver.AffineE6 and reflects isomorphism among
them.
The extended Dynkin quiver E₆~ has infinite representation type over every field. The
images of the nilpotent Jordan blocks (TauCeti.oneLoopNilpotentRep) are infinitely many pairwise
non-isomorphic finite-dimensional indecomposables.