Lecture notes for Introduction to Representation Theory

The first thing we can do is - as usual - split away the kernels of the maps A 1 , A 2 , A 3 . More
precisely, we split away the representations
0
0 0 0 0
• • • • • • • • •
ker A 1 0 0 0 0 ker A 3
0
•
•
•
0 ker A 2 0
These representations are multiples of the indecomposable objects
0 0 0 0 0
• •
• • • • • •
1 0 0
0 0
0
•
•
•
0 1 0
So we get to a situation where all of the maps A 1 , A 2 , A 3 are injective.
•
1
to reach a situation where
A
•
1 V A
•
3
•
V 1 V 3
A 2
•
V 2
As in 2, we can then identify the spaces V 1 , V 2 , V 3 with subspaces of V . So we get to the triple of
subspaces problem of classifying a triple of subspaces of a given space V .
The next step is to split away a multiple of
1
• • •
0 0
•
0
V 1 + V 2 + V 3 = V.
By letting Y = V 1 ∈ V 2 ∈ V 3 , choosing a complement V of Y in V , and setting V i
= V ∈ V i ,
i = 1, 2, 3, we can decompose this representation into
•
V1
V
•
Y
• • •
V3 Y
•
•
V2
Y
The last summand is a multiple of the indecomposable representation
1
• •
•
1 1
•
1
•
Y
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