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D. A. Shmelkin Independent University of Moscow Moscow, Russia mitia@mccme.ru

Geometric metho ds for semi-invariants of quivers
For a quiver Q with n vertices and a dimension vector Zn the set R(Q, ) of represen+ tations of Q with dimension is interesting for both Representation and Invariant Theories. From the point of view of Invariant Theory, R(Q, ) is a module over the reductive group GL() and the natural questions about the action GL() : R(Q, ) are the orbits, the orbit closures, the invariants etc. For example, the invariants are in general closely related with the closed orbits, and the orbit of a representation V is closed if and only if V is semi-simple and in this case the slice ґ etale at V is described in terms of a local quiver V , which is defined from the indecomposable summands of V (see [LBP]). However, in many ineresting cases all regular invariants are constant and the only semisimple representation is the zero point of R(Q, ). In all cases we have a more general ob ject, the regular semi-invariants of GL() or the invariants of S L(), the commutator subgroup in (GL()) GL(). The module k[R(Q, )] of semi-invariants of weight has a nice description in terms of Schofield's determinantal semi-invariants and perpendicular categories W and W (see [Sch]). Using these ideas, we suggested a geometical approach to semi-invariants based on: Theorem 1. ([Sh1]) Let V = m1 S1 + · · · + mt St R(Q, ) be a decomposition into indecomposable summands. The fol lowing properties of V are equivalent: (i) the S L()-orbit of V is closed in R(Q, ) (GL()) (ii) the GL()-orbit of V is closed in R(Q, )f , f k[R(Q, )] (iii) S1 , · · · , St are simple objects in W for a representation W . We call the representations fulfilling the above equivalent conditions local ly semi-simple. The condition (ii) of the Theorem allows to apply Luna ґ etale slice Theorem [Lu], and the slice at V is described by the local quiver V exactly in the same style as in semi-simple case. Luna slice ґ etale Theorem gives rise to a stratification of the quotient R(Q, )/ L() /S GL() by the locally-closed strata (R(Q, )/ L())(H ) , such that the stabilizers in GL() of the /S points in the closed orbit in the fiber are conjugate to H GL(). The locally semi-simple representations over a stratum have the same dimensions of indecomposable summands, so the strata are in 1-to-1 correspondance with the local ly semi-simple decompositions of and, similarly to the usual Luna stratification, there is a unique generic locally semi-simple decomposition. This generic stabilizer H gives rise to a version of the Luna-Richardson Theorem relating the semi-invariants of an affine action G : X with those for NG (H ) acting on X H , and this was applied in [Sh1] to the semi-invariants of tame quivers. So it is useful to calculate locally semi-simple decompositions, in particular, generic. Derksen and Weyman introduced in [DW] the notion of quiver Schur sequence, a sequence = (1 , · · · , k ) of Schur roots with special properties. Theorem 2. [Sh2] A decomposition with the summands from a quiver Schur sequence is local ly semi-simple. When the dimension vector is prehomogeneous, i.e., GL() has a dense orbit in R(Q, ), then the converse to the last Theorem is true and moreover Theorem 3. [Sh2] If is prehomogeneous, then there is an order preserving isomorphism GL() of the strata (R(Q, )/ L())(H ) with the faces of a simplicial cone. /S In [Sh2] we presented an algorithm for calculating the generic locally semi-simple decomposition for a quiver without oriented cycles and arbitrary dimension vector. This algorithm (and some other related ones) was implemented in the computer program TETIVA (see [Te]). 1


References [DW] H. Derksen and J. Weyman, The combinatorics of quiver representations, preprint arXiv:math.RT/0608288. [LBP] L. Le Bruyn and C. Procesi, Semisimple representations of quivers, Transactions of AMS, 317 (1990), 2, 585-598. [Lu] D. Luna, Slices ґ etales, Bull. Soc. Math. France 33 (1973), 81-105. [Sch] A. Schofield, Semi-invariants of quivers, J. London Math. Soc. 43 (1991), 383-395. [Sh1] D. A. Shmelkin, Locally semi-simple representations of quivers, Transf. Groups 12 (2007), 153-183. [Sh2] D. A. Shmelkin, Some algorithms for semi-invariants of quivers, arXiv:0708.4189v1 [Te] D. A. Shmelkin, TETIVA a computer program available at http://www.mccme.ru/~mitia

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