If \(\lambda=(\lambda_1\ge\lambda_2\ge\dots\ge\lambda_k)\) is a partition of \(n\), let \(A^\lambda\) denote the irreducible representation of the symmetric group \(S_n\) such that the Frobenius image of \(\chi^{A^\lambda}\) is the Schur function \(S_\lambda(x_1,\ldots,x_N)\) where \(N>n\).
Use the Murnaghnam-Nakayama rule to compute the values of the character \(A^{(4,1)}\) on the conjugacy classes of \(S_5\).
Express \(\chi^{A^{(1,4)}\downarrow^{S_5}_{S_3\times S_2}}\) as a sum of irreducible characters of \(S_3\times S_2\). Here \(S_3\times S_2\) is the Young subgroup of \(S_5\) consisting of all permutations \(\sigma\in S_5\) such that \[ \sigma(1),\sigma(2),\sigma(3)\in\{1,2,3\},\qquad \sigma(4),\sigma(5)\in\{4,5\}. \]
Solution
Write \(\chi=\chi^{(4,1)}\).
1. The values of \(\chi\) on the conjugacy classes of \(S_5\), ordered by cycle type\[ (1^5),\ (2,1^3),\ (2^2,1),\ (3,1^2),\ (3,2),\ (4,1),\ (5), \]are\[ \boxed{4,\ 2,\ 0,\ 1,\ -1,\ 0,\ -1}. \]
The Specht module \(S^{(4,1)}\) is the standard representation of \(S_5\), obtained from the permutation representation \(\mathbb C^5\) by removing the one-dimensional invariant line spanned by \(e_1+\cdots+e_5\). Hence for every \(\sigma\in S_5\), \[ \chi^{(4,1)}(\sigma)=\#\operatorname{Fix}(\sigma)-1. \] The numbers of fixed points for the seven cycle types above are respectively \[ 5,\ 3,\ 1,\ 2,\ 0,\ 1,\ 0, \] which yields the displayed row.
This agrees with the Murnaghan–Nakayama rule: removing rim hooks from the hook diagram \((4,1)\) gives exactly the same signed contributions for each cycle type.
2. The restriction of \(S^{(4,1)}\) to the Young subgroup \(S_3\times S_2\) decomposes as\[ \boxed{ S^{(4,1)}\!\downarrow_{S_3\times S_2}^{S_5} \cong (S^{(3)}\boxtimes S^{(2)}) \oplus (S^{(2,1)}\boxtimes S^{(2)}) \oplus (S^{(3)}\boxtimes S^{(1,1)}). } \]
Restrict first the permutation representation \(\mathbb C^5\). Under \(S_3\times S_2\), the first three basis vectors and the last two basis vectors span invariant subspaces, so \[ \mathbb C^5\!\downarrow_{S_3\times S_2} \cong \mathbb C^3\boxtimes \mathbf 1 \oplus \mathbf 1\boxtimes \mathbb C^2. \] The permutation modules decompose as \[ \mathbb C^3\cong S^{(3)}\oplus S^{(2,1)}, \qquad \mathbb C^2\cong S^{(2)}\oplus S^{(1,1)}. \] Therefore \[ \mathbb C^5\!\downarrow_{S_3\times S_2} \cong 2(S^{(3)}\boxtimes S^{(2)}) \oplus (S^{(2,1)}\boxtimes S^{(2)}) \oplus (S^{(3)}\boxtimes S^{(1,1)}). \] Finally, \[ \mathbb C^5\cong S^{(5)}\oplus S^{(4,1)}, \] and the restricted trivial representation \(S^{(5)}\) is one copy of \(S^{(3)}\boxtimes S^{(2)}\). Subtracting that copy gives the asserted decomposition.