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Algebraic Solutions of Differential Equations
41
itself by F~bs: k~ k) is given by
(3.2.1.1)
x ~p~
(g) (v  a) d~. x(g) (v)  a.
In matricial terms, the matrix coefficients of Z ~p) are the pth powers
of the matrix coefficients of Z. The character of z~P~ is the pth power of
the character of Z.
(3.2.2) Proposition.
Hypotheses as in
(3.2.1), /f Z
is an absolutely ir
reducible representation of G in a finitedimensional kspace, then its degree
divides the order of G, and in particular is prime to p.
Proof.
This will be a simple consequence of the analogous fact for
representation in characteristic zero. In fact, extending scalars if neces
sary, we may suppose that k is the residue field of a discrete valuation
ring ((9, p) whose fraction field K has characteristic zero, and is such that
every irreducible representation of G in a finitedimensional Kspace is
absolutely irreducible. In fact, we will prove
(3.2.2bis)
Proposition.
Let ((9,
p)
be a discrete valuation ring whose
fraction field K has characteristic zero, and whose residue field k has
characteristic p > O. Let G be a finite group, of order prime to p, such that
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 Fall '11
 NormanKatz

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