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

Abstract

Let $Q$ be a quiver without oriented cycles. For a dimension vector $\beta$ let $\operatorname {Rep}(Q, \beta )$ be the set of representations of $Q$ with dimension vector $\beta$. The group $\operatorname {GL}(Q, \beta )$ acts on $\operatorname {Rep}(Q, \beta )$. In this paper we show that the ring of semi-invariants $\operatorname {SI} (Q,\beta )$ is spanned by special semi-invariants $c^V$ associated to representations $V$ of $Q$. From this we show that the set of weights appearing in $\operatorname {SI}(Q,\beta )$ is saturated. In the case of triple flag quiver this reduces to the results of Knutson and Tao on the saturation of the set of triples of partitions for which the Littlewood-Richardson coefficient is nonzero.