The SL(V)-ample cone of product of flag varieties

Let k be an algebraically closed field, and V be a vector space of dimension n over k. For a set . = (d. (1), ..., d. (m)) of sequences of positive integers, denote by L . the ample line bundle corresponding to the polarization on the product X = . mi=1 Flag(V, n. (i)) of flag varieties of type n. (i) determined by .. We study the SL(V)-linearization of the diagonal action of SL(V) on X with respect to L .. We give a sufficient and necessary condition on . such that Xss(L .) . /0 (resp., Xs(L .) . /0). As a consequence, we characterize the SL(V)-ample cone (for the diagonal action of SL(V) on X), which turns out to be a polyhedral convex cone.