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P. 45
2.4. Flows on Graßmann manifolds
35
we set
m+n
ζ(a) := aν λν ,
ν=1
m+n b) Let N := n
be the set with
and let
A := {a1,...,aN} ⊂ {0,1}m+n
j < k ⇔
for some ε0 > 0, where ≤lex means lexicographical ordering.
(2.29)
R(a) := R(ζ(a)),
I(a) := I(ζ(a)).
aj = (aj1,...,ajm+n)
j 1 for i∈{ν1j,...,νnj},1≤ν1j <···<νnj ≤m+n,
and
ai := 0 else,
where {ν1j , . . . , νnj } ⊂ {1, 2, . . . , m + n}.
In the sequel we assume that the N elements of A are numbered such that
either R(ζ(aj)eiΦ) < R(ζ(ak)eiΦ), for 0 < Φ < ε0, or ζ(aj) = ζ(ak) and (ν1j,...,νnj) ≤lex (ν1k,...,νnk)
c) For
and with the determinants as used in (2.14), we set v˜(aj) := |v˜ν1j ,...,v˜νnj |,
vlα(aj) := |vν1j (l,α),...,vνnj (l,α)|, 1 ≤ l ≤ m, 1 ≤ α ≤ n, cνj
j .1 D(a,C):= . .
cνnj With these abbreviations (2.14) reads as
aj ∈A etζ(aj )vlα(aj )D(aj , C) wlα(t; WC ) = aj ∈A etζ(aj )v˜(aj )D(aj , C) .
c1
j . (m+n)×n
a ∈ A and C = . ∈ C cm+n
(2.30)
With this formula we shall now study the asymptotic behavior of (2.30) along specific rays in the complex plane. (2.30) is nothing else than a quotient of so-called exponential sums; there exists a vast literature on the study of such
