Formula for calculating effect of Schur functor on character

Statement

Suppose $G$ is a group, $K$ is a field (which we will assume for simplicity to have characteristic zero) and $\varphi :G\to GL(d,K)$ is a linear representation of $G$ over $K$ with character $\chi$. Suppose $\lambda$ is a partition of a positive integer $n$ and denote by $\mathbb {S} _{\lambda }$ the Schur functor corresponding to $\lambda$. We can consider a new representation $\mathbb {S} _{\lambda }(\varphi )$ and we denote its character by $\mathbb {S} _{\lambda }(\chi )$. Our goal is to provide an explicit description of $\mathbb {S} _{\lambda }(\chi )$.

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 $n$. Explicitly, the polynomial is defined as follows. Denote by $C(S_{n})$ the set of conjugacy classes of the symmetric group $S_{n}$. Let $\nu :C(S_{n})\to \mathbb {N}$ be the function that outputs the number of cycles of a given conjugacy class. Denote by $\theta _{\lambda }$ the character of the irreducible representation of $S_{n}$ corresponding to the partition $\lambda$ of $n$. Then, the polynomial is:

${\frac {1}{n!}}\sum _{c\in C(S_{n})}\theta _{\lambda }(c)|c|d^{\nu (c)}$

Explicit description of character of representation

The character of the representation is given as follows. Denote by $C(S_{n})$ the set of conjugacy classes of the symmetric group $S_{n}$. Let $\nu :C(S_{n})\to \mathbb {N}$ be the function that outputs the number of cycles of a given conjugacy class. Denote by $\theta _{\lambda }$ the character of the irreducible representation of $S_{n}$ corresponding to the partition $\lambda$ of $n$. Then, the character is:

$\mathbb {S} _{\lambda }(\chi )(g)={\frac {1}{n!}}\sum _{c\in C(S_{n})}\theta _{\lambda }(c)|c|\left(\prod _{{\mbox{parts in the cycle type of }}c}\chi (g^{\mbox{size of the part}})\right)$

For instance, for a conjugacy class in $S_{9}$ with cycle type $3+3+2+1$, the product on the inside reads $\chi (g^{3})\chi (g^{3})\chi (g^{2})\chi (g)$.

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 $d$	Formula for computing character of the representation obtained after applying this functor in terms of the original character $\chi$, on an element $g$	Explanation for character formula
1	1	identity functor	$d$	$\chi (g)$	$n=1$, so the only summand corresponds to the partition 1 of 1, and is ${\frac {1}{1!}}(1)(1)\chi (g^{1})=\chi (g)$.
2	2	symmetric square	$d(d+1)/2$	$(\chi (g)^{2}+\chi (g^{2}))/2$	$n=2$ and $\lambda$ is the partition 2, so the summation is carried out over the partitions of 2. The representation $\theta _{\lambda }$ is the trivial representation. The two parts are: partition 1 + 1 of 2, corresponding to conjugacy class of identity element: the summand is ${\frac {1}{2!}}(1)(1)\chi (g)\chi (g)=(\chi (g))^{2}/2$. partition 2 of 2, corresponding to conjugacy class of non-identity element: the summand is ${\frac {1}{2!}}(1)(1)\chi (g^{2})=\chi (g^{2})/2$. Adding, we get the result.
2	1 + 1	exterior square or alternating square	$d(d-1)/2$	$(\chi (g)^{2}-\chi (g^{2}))/2$
3	3	symmetric cube	$d(d+1)(d+2)/6$	$(\chi (g)^{3}+3\chi (g^{2})\chi (g)+2\chi (g^{3}))/6$
3	2 + 1	?	$d(d+1)(d-1)/3$	$(\chi (g)^{3}-\chi (g^{3}))/3$
3	1 + 1 + 1	exterior cube	$d(d-1)(d-2)/6$	$(\chi (g)^{3}-3\chi (g^{2})\chi (g)+2\chi (g^{3}))/6$