Geometric characterizations of the representation type of hereditary algebras and of canonical algebras

Abstract

We show that a finite connected quiver Q with no oriented cycles is tame if and only if for each dimension vector d and each integral weight θ of Q, the moduli space M(Q, d) ss θ of θ-semi-stable d-dimensional representations of Q is just a projective space. In order to prove this, we show that the tame quivers are precisely those whose weight spaces of semi-invariants satisfy a certain log-concavity property. Furthermore, we characterize the tame quivers as being those quivers Q with the property that for each Schur root d of Q, the field of rational invariants k(rep(Q, d)) GL(d) is isomorphic to k or k(t). Next, we extend this latter description to canonical algebras. More precisely, we show that a canonical algebra Λ is tame if and only if for each generic root d of Λ and each indecomposable irreducible component C of rep (Λ, d), the field of rational invariants k(C) GL(d) is isomorphic to k or k(t). Along the way, we establish a general reduction technique for studying fields of rational invariants on Schur irreducible components of representation varieties.

Semi-invariants for concealed-canonical algebras

“…Then [24, Proposition 2.3] (see also [9,Proposition 5.2]) implies that there exists a regular map Φ ′ :…”

On the invariant theory for acyclic gentle algebras

“…Moreover, if K/k is a field extension and m is a positive integer, we define S m (K/k) to be the field (Quot(K ⊗m )) Sm which is, in fact, the same as Quot((K ⊗m ) Sm ) since S m is a finite group. Now we are ready to prove the following reduction result (compare to [14,Proposition 4.7]):…”

On the equations and classification of toric quiver varieties

“…Thanks to [1] these results can be applied to decide smoothness of the GIT moduli spaces of quiver representations mentioned above. Furthermore, it was shown in [13] that a connected quiver is Dynkin or extended Dynkin if and only if all moduli spaces of its representations are smooth (see also [10], [5], [9] for related work). A key tool to develop results in this direction is to establish methods of simplifying the structure of the quiver without altering the property of being smooth.…”