Problem 5.1. Field embeddings. Recall that k (y 1 , ., ym ) denotes the eld of rational functions
of y1 , ., ym over a eld k . Let f : k [x1 , ., xn ] k (y1 , ., ym ) be an injective k -algebra homomor
phism. Show tha
Representations of nite groups: further results
Suppose that G is a nite group and V is an irreducible representation of G over C. We say that
- of complex type, if V V ,
- of real type, if V has a nondegenerate symmet
Representations of nite groups: basic results
Recall that a representation of a group G over a eld k is a k -vector space V together with a
group homomorphism : G GL(V ). As we have explained above, a representation of a group G
over k is the same thing
General results of representation theory
Subrepresentations in semisimple representations
Let A b e an algebra.
Denition 2.1. A semisimple (or completely reducible) representation of A is a direct sum of
Example. Let V b
Basic notions of representation theory
What is representation theory?
In technical terms, representation theory studies representations of associative algebras. Its general
content can b e very briey summarized as follows.
An associative algebra ove