Semi-invariants of Quivers and Saturation for Littlewood-richardson Coefficients

Σ(Q,α) is defined in the space of all weights by one homogeneous linear equation and by a finite set of homogeneous linear inequalities. In particular the set Σ(Q,α) is saturated, i.e., if nσ ∈ Σ(Q,α), then also σ ∈ Σ(Q,α). These results, when applied to a special quiver Q = Tn,n,n and to a special dimension vector, show that the GLn-module Vλ appears in Vμ ⊗ Vν if and only if the partitions λ, μ and ν satisfy an explicit set of inequalities. This gives new proofs of the results of Klyachko ([7, 3]) and Knutson and Tao ([8]). The proof is based on another general result about semi-invariants of quivers (Theorem 1). In the paper [10], Schofield defined a semi-invariant cW for each indecomposable representation W of Q. We show that the semi-invariants of this type span each weight space in SI(Q,α). This seems to be a fundamental fact, connecting semi-invariants and modules in a direct way. Given this fact, the results on sets of weights follow at once from the results in another paper of Schofield [11].