Page 133 - Textos de Matemática Vol. 39

PARTIAL SPECTRA OF HERMITIAN SUMS 123
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2x+y=29
Conditions (6.1) and (6.2) can be found by scanning Horn’s list for in- equalities that, together with the ordering of the γ’s, imply extra restrictions involving the partial spectrum γ = (γ2,γ3). (That they are exactly what is needed requires a separate proof.)
This suggests a way to identify the extra conditions in other cases.
For example, again for n = 4, if γ = (γ1, γ2) we should add to the explicit upper and lower bounds for γ1, γ2 and γ1 + γ2 coming from Horn’s list the following conditions:
⎧⎪⎨α1 +α2 +α3 +β2 +β3 +β4
γ1 +2γ2 ≥ α1 +α2 +α4 +β1 +β3 +β4 ⎪⎩α1 +α3 +α4 +β1 +β2 +β4 α2 +α3 +α4 +β1 +β2 +β3
γ1+3γ2 ≥α1+α2+α3+α4+β1+β2+β3+β4 For γ = (γ1, γ3) the extra conditions are:
α1 +α3 +α4 +β2 +β3 +β4
α +α +α +β +β +β⎧ ≤γ1+2γ3 ≤α1+α2+α3+β1+β2+β3
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⎪⎨⎪⎩α1 +α2 +α3 +β2 +β3 +β4
2γ1 +γ3 ≥ α1 +α2 +α4 +β1 +β3 +β4 ⎪α1 +α3 +α4 +β1 +β2 +β4 α2 +α3 +α4 +β1 +β2 +β3