Dr. Katz DEq Homework Solutions 41

Dr. Katz DEq Homework Solutions 41 - Algebraic Solutionsof...

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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 p-th powers of the matrix coefficients of Z. The character of z~P~ is the p-th 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 finite-dimensional k-space, 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 finite-dimensional K-space 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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