Locally semi-simple representations of quivers

Abstract

We suggest a geometrical approach to the semi-invariants of quivers based on Luna's slice theorem and the Luna-Richardson theorem. The locally semi-simple representations are defined in this spirit but turn out to be connected with stable representations in the sense of GIT, Schofield's perpendicular categories, and Ringel's regular representations. As an application of this method we obtain an independent short proof of a theorem of Skowronski and Weyman about semi-invariants of the tame quivers.

Some algorithms for semi-invariants of quivers

“…Hence, such a decomposition (we call them locally semi-simple) yields a conjugate class (H) of subgroups in GL(α) and a stratum (R(Q , α)//SL(α)) (H) consisting of all ξ ∈ R(Q , α)//SL(α) such that the automorphism group of the locally semi-simple representations in π −1 SL(α) (ξ ) belong to (H). In [11] we proved that this is a stratification of R(Q , α)//SL(α) by locally closed smooth subsets, and moreover, this is a refinement of Luna's stratification of R(Q , α)//SL(α) in the sense of [7]. So we call the set of the locally semisimple decompositions of α the GL(α)-stratification of R(Q , α)//SL(α).…”

“…By [11,Theorem 19] the multiplicities of imaginary roots in generic locally semi-simple decomposition are equal to 1. Moreover, if q Q (β) < 0, then by [10,Theorem 3.7] nβ is again a Schur root, so the generic locally semi-simple decomposition is almost loopless.…”

“…The idea of the latter goes back to [6], where it was applied for semi-simple representations and we used it in [11] for locally semi-simple ones, too. Moreover, the definition works for any representation V = m 1 S 1 + · · · + m t S t satisfying condition (4) and we define Σ V to be the quiver with t vertices corresponding to the summands S 1 , .…”

“…So it is possible and even more correct to denote the quiver Σ α . This local quiver plays a crucial role in [6] and [11] because for locally semi-simple V the slice representation at V is (by formula (9) in [11]):…”

“…In [11] we introduced a class of representations helping to study the semi-invariants of quiver from the geometrical point of view: [11].) Let V = m 1 S 1 + · · · + m t S t ∈ R(Q , α) be a decomposition into indecomposable summands.…”