For a partition $\lambda \vdash n$, the Specht module $S^\lambda$ is a $\mathbb{C}[S_n]$-module constructed from Young tableaux of shape $\lambda$.

Given a Young tableau $T$ of shape $\lambda$, define:

• $R(T)$ = subgroup of $S_n$ preserving rows of $T$
• $C(T)$ = subgroup of $S_n$ preserving columns of $T$

The row stabilizer $R(T)$ and column stabilizer $C(T)$ generate symmetrizers: $a_T = \sum_{\sigma \in R(T)} \sigma, \quad b_T = \sum_{\tau \in C(T)} \text{sgn}(\tau) \tau$

The Young symmetrizer is $c_T = b_T a_T$, and $S^\lambda = \mathbb{C}[S_n] c_T$ is the Specht module.

For a standard Young tableau $T$, the standard polytabloid $e_T$ is the element: $e_T = \sum_{\tau \in C(T)} \text{sgn}(\tau) \tau \{T\}$ where $\{T\}$ is the tabloid (equivalence class under row permutations).

The Specht module has basis $\{e_T : T \text{ standard of shape } \lambda\}$.