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 \(N>n\).
Let \(T\) be the trivial representation. Decompose \(T\uparrow_{S_3\times S_3}^{S_6}\) as a sum of irreducible representations of \(S_6\) where \(S_3\times S_3\) is the Young subgroup of \(S_6\) consisting of all permuations \(\sigma\in S_6\) such that \[ \sigma(1),\sigma(2),\sigma(3)\in\{1,2,3\},\qquad \sigma(4),\sigma(5),\sigma(6)\in\{4,5,6\}. \]
Decompose \(A^{(2,4)}\otimes A^{(1,5)}\) as a sum of irreducible representations of \(S_6\) where \(\otimes\) represents the Kronecker product of the representations.
Decompose \(A^{(1,2)}\times A^{(1,2)}\uparrow_{S_3\times S_3}^{S_6}\) as a sum of irreducible representations of \(S_6\). (Here \(S_3\times S_3\) is group described in part (a).)
Solution
(a).
1. \(T \uparrow_{S_3 \times S_3}^{S_6}\) is the permutation representation on the cosets of \(S_3 \times S_3\), i.e. on the \(\binom{6}{3} = 20\) three-element subsets of \(\{1, \ldots, 6\}\).
the cosets of the Young subgroup \(S_3 \times S_3\) correspond to \(3\)-subsets.
2. By the Pieri rule (or the Littlewood–Richardson rule), the induced trivial representation decomposes as\[T \uparrow_{S_3 \times S_3}^{S_6} = A^{(6)} \oplus A^{(5,1)} \oplus A^{(4,2)} \oplus A^{(3,3)}.\]
the permutation representation on \(3\)-subsets decomposes into the Specht modules indexed by partitions of \(6\) with at most \(2\) parts; these are exactly \((6), (5,1), (4,2), (3,3)\).
(b).
1. \(A^{(2,4)} = A^{(4,2)}\) and \(A^{(1,5)} = A^{(5,1)}\).
\((2,4)\) and \((4,2)\) are the same partition (written in nonincreasing order), and similarly \((1,5) = (5,1)\).
2. The Kronecker product \(A^{(4,2)} \otimes A^{(5,1)}\) decomposes (by the Kronecker coefficients) as\[A^{(4,2)} \otimes A^{(5,1)} = A^{(6)} \oplus A^{(5,1)} \oplus A^{(4,2)} \oplus A^{(4,1,1)} \oplus A^{(3,2,1)} \oplus A^{(3,1,1,1)} \oplus A^{(2,2,1,1)}.\]
computation of the Kronecker product of the two Specht modules (via the Murnaghan–Nakayama / character inner product).
(c).
1. \(A^{(1,2)} = A^{(2,1)}\) is the standard \(2\)-dimensional representation of \(S_3\).
\((1,2) = (2,1)\) is the partition of \(3\) with two parts.
2. \(A^{(2,1)} \times A^{(2,1)}\) is the outer (tensor) product, which by the Pieri rule is\[A^{(2,1)} \times A^{(2,1)} = A^{(4,2)} \oplus A^{(4,1,1)} \oplus A^{(3,3)} \oplus A^{(3,2,1)} \oplus A^{(2,2,2)}.\]
Littlewood–Richardson rule for the outer product of two copies of \((2,1)\).
3. Inducing from \(S_3 \times S_3\) to \(S_6\) (which is the same as the outer product, since induction from a Young subgroup is the outer product), the decomposition is exactly the one in
2.
induction from \(S_3 \times S_3\) to \(S_6\) of an outer product is the outer product itself.
4. Q.E.D.
2 (a), 2 (b), <1>2–<1>3 (c).