Construction of Hyperkähler Metrics for Complex Adjoint Orbits

3.3 Regular Nahm Solutions and Line bundles on S 36
matrix ⎛
⎜ · · ·
⎝
1 λ1 · · · λ k−1
1
1 λ2 · · · λ k−1
2
1 λk · · · λ k−1
k
since the Vandermonde matrix is non-singular for distinct λ ′ is, these j + 1
⎞
⎟ ;
⎟
⎠
colums are linearly independent, so that condition (3.22) says precisely that
the first column is a unique linear combination of the others, as we wanted.
We must also require that, for j ≥ k − 1,
⎛
f
⎜
rank ⎜
⎝
(j)
1 (0) 1 λ1 · · · λ k−1
f
1
(j)
2 (0) 1 λ2 · · · λ k−1
· · ·
2
f (j)
k (0) 1 λk · · · λ k−1
⎞
⎟ = k;
⎟
⎠
k
however these conditions are always satisfied, since the Vandermonde matrix
is a k × k non-singular minor of the matrix above. Now we observe that if
the conditions (3.22) are satisfied, we may find aij ′ s satisfying (3.21), define
gj(ζ) by (3.20) and σ 0 by (3.19).
Claim. We may rewrite the condition (3.22) as
⎛
f
⎜
det ⎜
⎝
(j)
1 (0) 1 λ1 · · · λ j
1
· · ·
f (j)
j+1 (0) 1 λj+1 · · · λ j
j+1
f (j)
m (0) 1 λm · · · λj ⎞
⎟ = 0,
⎟
⎠
(3.23)
m
for m = j + 2, . . . , k.
Proof of Claim. The (j + 2) × (j + 2) determinant in (3.23) must vanish
since otherwise the rank in (3.22) would be ≥ j +2. Conversely, the relations