The Relation Of Irreducible Representation Over Different Fields And K Type Representation

The representation of finite group is an very important branch in algebra and there are lots of conclusions and results in this field.In the complex field C,all the irreducible representations of groups are divided into three types by their Frobenius-Schur indicator's values:real type,complex type and quaternionic type.Similarly,in a more general situation,for a field extension L/K,we define the K-type representation in L.Then we give some useful properties of K-type representations.1.Let G be a finite group,L an extension field of a field K,and Gal(L/K)its Galois group.For any ? ? Gal(L/K),we define a new representation(??,V?)over the field K.Through the concept of Gal(L/K)-structure,we get an equivalent condition for K-type representation:An irreducible representation has a Gal(L/K)-structure if and only if it is K-type.2.For a cyclic group with order n,we calculate out the classification of irreducible representations over complex number field and real number field.Over complex number field if the order of G is odd,one irreducible representation is of real type,and the orther n-1 representations are of complex types.If the order of G is even,there are two real types of irreducible representations,n-2 complex type irreducible representations.Over the rational number field,we give a method to judge whether an irreducible representation is of rational type.3.In addition,for a cyclic group G with special order,we calculate out its irreducible representations over prime field Fp and extension Fpn.