Supergroup Representation Theory

Based on the understanding of Group Representation,this paper mainly generalizes and studies some conclusions and properties of Group Representation in Super Representation.The first part of the paper mainly introduces some basic concepts and related conclusions,which is the foundation of the whole article.The second part mainly explore the matrix representation of supergroups.Firstly,we define a super linear representation of the super group G on the super space V,secondly,we discuss the induction of relationship between the matrix representation of group and the matrix representation of supergroup.In the third chapter,we discuss the property theorem of supermodel which is closely related to the super group algebra.We construct a super module structure and define an inner product operation<,>on the super space V,discuss its property of invariance under the action of G,In the meantime,the Maschkes' theorem and Schur's lemma have been generalized to the super case.In the fourth part,we mainly study the character theory.Give the definition of the supercharacter and its inner product operation,and prove the supercharacter is orthogonal to the inner product,which interprets the meaning of supercharacter.