Representation pullbackability theorem
This fact is related to: representation theory
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Let be a finite group and a finite field whose characteristic does not divide the order of . Let be a local ring with residue field . Then, any representation of over pulls back to a representation of over .
In other words, given any map:
there exists a map:
such that , composed with the residue map modulo the maximal ideal, gives .
has order where , not dividing the order of . Let be the map from to , that involves taking each matrix entry modulo the maximal ideal.
The kernel of is, as can easily be checked, the subgroup comprising matrices which are congruent entry-wise to the identity matrix. The cardinality of this kernel is thus .
Then, consider . The map:
has, as kernel, a -group. Since does not divide the order of , it does not divide the order of , os the kernel is a normal Sylow subgroup. We can thus apply Schur-Zassenhaus theorem to find a section for this map. Composing this section with gives us the pulled back representation .