On The Green Ring Of The Symmetric Groups S₄

In this paper,the power formulas of two-dimensional and three-dimensional irreducible representations of the symmetric group S4 are calculated by using the character theory of finite groups,In addition,by decomposing tensor products of irreducible representations of S4,we characterize the representation ring r(S4)of S4.Based on this,we use the regular representation of rQ(S4)on itself over the field Q,of rational numbers to obtain five one-dimensional representations and central primitive idempotents of rQ(S4).The full text is divided into four chapters.In the first chapter,we briefly introduce the history of the development of finite group representation theory,the development background and research status of this paper,and introduce the main research contribution of this paper in detail.In Chapter 2,we introduce some basic definitions in the representation theory of finite groups.In Chapter 3,with the help of a set of lemmas,we calculate all the irreducible characters of S4 and the tensor product decomposition of irreducible representations,obtain the corresponding matrix with the help of the recurrence relation of tensor product decomposition,calculate the eigenvalues and eigenvectors of the matrix,and give the power formulas of two-dimensional and three-dimensional irreducible representation of S4.In Chapter 4,we discuss the representation ring r(S4);of S4and find five onedimensional representations of the representation algebra rQ(S4)over the field Q.On this basis,we calculate the five central primitive idempotents ofrQ(S4),and obtain the general expression of regular elements of rQ(S4).