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 gradedcommutative 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 integer partition 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:
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.[4] 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
^MacDonald, Ian Grant (2015). Symmetric functions and Hall polynomials. Oxford University Press; 2nd edition. p. 112. ISBN9780198739128.
^Macdonald, Ian Grant (2015). Symmetric functions and Hall polynomials. Oxford University Press; 2nd edition. p. 63. ISBN9780198739128.
^Stanley, Richard (1999). Enumerative Combinatorics: Volume 2 (Cambridge Studies in Advanced Mathematics Book 62). Cambridge University Press. p. 349. ISBN9780521789875.
^Stanley, Richard (1999). Enumerative Combinatorics: Volume 2 (Cambridge Studies in Advanced Mathematics Book 62). Cambridge University Press. p. 352. ISBN9780521789875.