The four subspace quiver has infinite representation type

The subspace quiver with four outer vertices, whose underlying graph is the extended Dynkin diagram D4~, has infinitely many finite-dimensional indecomposable representations over every field. The family exhibited here is the nilpotent Jordan block in the guise of four subspaces: writing A = k[X]/(Xⁿ⁺¹), the representation TauCeti.subspaceJordanRep k n places A × A at the centre and A at each outer vertex, and the four arrows embed A in A × A as

x ↦ (x, 0), x ↦ (0, x), x ↦ (x, x), x ↦ (x, X x),

that is as the two coordinate axes, the diagonal, and the graph of multiplication by X.

An endomorphism is forced, by naturality along the first two arrows, to act on the centre coordinatewise through its components at the first two outer vertices; naturality along the diagonal equates those two components, and naturality along the graph of X equates the fourth with them and makes them commute with multiplication by X. So an endomorphism is multiplication by a single element of A, its value at 1, and the local ring A has no idempotent but 0 and 1: each representation is indecomposable. Their dimension vectors differ, so they are pairwise non-isomorphic, and TauCeti.not_isFiniteRepType_subspace_fin_four follows over every field, with no algebraic closedness and no infinitude of k.

The scalar family of the four subspace problem, the cross-ratio lines through (1, c) in k² for c ∉ {0, 1}, would only settle the case of an infinite field.

Main declarations

• TauCeti.subspaceJordanRep: the Jordan block k[X]/(Xⁿ⁺¹) as a configuration of four subspaces of (k[X]/(Xⁿ⁺¹))².

Main results

• TauCeti.indecomposable_subspaceJordanRep: these representations are indecomposable.
• TauCeti.nonempty_subspaceJordanRep_iso_iff: two of them are isomorphic exactly when their sizes agree.
• TauCeti.not_isFiniteRepType_subspace_fin_four: the four subspace quiver has infinite representation type over every field.

Implementation notes

TauCeti.subspaceJordanRep carries @[expose] for the reason recorded on TauCeti.oneLoopNilpotentRep: a functor built by CategoryTheory.Paths.lift reveals its vertex spaces only through its definition, and the statements below need them to reduce.

References

• I. M. Gelfand, V. A. Ponomarev, Problems of linear algebra and classification of quadruples of subspaces in a finite-dimensional vector space, Colloq. Math. Soc. János Bolyai 5 (1972), 163--237.
• I. Assem, D. Simson, A. Skowroński, Elements of the Representation Theory of Associative Algebras, Volume I, Chapter VII.

noncomputable def TauCeti.subspaceJordanRep (k : Type u) [Field k] (n : ℕ) : QuiverRep k (Quiver.Subspace (Fin 4))

The Jordan block of size n + 1 as four subspaces: with A = k[X]/(Xⁿ⁺¹), the representation of the four subspace quiver with A × A at the centre, A at each outer vertex, and the four arrows acting by x ↦ (x, 0), x ↦ (0, x), x ↦ (x, x) and x ↦ (x, X x).

@[simp]

theorem TauCeti.subspaceJordanRep_obj_center {k : Type u} [Field k] {n : ℕ} : (subspaceJordanRep k n).obj Quiver.Subspace.center = ↧(AdjoinRoot (Polynomial.X ^ (n + 1)) × AdjoinRoot (Polynomial.X ^ (n + 1)))

The centre of TauCeti.subspaceJordanRep carries (k[X]/(Xⁿ⁺¹))².

@[simp]

theorem TauCeti.subspaceJordanRep_obj_outer {k : Type u} [Field k] {n : ℕ} (i : Fin 4) : (subspaceJordanRep k n).obj (Quiver.Subspace.outer i) = ↧(AdjoinRoot (Polynomial.X ^ (n + 1)))

Each outer vertex of TauCeti.subspaceJordanRep carries k[X]/(Xⁿ⁺¹).

@[simp]

theorem TauCeti.subspaceJordanRep_map_arrow_zero_apply {k : Type u} [Field k] {n : ℕ} (x : AdjoinRoot (Polynomial.X ^ (n + 1))) : (ModuleCat.Hom.hom ((subspaceJordanRep k n).map (Quiver.Subspace.arrow 0).toPath)) x = (x, 0)

The first arrow embeds k[X]/(Xⁿ⁺¹) as the first coordinate axis.

@[simp]

theorem TauCeti.subspaceJordanRep_map_arrow_one_apply {k : Type u} [Field k] {n : ℕ} (x : AdjoinRoot (Polynomial.X ^ (n + 1))) : (ModuleCat.Hom.hom ((subspaceJordanRep k n).map (Quiver.Subspace.arrow 1).toPath)) x = (0, x)

The second arrow embeds k[X]/(Xⁿ⁺¹) as the second coordinate axis.

@[simp]

theorem TauCeti.subspaceJordanRep_map_arrow_two_apply {k : Type u} [Field k] {n : ℕ} (x : AdjoinRoot (Polynomial.X ^ (n + 1))) : (ModuleCat.Hom.hom ((subspaceJordanRep k n).map (Quiver.Subspace.arrow 2).toPath)) x = (x, x)

The third arrow embeds k[X]/(Xⁿ⁺¹) as the diagonal.

@[simp]

theorem TauCeti.subspaceJordanRep_map_arrow_three_apply {k : Type u} [Field k] {n : ℕ} (x : AdjoinRoot (Polynomial.X ^ (n + 1))) : (ModuleCat.Hom.hom ((subspaceJordanRep k n).map (Quiver.Subspace.arrow 3).toPath)) x = (x, AdjoinRoot.root (Polynomial.X ^ (n + 1)) * x)

The fourth arrow embeds k[X]/(Xⁿ⁺¹) as the graph of multiplication by X.

theorem TauCeti.isFinDim_subspaceJordanRep {k : Type u} [Field k] {n : ℕ} : IsFinDim k (Quiver.Subspace (Fin 4)) (subspaceJordanRep k n)

TauCeti.subspaceJordanRep k n is finite-dimensional: its vertex spaces are k[X]/(Xⁿ⁺¹) and its square.

theorem TauCeti.dimVector_subspaceJordanRep_center {k : Type u} [Field k] {n : ℕ} : dimVector (subspaceJordanRep k n) Quiver.Subspace.center = 2 * (n + 1)

The dimension vector of TauCeti.subspaceJordanRep k n is 2 (n + 1) at the centre.

theorem TauCeti.dimVector_subspaceJordanRep_outer {k : Type u} [Field k] {n : ℕ} (i : Fin 4) : dimVector (subspaceJordanRep k n) (Quiver.Subspace.outer i) = n + 1

The dimension vector of TauCeti.subspaceJordanRep k n is n + 1 at each outer vertex.

Endomorphisms

theorem TauCeti.indecomposable_subspaceJordanRep {k : Type u} [Field k] {n : ℕ} : CategoryTheory.Indecomposable (subspaceJordanRep k n)

TauCeti.subspaceJordanRep k n is indecomposable: it is not a direct sum of two nonzero subrepresentations. Its endomorphism ring is isomorphic to the local ring k[X]/(Xⁿ⁺¹), so its only idempotent endomorphisms are 0 and 1.

theorem TauCeti.eq_of_nonempty_subspaceJordanRep_iso {k : Type u} [Field k] {m n : ℕ} (h : Nonempty (subspaceJordanRep k m ≅ subspaceJordanRep k n)) : m = n

Jordan blocks of different sizes give non-isomorphic configurations: their dimension vectors differ.

@[simp]

theorem TauCeti.nonempty_subspaceJordanRep_iso_iff {k : Type u} [Field k] {m n : ℕ} : Nonempty (subspaceJordanRep k m ≅ subspaceJordanRep k n) ↔ m = n

Two of the representations TauCeti.subspaceJordanRep k n are isomorphic exactly when their sizes agree.

theorem TauCeti.not_isFiniteRepType_subspace_fin_four (k : Type u) [Field k] : ¬IsFiniteRepType k (Quiver.Subspace (Fin 4))

The four subspace quiver has infinite representation type over every field. Its underlying graph is the extended Dynkin diagram D4~. The representations TauCeti.subspaceJordanRep k n are finite-dimensional, indecomposable and pairwise non-isomorphic, so ℕ indexes an infinite family of them.