Semi-invariants of quivers and saturation for Littlewood-Richardson coefficients.

*(English)*Zbl 0993.16011The problem of classifying finite dimensional modules over an associative \(K\)-algebra, where \(K\) is an algebraically closed field, leads naturally to the study of semi-invariants of representations of quivers. Let \(Q\) be a quiver (that is, a finite directed graph) and let \(\beta\) be a dimension vector (i.e., a non-negative integer valued function on the set of vertices of \(Q\)). The representations of \(Q\) with dimension vector \(\beta\) are parameterized by an affine \(K\)-space (whose points are collections of matrices of various sizes) endowed with a linear action (a kind of conjugation action) of \(\text{GL}(\beta)\), a product of general linear groups, such that the orbits are in bijective correspondence with the isomorphism classes of representations.

The authors prove that if \(Q\) has no oriented cycles, then \(\text{SI}(Q,\beta)\), the ring of semi-invariants (or in other words, the algebra of polynomial \(\text{SL}(\beta)\)-invariants) for this action is spanned by determinantal semi-invariants \(c^V\), assigned to representations \(V\) of \(Q\) by A. Schofield [J. Lond. Math. Soc., II. Ser. 43, No. 3, 385-395 (1991; Zbl 0779.16005)]. This result connects semi-invariants and modules in a direct way. Moreover, as an algebra, \(\text{SI}(Q,\beta)\) is generated by the \(c^V\) where \(V\) is indecomposable and the dimension vector of \(V\) is a Schur root. A reciprocity property relating weight spaces of semi-invariants for certain pairs of dimension vectors is also deduced.

It is shown that the semigroup of weights of non-zero homogeneous semi-invariants is defined by one homogeneous linear equation and a finite set of linear inequalities. Applied to a special quiver this yields a new proof for the saturation of Littlewood-Richardson coefficients [A. Knutson and T. Tao, J. Am. Math. Soc. 12, No. 4, 1055-1090 (1999; Zbl 0944.05097)]. Spanning sets of \(\text{SI}(Q,\beta)\) were also obtained independently with different approaches by A. Schofield and M. Van den Bergh [Indag. Math., New Ser. 12. No. 1, 125-138 (2001; Zbl 1004.16012)] (assuming that \(K\) is of characteristic zero) and by M. Domokos and A. N. Zubkov [Transform. Groups 6, No. 1, 9-24 (2001; Zbl 0984.16023)].

The authors prove that if \(Q\) has no oriented cycles, then \(\text{SI}(Q,\beta)\), the ring of semi-invariants (or in other words, the algebra of polynomial \(\text{SL}(\beta)\)-invariants) for this action is spanned by determinantal semi-invariants \(c^V\), assigned to representations \(V\) of \(Q\) by A. Schofield [J. Lond. Math. Soc., II. Ser. 43, No. 3, 385-395 (1991; Zbl 0779.16005)]. This result connects semi-invariants and modules in a direct way. Moreover, as an algebra, \(\text{SI}(Q,\beta)\) is generated by the \(c^V\) where \(V\) is indecomposable and the dimension vector of \(V\) is a Schur root. A reciprocity property relating weight spaces of semi-invariants for certain pairs of dimension vectors is also deduced.

It is shown that the semigroup of weights of non-zero homogeneous semi-invariants is defined by one homogeneous linear equation and a finite set of linear inequalities. Applied to a special quiver this yields a new proof for the saturation of Littlewood-Richardson coefficients [A. Knutson and T. Tao, J. Am. Math. Soc. 12, No. 4, 1055-1090 (1999; Zbl 0944.05097)]. Spanning sets of \(\text{SI}(Q,\beta)\) were also obtained independently with different approaches by A. Schofield and M. Van den Bergh [Indag. Math., New Ser. 12. No. 1, 125-138 (2001; Zbl 1004.16012)] (assuming that \(K\) is of characteristic zero) and by M. Domokos and A. N. Zubkov [Transform. Groups 6, No. 1, 9-24 (2001; Zbl 0984.16023)].

Reviewer: Matyas Domokos (Edinburgh)

##### MSC:

16G20 | Representations of quivers and partially ordered sets |

13A50 | Actions of groups on commutative rings; invariant theory |

14L24 | Geometric invariant theory |

15A72 | Vector and tensor algebra, theory of invariants |

20G05 | Representation theory for linear algebraic groups |

05E05 | Symmetric functions and generalizations |

14L30 | Group actions on varieties or schemes (quotients) |

16R30 | Trace rings and invariant theory (associative rings and algebras) |

##### Keywords:

representations of quivers; semi-invariants; Littlewood-Richardson rule; Klyachko cones; saturation; general linear groups; orbits; weight spaces; dimension vectors
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\textit{H. Derksen} and \textit{J. Weyman}, J. Am. Math. Soc. 13, No. 3, 467--479 (2000; Zbl 0993.16011)

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##### References:

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