.
More generally, one could say that the Auslander-Reiten translate in
categorifies the twist automorphism
of
[
].
This allows us to compare the above mentioned cluster character
with the Caldero-Fu-Keller cluster character.
of the Weyl group 
the cluster algebra structure on the coordinate ring
[]
of the unipotent cell
is categorified by a subcategory
of the modules over the corresponding preprojective algebra.
Under the cluster character coming from Lusztig's construction
of the semicanonical basis, the initial seed consisting of certain
generalized minors corresponds to a canonical cluster tilting object
in
.
In order to solve in this context the factorization problem,
Berenstein, Fomin and Zelevinsky introduced twisted minors. We show that
these twisted minors correspond essentially to the inverse of
Auslander-Reiten translate of the summands of
.
More generally, one could say that the Auslander-Reiten translate in
categorifies the twist automorphism
of
[].
This allows us to compare the above mentioned cluster character
with the Caldero-Fu-Keller cluster character.