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C ∈ R eew eto few a mention we here ; r,adhsreappeared has and ury, tcslto oteprob- the to solution otic nthree on 3 Z r n ojcue that conjectured and , 0 6= ≥ 3 r 0 naihn of onvanishing aif i inequal- his satisfy obndwith Combined . r c × λ,µ ν r Hermitian 0 6= then , 9] λ The main goal of this paper is to provide further evidence of the naturality of the con- nection of Horn’s problem to Schubert calculus. We demonstrate how the connection persists for the following extension of this eigenvalue problem. Recall that a Hermitian matrix M majorizes another Hermitian matrix M ′ if M − M ′ is positive semidefinite (its eigenvalues are all nonnegative). In this case, we write M ≥ M ′. S. Friedland [Fr00] considered the following question: Which eigenvalues (λ,µ,ν) can occur if A + B ≥ C? His solution is in terms of linear inequalities, which includes Klyachko’s inequalities, a trace inequality and some additional inequalities. Later, W. Fulton [Fu00a] proved the additional inequalities are unnecessary. See followup work by A. Buch [Bu06] and by C. Chindris [Ch06] (who extends the work of H. Derksen-J. Weyman [DeWe00]). Our finding is that the solution to S. Friedland’s problem also governs the equivariant Schubert calculus of Grassmannians. This parallels the Horn problem’s connection to classical Schubert calculus, but separates the problem from GLn-representation theory. ν Let Cλ,µ be the equivariant Schubert structure coefficient (defined in Section 1.2). The analogy with the earlier results is illustrated by: ν N·ν N Theorem 1.1 (Equivariant saturation). Cλ,µ =06 if and only if CN·λ,N·µ =06 for any N ∈ .

ν ν When |λ| + |µ| = |ν| then Cλ,µ = cλ,µ. Hence Theorem 1.1 actually generalizes the saturation theorem. That said, our proofs rely on the classical Horn inequalities and so do not provide an independent proof of the earlier results. In addition, we use the recent ν 1 combinatorial rule for Cλ,µ developed by H. Thomas and the third author [ThYo12].

n 1.2. Equivariant cohomology of Grassmannians. Let Grr(C ) denote the Grassmannian of r-dimensional subspaces V ⊆ Cn. This space comes with an action of the torus T = (C∗)n (induced from the action of T on Cn). Therefore, it makes sense to discuss the ∗ Cn Z equivariant cohomology ring HT Grr( ). This ring is an algebra over [t1,...,tn]. (A more complete exposition of equivariant cohomology may be found in, e.g., [Fu07].) Z ∗ Cn As a [t1,...,tn]-module, HT Grr( ) has a basis of Schubert classes. To define these, fix the flag of subspaces n F• :0 ⊂ F1 ⊂ F2 ⊂···⊂ Fn = C , where Fi is the span of the standard basis vectors en, en−1, ...,en+1−i. For each Young diagram λ inside the r × (n − r) rectangle, which we denote by Λ, there is a corresponding Schubert variety, defined by Cn Xλ := {V ⊆ | dim(V ∩ Fn−r+i−λi ) ≥ i, for 1 ≤ i ≤ r}.

Since Xλ is invariant under the action of T , and has codimension 2|λ|, it determines a class 2|λ| Cn [Xλ] in HT Grr( ). As λ varies over all Young diagrams inside Λ, the classes [Xλ] form ∗ Cn Z ∗ Cn a basis for HT Grr( ) over [t1,...,tn]. Therefore in HT Grr( ) we have ν [Xλ] · [Xµ]= Cλ,µ[Xν], Xν⊆Λ

1 Cν CN·ν The easy direction of (equivariant) saturation, λ,µ 6= 0 ⇒ N·λ,N·µ 6= 0, can be proved directly by using this rule (or others). However, as in the classical situation, it is the converse that is nonobvious.

2 ν Z where the coefficients Cλ,µ ∈ [t1,...,tn] are the equivariant Schubert structure coeffi- ν cients. By homogeneity, Cλ,µ is a polynomial of degree |λ| + |µ|−|ν|. In particular, this coefficient is zero unless |λ| + |µ|≥|ν|. ν The polynomials Cλ,µ depend on the parameters r and n, but our notation drops this dependency, with the following justification. First, we already fixed r. Next, the stan- Cn Cn+1 ∗ ∗ Cn+1 ∗ Cn .( )dard embedding ι: Grr( ) ֒→ Grr( ) induces a map ι : HT Grr( ) → HT Grr −1 (n+1) (n) Using superscripts to indicate where a subvariety lives, we have ι Xλ = Xλ , and ∗ (n+1) (n) ∗ therefore