# Formula for calculating effect of Schur functor on character

## Contents

## Statement

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 | 3 | symmetric cube | |||

3 | 2 + 1 | ? | |||

3 | 1 + 1 + 1 | exterior cube |