Since I am not a superhuman, my blog lengths will significantly reduce in size from now on. Let be the space of functions on a finite group . We can define the usual inner product on it, and the convolution makes it into a group algebra; one can associate each function with an element of the group algebra in the obvious way. The center of consists of class functions, as testified by the indicator functions of single elements. Thus a basis of the center is given by , where ranges over conjugacy classes. For class functions on ,
Definition 9 Let to be the graded algebra of class functions, where multiplication is defined by
Further let called the characteristic map be defined by
Proposition 10 Ch is an isometry.
The induced map is defined by the Frobenius relation:
where are representative of right cosets of in , and equals f on H and 0 o.w.
The proposition above is easy, next we show
Proposition 11 Ch is an algebra isomorphism.
Proof: One can write
where given by . Note is a function on with value in the space of symmetric functions. This coupled with Frobenius reciprocity easily prove the assertion.
Need some representation theory of . For a G-set X (G acts transitively), one has the permutation reprsentation on it. For set partitions of given by a partition , called partial rankings, acts on them in the natural way, i.e., permuting the labels. This representation is called Specht module.
Fact: , where . But these representations are reducible, for instance:
Let be the character of .
Proof: First we establish it for . So we basically need . Recall . Setting , , then . Then
Next we find linear combinations of giving an irreducible character. Recall Munagen-Nakayama:
is the number of tableaux of shape and type . Define the following linear combination of products of ‘s:
in analogy of Jacobi-Trudi formula for Schur functions. Then using isometry and algebra isomorphism of , we see
Since the ‘s are characters of possibly reducible representations, which in turn are -linear combinations of irreps, we know that ‘s must be irreducible characters. And since the number of SSYT’s which is positive, we know that is the irreducible characters (they evaluate to a positive trace on identity) and can be identified with .