Assume the system of differential equations
E4(
a,
b,
c,
c′;
X,
Y) satisfied by Appell’s hypergeometric function
F4(
a,
b,
c,
c′;
X,
Y) has a finite irreducible monodromy group
M4(
a,
b,
c,
c′). The monodromy matrix Γ
3∗ derived from a loop Γ
3 surrounding once the irreducible component
C ={(
X,
Y)|(
X -
Y)
2 -2(
X +
Y) +1 =0} of the singular locus of
E4 is a complex reflection. The minimal normal subgroup
NC of
M4 containing Γ
3∗ is, by definition, a finite complex reflection group of rank four. Let
P(
G) be the projective monodromy group of the Gauss hypergeometric differential equation
2E1(
a,
b,
c). It is known that
NC is reducible if ε :=
c +
c′-
a -
b -1 ∉
Z or if ε ∈
Z and
P(
G) is a dihedral group. We prove that, if ε ∈
Z,then
NC is the (irreducible) Coxeter group
W(
D4),
W(
F4)and
W(
H4) according as
P(
G) is the tetrahedral, octahedral and icosahedral group, respectively.
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