Formula for calculating effect of Schur functor on character
Suppose is a group, is a field (which we will assume for simplicity to have characteristic zero) and is a linear representation of over with character . Suppose is a partition of a positive integer and denote by the Schur functor corresponding to . We can consider a new representation and we denote its character by . Our goal is to provide an explicit description of .
Explicit description of degree of representation
The new degree is a polynomial in the old degree. The degree of the polynomial (not to be confused with degrees of representations) equals . Explicitly, the polynomial is defined as follows. Denote by the set of conjugacy classes of the symmetric group . Let be the function that outputs the number of cycles of a given conjugacy class. Denote by the character of the irreducible representation of corresponding to the partition of . Then, the polynomial is:
Explicit description of character of representation
The character of the representation is given as follows. Denote by the set of conjugacy classes of the symmetric group . Let be the function that outputs the number of cycles of a given conjugacy class. Denote by the character of the irreducible representation of corresponding to the partition of . Then, the character is:
For instance, for a conjugacy class in with cycle type , the product on the inside reads .
Description for small Schur functors
|Size of set being partitioned||Partition for Schur functor||Name of functor||Formula for computing degree if original representation has degree||Formula for computing character of the representation obtained after applying this functor in terms of the original character , on an element||Explanation for character formula|
|1||1||identity functor||, so the only summand corresponds to the partition 1 of 1, and is .|
|2||2||symmetric square|| and is the partition 2, so the summation is carried out over the partitions of 2. The representation is the trivial representation. The two parts are:|
partition 1 + 1 of 2, corresponding to conjugacy class of identity element: the summand is .
partition 2 of 2, corresponding to conjugacy class of non-identity element: the summand is .
Adding, we get the result.
|2||1 + 1||exterior square or alternating square|
|3||2 + 1||?|
|3||1 + 1 + 1||exterior cube|