Page 128 - Textos de Matemática Vol. 39

118 JOA˜O FILIPE QUEIRO´
4. Note on other values of s
In light of the last comment in the previous section, a natural conjecture would be that, for any s, the restrictions involving γ = (γk1,...,γks) in our problem would be precisely those appearing in Horn’s list as explicit bounds on sums of entries in γ (both upper and lower bounds, the latter obtained using the trace condition).
Again as remarked in [4], this conjecture is not true. The example given there is n = 3, s = 2, γ = (γ1, γ3). The explicit bounds from Horn’s list in this case are:
α1+β3,α2+β2,α3+β1 ≤γ1 ≤α1+β1
α3+β3 ≤γ3 ≤α1+β3,α2+β2,α3+β1
α1+α3+β2+β3 , α2+α3+β1+β3 ≤ γ1+γ3 ≤ α1+α2+β1+β3 , α1+α3+β1+β2
But these conditions are not sufficient, as they do not force the second eigen- value of the sum (which must be equal to α1 +α2 +α3 +β1 +β2 +β3 −γ1 −γ3) to be between γ1 and γ3.
It turns out that the introduction of the (obviously necessary) inequalities needed to solve that ordering problem,
γ1+2γ3 ≤α1+α2+α3+β1+β2+β3 ≤2γ1+γ3,
yields the complete answer in this case.
The following illustrates the polygon of realizable γ when the given spectra
are α = (6,4,2),β = (7,4,1):
8
7
6
5
4
3
2
8 9 10 11 12 13 14
2x+y=24
x+2y=24