Formula for calculating effect of Schur functor on character

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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