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 $$\chi(g^3)\chi(g^3)\chi(g^2)\chi(g)$$.