Xueyu Luo

is isomorphic to the following R-order:
⎡
R R R R (x −1 ⎤
) R R
(x) R R R R R R
P(Q, W, F ) ∼ (x 2 ) (x) R R R (x) (x)
=
(x 2 ) (x 2 ) (x) R R (x) (x)
⎢(x 2 ) (x 2 ) (x 2 ) (x) R (x 2 ) (x)
⎥
⎣
(x) (x) R R R R R
⎦
(x) (x) (x) R R (x) R
It is clear that e F P(Q, W, F )e F
∼ = Λ5 holds in this case. The Auslander-Reiten quiver of
Λ 5 is the following:
⎡ ⎤
R
(x)
⎢(x 2 )
⎥
⎣(x 2 ) ⎦
(x 2 )
⎡ ⎤
R
R
⎢ R
⎥
⎣(x)
⎦
(x)
⎡ ⎤
R
(x)
⎢ (x)
⎥
⎣(x 2 ) ⎦
(x 2 )
⎡ ⎤
R
R
⎢ R
⎥
⎣ R ⎦
(x)
⎡ ⎤
R
R
⎢ (x)
⎥
⎣(x 2 ) ⎦
(x 2 )
⎡ ⎤
R
(x)
⎢ (x)
⎥
⎣ (x) ⎦
(x 2 )
⎡ ⎤
R
R
⎢ (x)
⎥
⎣ (x) ⎦
(x 2 )
⎡ ⎤
R
(x)
⎢(x)
⎥
⎣(x)
⎦
(x)
⎡
⎢
⎣
R
R
R
(x)
(x 2 )
⎤
⎥
⎦
⎡ ⎤
R
R
⎢(x)
⎥
⎣(x)
⎦
(x)
⎡ ⎤
R
R
⎢ R
⎥
⎣(x)
⎦
(x)
⎡ ⎤
R
(x)
⎢(x 2 )
⎥
⎣(x 2 ) ⎦
(x 2 )
As a Λ 5 -module, e F P(Q, W, F ) is isomorphic to
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤
R
(x)
(x 2 )
(x 2 )
(x 2 )
⎢
⎣
⎥ ⎢
⎦ ⊕ ⎣
R
R
(x)
(x 2 )
(x 2 )
⎥ ⎢
⎦ ⊕ ⎣
R
R
R
(x)
(x 2 )
⎥ ⎢
⎦ ⊕ ⎣
R
R
R
R
(x)
⎥
⎦ ⊕
⎡
R
(x)
(x)
(x)
(x)
⎢
⎣
⎤
⎡
⎥ ⎢
⎦ ⊕ ⎣
⎤
R
R
(x)
(x)
(x)
⎡
⎥ ⎢
⎦ ⊕ ⎣
which is cluster tilting in CM(Λ 5 ). As expected, its summands correspond bijectively to
the (internal and external) edges of the triangulation, or equivalently to the vertices of
the corresponding quiver.
–97–
R
R
(x)
(x)
(x 2 )
⎤
⎥
⎦