Page 126 - Textos de Matemática Vol. 39

116 JOA˜O FILIPE QUEIRO´
In 1962, A. Horn conjectured that E(α,β) is completely described by a family of inequalities of the type
γk1 +···+γkr ≤αi1 +···+αir +βj1 +···+βjr
where r ∈ {1, ..., n} and i1 < . . . < ir , j1 < . . . < jr , k1 < . . . < kr , or, in short,
ΣγK ≤ΣαI +ΣβJ
where I = (i1,...,ir),J = (j1,...,jr),K = (k1,...,kr). A consequence of this would be that E(α, β) is a convex polytope.
The question is to identify the right triples (I,J,K). Horn makes an elab- orate conjecture on this, which, in sightly changed form, reads as follows:
Write λ(I) = (ir −r,...,i2 −2,i1 −1) and similarly for λ(J) and λ(K). Then Horn’s conjecture is that γ ∈ E(α, β) if and only if
Σγ = Σα + Σβ and
ΣγK ≤ΣαI +ΣβJ wheneverλ(K) ∈ E[λ(I),λ(J)] (forallr,1≤r<n).
This means that the set E(α, β) is described recursively from lower dimensions. We present below the list of Horn inequalities for n = 2 and n = 3 (apart
from the trace equalities).
n=2: n=3:
γ2 ≤α1 +β2 γ2 ≤α2 +β1
γ1+γ2≤α1+α2+β1+β2 γ1+γ3≤α1+α2+β1+β3 γ2+γ3≤α1+α2+β2+β3 γ1+γ3≤α1+α3+β1+β2 γ2+γ3≤α1+α3+β1+β3 γ2+γ3≤α2+α3+β1+β2
γ1 ≤α1 +β1
γ1≤α1+β1 γ2≤α1+β2 γ3≤α1+β3 γ2≤α2+β1 γ3≤α2+β2 γ3≤α3+β1
The conjecture was proved in the late 1990s, as a result of work of Kly- achko in [7] and Knutson and Tao in [9]. This work involves, among other subjects, the intersection of Schubert varieties as well the representations of the general linear group, including the combinatorics of tableaux.
Other references on the problem are [3, 8, 12, 13].
The number of inequalities in Horn’s list grows very rapidly with n. In [1] and [10], the question of the independence of these inequalities for each n was studied.