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Given a finite dimensional hereditary algebra A and let $X_A$ denote the set of tilting $A$-modules.

Questions:

1.Is there a "canonical" bijection from $X_A$ to $X_A$ that sends $A$ to $D(A)$? Canonical here means that it should be defined in some fixed way for any hereditary algebra (or at least any representation-finite hereditary algebra).

  1. Are there other canonical tilting modules for hereditary algebras besides A and D(A) that can be written down in a nice way (for example without choosing orientation of the underlying quiver).
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I am going to give a negative answer for the first question, under a stronger notion of canonicity.

The approach I want to take is to consider the poset of tilting modules. They are ordered by inclusion of the corresponding torsion classes (all quotients of direct sums of copies of the tilting module). $A$ is the maximum element (the corresponding torsion class being the whole module category), and $DA$ is the minimum element (the corresponding torsion class consisting only of the injectives).

Now, my proposal is that the canonical map one would naturally want would be an anti-automorphism of this poset. However, this poset need not have any anti-automorphism.

This can be established as follows. The cover relations in this poset are given by mutations, in which one indecomposable summand of the tilting module is replaced by a different one.

Consider, for example, an $D_4$-type algebra oriented towards its central node. The top element of the poset admits only one mutation, but the bottom element admits three mutations. Thus, there is no poset anti-automorphism.

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