The extended Dynkin quiver E₇~ has infinite representation type #
The quiver TauCeti.Quiver.AffineE7, whose underlying graph is the extended Dynkin diagram E₇~,
has infinitely many finite-dimensional indecomposable representations over every field
(TauCeti.not_isFiniteRepType_affineE7).
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.affineE7LoopRep turns it into the representation of E₇~ with
V⁴ at the centre, V³, V² and V along each long arm, and V² at the short vertex. All its
arrows are injective, so it is a configuration in V⁴ of two flags and one subspace:
- the flag
V × 0 × 0 × 0 ⊆ V × V × 0 × 0 ⊆ V × V × V × 0along the first long arm; - the flag
0 × 0 × 0 × V ⊆ 0 × 0 × V × V ⊆ 0 × V × V × Valong the second long arm; - the subspace
{(a, b, a + b, a + f b)}at the short vertex.
This construction is fully faithful (TauCeti.fullyFaithfulAffineE7LoopFunctor). A morphism
between two such configurations preserves both flags, so its centre component preserves the four
coordinate axes and acts on them separately; preserving the subspace at the short vertex forces
the four coordinate actions to agree, and 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 E₇~. 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.affineE7LoopRep: the representation ofE₇~attached to a representation of•↺.TauCeti.affineE7LoopFunctor: the same construction, as a functor.
Main results #
TauCeti.fullyFaithfulAffineE7LoopFunctor: the functor is fully faithful.TauCeti.IsFiniteRepType.oneLoop_of_affineE7: finite representation type ofE₇~would force it on•↺.TauCeti.not_isFiniteRepType_affineE7:E₇~has infinite representation type over every field.
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 E₇~. For a representation
M of •↺, with vertex space V on which the loop acts by f, this is the representation of
TauCeti.Quiver.AffineE7 with V⁴ at the centre, V³, V² and V at the inner, middle and
outer vertex of each long arm, and V² at the short vertex. Along the first long arm the arrows
are x ↦ (x, 0), (a, b) ↦ (a, b, 0) and (a, b, c) ↦ (a, b, c, 0); along the second they are
x ↦ (0, x), (c, d) ↦ (0, c, d) and (b, c, d) ↦ (0, b, c, d); the short arrow is
(a, b) ↦ (a, b, a + b, a + f b).
Instances For
The centre of TauCeti.affineE7LoopRep M carries the fourth power of the vertex space of
M.
The short vertex of TauCeti.affineE7LoopRep M carries the square of the vertex space of
M.
Each inner vertex of TauCeti.affineE7LoopRep M carries the cube of the vertex space of
M.
Each middle vertex of TauCeti.affineE7LoopRep M carries the square of the vertex space of
M.
Each outer vertex of TauCeti.affineE7LoopRep M carries the vertex space of M.
The outer arrow of the first long arm sends x to (x, 0).
The outer arrow of the second long arm sends x to (0, x).
The middle arrow of the first long arm sends (a, b) to (a, b, 0).
The middle arrow of the second long arm sends (c, d) to (0, c, d).
The inner arrow of the first long arm sends (a, b, c) to (a, b, c, 0).
The inner arrow of the second long arm sends (b, c, d) to (0, b, c, d).
The short arrow sends (a, b) to (a, b, a + b, a + f b), for the endomorphism f by which
the loop acts on M.
A finite-dimensional representation of •↺ gives a finite-dimensional representation of
E₇~: the vertex spaces of TauCeti.affineE7LoopRep M are powers of the vertex space of M.
The functor #
A vector space with an endomorphism, as a representation of E₇~, as a functor: a
morphism with component ψ acts by ψ at the outer vertices, by ψ × ψ at the middle and short
vertices, by ψ × ψ × ψ at the inner vertices and by ψ × ψ × ψ × ψ at the centre.
Instances For
The functor TauCeti.affineE7LoopFunctor sends a representation to its E₇~ representation.
At the centre, TauCeti.affineE7LoopFunctor maps a morphism by the fourfold product of its
component at the loop vertex, transported to the functor's vertex spaces.
At the short vertex, TauCeti.affineE7LoopFunctor maps a morphism by the twofold product of
its component at the loop vertex, transported to the functor's vertex spaces.
At each inner vertex, TauCeti.affineE7LoopFunctor maps a morphism by the threefold product
of its component at the loop vertex, transported to the functor's vertex spaces.
At each middle vertex, TauCeti.affineE7LoopFunctor maps a morphism by the twofold product
of its component at the loop vertex, transported to the functor's vertex spaces.
At each outer vertex, TauCeti.affineE7LoopFunctor maps a morphism by its component at the
loop vertex, transported to the functor's vertex spaces.
Full faithfulness #
The seven naturality squares of g, read on elements. Each is
CategoryTheory.NatTrans.naturality_apply along one arrow, whose action on elements is given by
the corresponding TauCeti.affineE7LoopRep_map_*_apply lemma.
The functor TauCeti.affineE7LoopFunctor is fully faithful. A morphism between images is
determined by its component ψ at the first outer vertex: by naturality along the seven arrows,
every other component is a power of ψ, and ψ commutes with the loop.
Instances For
The functor TauCeti.affineE7LoopFunctor 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 E₇~ would force it on the loop quiver: the functor
TauCeti.affineE7LoopFunctor carries the finite-dimensional indecomposables of •↺ to
finite-dimensional indecomposables of TauCeti.Quiver.AffineE7 and reflects isomorphism among
them.
The extended Dynkin quiver E₇~ 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.