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

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

2010-09-17

arxiv.org/abs/1009.3328v2

Mathematics

Representation Theory

27 pages. Fixed typos, changes/corrections to Section 6, few paragraphs about tame concealed algebras added

Scientific paper

We show that a finite connected quiver Q with no oriented cycles is tame if and only if for each dimension vector $\mathbf{d}$ and each integral weight $\theta$ of Q, the moduli space $\mathcal{M}(Q,\mathbf{d})^{ss}_{\theta}$ of $\theta$-semi-stable $\mathbf{d}$-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 $\mathbf{d}$ of Q, the field of rational invariants $k(rep(Q,\mathbf{d}))^{GL(\mathbf{d})}$ is isomorphic to $k$ or $k(t)$. Next, we extend this latter description to canonical algebras. More precisely, we show that a canonical algebra $\Lambda$ is tame if and only if for each generic root $\mathbf{d}$ of $\Lambda$ and each indecomposable irreducible component C of $rep(\Lambda,\mathbf{d})$, the field of rational invariants $k(C)^{GL(\mathbf{d})}$ 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.

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