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 Incoming links: Three-subspace decomposition

Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 3 / 2 / c / Solution Created 2026-10-03 Updated 2026-10-06

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First split off the kernel of each source-to-sink arrow. Each kernel contributes copies of the simple representation supported at that source. The remaining arrows are injective, so identify the three source spaces with subspaces

U_{1}, U_{2}, U_{3}

of the sink space

V

. We now give an elementary three-subspace decomposition.

Split off the common intersection

U_{1} \cap U_{2} \cap U_{3}

: any projection onto it preserves all three subspaces. Next split off each pairwise intersection. For example, once the triple intersection has gone,

W = U_{1} \cap U_{2}

is disjoint from

U_{3}

; a projection onto

W

that kills

U_{3}

preserves

U_{1}, U_{2}, U_{3}

. This extracts blocks whose membership is exactly

{1, 2}

. Repeating for the other pairs leaves pairwise disjoint subspaces.

For each

i

, choose a complement

W_{i}

U_{i} \cap (U_{j} + U_{ℓ})

U_{i}

. A projection onto

W_{i}

killing

U_{j} + U_{ℓ}

preserves the triple, so these complements split off as blocks of membership exactly

{i}

. In the remaining triple, every

U_{i}

lies in the sum of the other two and all pairwise intersections are zero. Thus the sum is

U_{1} \oplus U_{2}

, and both projections of

U_{3}

onto

U_{1}, U_{2}

are isomorphisms. Hence

U_{3}

is the graph of an isomorphism

T : U_{1} \to U_{2}

. Choose a basis

u_{a}

U_{1}

and the basis

T u_{a}

U_{2}

; this graph decomposes into two-dimensional blocks with three distinct lines. A complement to

U_{1} + U_{2} + U_{3}

V

contributes sink-only simple blocks.

The complete list of indecomposable blocks is therefore:

Three source simples, with zero sink space.
Eight blocks with one-dimensional sink, one for each subset $S \subseteq {1, 2, 3}$ : the source space is $k$ for $i \in S$ and zero otherwise, and every nonzero arrow is the identity.
One exceptional block, with all three source spaces $k$ , sink $k^{2}$ , and arrows $1 \mapsto e_{1}$ , $1 \mapsto e_{2}$ , $1 \mapsto e_{1} + e_{2}$ .

Each of the first eleven blocks has endomorphism ring

k

. For the exceptional indecomposable of the three-subspace quiver, an endomorphism of

k^{2}

preserving the first two lines is diagonal; preserving the third makes its two diagonal entries equal. Its endomorphism ring is also

k

, so all twelve are indecomposable. Their dimension vectors of quiver representations distinguish them. The decomposition argument proves completeness, and hence there are exactly $12$ isomorphism classes. No general classification theorem is needed.

Figure 1.
Exceptional three-subspace representation with a two-dimensional sink and three distinct image lines
.

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Three-subspace quiver 2026-10-06

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The subspace quiver with three sources reduces to the problem of three subspaces in one vector space. The three-subspace decomposition gives eleven indecomposables with vertex dimensions at most one, and one exceptional indecomposable of the three-subspace quiver.

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