Geometric invariant theory and the generalized eigenvalue problem

GIT-cones and quivers

“…We also improve results of [16] on GIT-cones. In particular, Theorem 1 and 2 are improvements of [16,Theorem 4] and [16,Theorem 7], respectively.…”

“…Since L → μ L (C, λ) is a group morphism, it induces a linear map from Pic G (X ) Q to Q, also denoted by μ L (C, λ). By [16,Lemma 2], T C G (X ) is contained in the half-space μ L (C, λ) ≤ 0. In particular, intersecting T C G (X ) with the hyperplane μ L (C, λ) = 0, we obtain a face F (C) of T C G (X ).…”

“…This point of view was used in [1,2,13,16]. While the GIT and the quiver approaches to the Horn problem have traditionally been distinct, we hope to show that the GIT approach is more universal when studying (Q, β).…”

“…Moreover the current proof does not use the fact that a spanning set of the space of semi-invariant functions on Rep(Q, β) is known. Another difference is that in [16] the points outside the cone play a crucial role.…”

On the asymptotics of Kronecker coefficients

“…Ressayre proved in [23], in a more general setting, that the maximal codimension of a regular face is the rank of the group which in our case is GL(V ) × GL(W ). This exactly matches with the codimension of f T .…”