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	$d(d - 1)(d - 2)/6$	$(\chi(g)^3 - 3\chi(g^2)\chi(g) + 2\chi(g^3))/6$