Frobenius characteristic map
In mathematics, especially representation theory and combinatorics, a Frobenius characteristic map is an isometric isomorphism between the ring of characters of symmetric groups and the ring of symmetric functions. It builds a bridge between representation theory of the symmetric groups and algebraic combinatorics. This map makes it possible to study representation problems with help of symmetric functions and vice versa. This map is named after German mathematician Ferdinand Georg Frobenius. Definition

The ring of characters 

Let  be the -module generated by all irreducible characters of  over . In particular  and therefore . The ring of characters is defined to be the direct sumwith the following multiplication to make  a graded commutative ring. Given  and , the product is defined to bewith the understanding that  is embedded into  and  denotes the induced character. Frobenius characteristic map 
For , the value of the Frobenius characteristic map  at , which is also called the Frobenius image of , is defined to be the polynomial

Remarks 
Here,  is the partition of integers determined by . For example, when  and ,   corresponds to the partition . Conversely, a partition  of  (written as ) determines a conjugacy class  in . For example, given ,  is a conjugacy class. Hence by abuse of notation  can be used to denote the value of  on the conjugacy class determined by . Note this always makes sense because  is a class function. Let  be a partition of , then  is the product of power sum symmetric polynomials determined by  of  variables. For example, given , a partition of ,

Finally,  is defined to be , where  is the cardinality of the conjugacy class . For example, when , . The second definition of  can therefore be justified directly:

Properties

Inner product and isometry

Hall inner product 
The inner product on the ring of symmetric functions is the Hall inner product. It is required that  . Here,  is a monomial symmetric function and  is a product of completely homogeneous symmetric functions. To be precise, let  be a partition of integer, thenIn particular, with respect to this inner product,  form a orthogonal basis: , and the Schur polynomials  form a orthonormal basis: , where  is the Kronecker delta. Inner product of characters 
Let , their inner product is defined to be

If , then

Frobenius characteristic map as an isometry 
One can prove that the Frobenius characteristic map is an isometry by explicit computation. To show this, it suffices to assume that :

Ring isomorphism 

The map  is an isomorphism between  and the -ring . The fact that this map is a ring homomorphism can be shown by Frobenius reciprocity. For  and ,

Defining  by , the Frobenius characteristic map can be written in a shorter form:

In particular, if  is an irreducible representation, then  is a Schur polynomial of  variables. It follows that  maps an orthonormal basis of  to an orthonormal basis of . Therefore it is an isomorphism. Example

Computing the Frobenius image 
Let  be the alternating representation of , which is defined by , where  is the sign of the permutation . There are three conjugacy classes of , which can be represented by  (identity or the product of three 1-cycles), (transpositions or the products of one 2-cycle and one 1-cycle) and  (3-cycles). These three conjugacy classes therefore correspond to three partitions of  given by , , . The values of  on these three classes are  respectively. Therefore:Since  is an irreducible representation (which can be shown by computing its characters), the computation above gives the Schur polynomial of three variables corresponding to the partition . References 

Representation_theory
Combinatorics