For a Young diagram $\lambda$, let $V_\lambda$ be an irreducible representation of $\mathfrak{S}_n$ corresponding to $\lambda$ (over $\mathbb{C}$). And denote the transpose of $\lambda$ by $\lambda^T$.
I wonder if $V_{\lambda^T} \cong V \otimes_{\mathbb{C}} \mathbb{C}^\vee$, where $\mathbb{C}^\vee$ is an alternating representation of $\mathfrak{S}_n$ (i.e., $\mathrm{sgn} : \mathfrak{S}_n \to \mathrm{GL} (1, \mathbb{C})$). I know that this is correct for $n \le 5$ by calculating the character table for $\mathfrak{S}_n$.
But I am not sure how to prove (or counter-prove) that. Is the statement $V_{\lambda^T} \cong V \otimes_{\mathbb{C}} \mathbb{C}^\vee$ true? If not in general, how about the standard representation $V = \mathbb{C}^n / \{ x_1 = \dotsb = x_n \}$?
Updated:
An "irreducible representation corresponding to $\lambda$" is such that
- the "vertical" one-column Young diagram corresponds to $\mathbb{C}^\vee$,
- the "horizontal" one-row Young diagram to $\mathbb{C}$.
Explicitly, $V_\lambda = \bigoplus_{T} \mathbb{C} \Delta_T \subset \mathbb{C} [X_1, \dotsc, X_n]$, where $T$ running over all standard tableaux of $\lambda$, and where $\Delta_T = \prod_{i \lt k} (X_{n_{ij}} - X_{n_{kj}})$ if $$ T = \begin{bmatrix} n_{11} = 1 & n_{12} & \cdots \\ n_{21} & n_{22} & \cdots \\ \vdots & \vdots & 0 \end{bmatrix}. $$
