Given Schubert varieties \(X, X', \ldots, X''\) of a flag variety the authors are concerned with the following question: when is the intersection of the translates \( gX \cap g'X' \cap \ldots \cap g'' X''\) non-empty? It is well known that for the Grassmannian it is non-empty if and only if the indices of the Schubert varieties satisfy the linear Horn inequalities. The authors answer to this question is based upon the work of A. A. Klyachko [Sel. Math. 4, 419–445 (1998; Zbl 0915.14010)], A. Knutson and T. Tao [J. Am. Math. Soc. 12, No. 4, 1055–1090 (1999; Zbl 0944.05097)] and A. Horn [Pac. J. Math. 225–241 (1962; Zbl 0112.01501)].
The authors’ main theorem, theorem 4, states that for a collection \(M(P)\) of parabolic subgroups \(Q \subset L\) , assuming that \(G/P\) is a cominuscule flag variety, such an intersection is non-empty if and only if for every \(Q \in M(P) \) and every list of Schubert varieties \(X, X', \ldots, X''\) of \( L/Q\) whose general translates have non-empty intersection a set of necessary inequalities here numbered as (8) which depend on the combinatorial condition on the tangent space to \(G/R\), the corresponding lie algebra to \(R\) and the natural embedding of the center of the nilradical of R must be satisfied.
The paper is organized as follows. Section one gives the basic definitions and notations used throughout the paper, starting with section 1.1 about linear algebraic groups and their flag varieties. Section 1.2 deals with Schubert varieties and their tangent spaces. Section 1.3 is about transversality and section 1.4 is about cominuscule flag varieties; stating four equivalent characterizations of cominuscule flag varieties. In subsection 2.1, the main theorem, here theorem 4 is precisely stated and derives a very general inequality here numbered (7) in theorem 2 of this section and an inequality derived from (7) here numbered as (8) used to formulate theorem 4. The set of inequalities which determine the non-emptiness of the Schubert varieties are recursive in the sense that they come from similar non-empty intersections on smaller cominuscule varieties namely those given by \(L/Q\) where \( Q \in M(P)\) where the latter is the set of standard parabolic subgroups of \(L\) which are equal to the stabilizer of the tangent space to some \(L\)-orbit on the lie algebra of \(G/P\). Section 3 is devoted to the proof of theorem 4 relying upon technical results about root systems which are given in appendix A of this paper.
In section 4 the authors examine the cominuscule relation in more detail, describing it on a case-by-case basis. It is interesting to note that the inequalities given by the inequality (8) might be redundant and do not in fact give a set of irredundant inequalities as the authors show in the examples of this section. As for example, in section 4.3 the case for type \(B_n\), \(G = SO(2n+1)\), \(G/P\) an odd-dimensional quadric they show that there is only one inequality needed namely the basic codimension inequality. In section 5, the authors discuss how the classical Horn inequalities arise form the inequalities from theorem 2 and show how to modify the proof of theorem 4 to prove their sufficiency in proposition 24 and 25.
Proposition 24 in fact recalls the Horn recursion formula obtained already by W. Fulton [Bull. Am. Math. Soc. 37, No. 3, 209–249 (2000; Zbl 0994.15021)]. The authors show that both proposition 24 and 25 give exactly the same recursion formulas and in theorem 26 the authors obtain a different set of necessary inequalities for feasibility on \(G/P\) which they call naive inequalities using proposition 11. The naive inequalities were already given by P. Belkale and S. Kumar [Invent. Math. 166, No. 1, 185–228 (2006; Zbl 1106.14037)]. Finally in section 5.3, the authors obtain a set of necessary inequalities for the Lagrangian Grassmannian which are derived from the naive inequalities of theorem 26. The obtained inequalities are different from those of corollary 5 to main theorem 4 of this paper.