10.2 Tableaux, Tabloids and Specht Module

Definition

Let $\lambda ⊢n$ be a partition of $n$ with $\lambda =({\lambda }_1,\cdots ,{\lambda }_{\ell })$, then define corresponding Young subgroup as

$$S_{\lambda }=S_{\{1,\cdots ,{\lambda }_1\}}\times \cdots \times S_{\{n-{\lambda }_{\ell },\cdots ,n\}}.$$

For a trivial representation of $S_{\lambda }$, we have that

$$\dim \mathrm{tr}{\uparrow }_{S_{\lambda }}^{S_n}=\frac{|S_n|}{|S_{\lambda }|}=\frac{n!}{{\lambda }_1!\cdots {\lambda }_{\ell }!}.$$

tableau

Let $\lambda ⊢n$. A Young tableau of shape $\lambda$ is an array obtained by putting $1,\cdots ,n$ into $\lambda$-tableau.

For example, see here. Note that $\tau \in S_n$ can naturally act on $\lambda$-tableaus.

Definition

Two tableaux are equivalent if they are of the same $\lambda$-shape and they have the same element in each row. Here is an example.

tabloid

Given a tableau $t$, define the tabloid or $\lambda$-tabloid as the set $\{t^{\prime }:t^{\prime }\sim t\}$.

the Corresponding Module $M^{\lambda }$

Definition

For every partition $\lambda ⊢n$ the corresponding module $M^{\lambda }$ is the linear space of all tabloids of shape $\lambda$, $M^{\lambda }\simeq \mathbb{C}[\{t_1\},\cdots .,\{t_n\}]$ where $\dim M^{\lambda }=\frac{n!}{\lambda !}$ and $\lambda !={\lambda }_1!\cdots {\lambda }_{\ell }!$.

Examples.

• Let $\lambda =(n)$, then $M^{(n)}=\mathbb{C}[\{t\}]$ and $\dim M^{(n)}=1$.
• Let $\lambda =(1,1,\cdots ,1)$, then $M^{\lambda }=\mathbb{C}[\{t_1\},\cdots ,\{t_{n!}\}]$ and this action is regular, as $G$ acting on $\mathbb{C}G$.
• Let $\lambda =(n-1,1)$, then $M^{\lambda }=\mathbb{C}[\{t_1\},\cdots ,\{t_n\}]$ and this action deduces the permutation representation of $S_n$.

Proposition

Let $\lambda ⊢n$ be a partition, and let $S_{\lambda }<S_n$ be the subgroup corresponding to $\lambda$. Then $\mathrm{tr}{\uparrow }_{S_{\lambda }}^{S_n}\simeq M^{\lambda }$.

\begin{proof} Define $\phi :{\pi }_i\otimes W\mapsto \{{\pi }_i{t}^{\lambda }\}$ where $W$ is the trivial $S_{\lambda }$-module and $t^{\lambda }$ is the tabloid whose $\ell$ rows are $\{1,\cdots ,{\lambda }_1\}$, $\{{\lambda }_1+1,\cdots ,{\lambda }_2\}$, …, $\{n-{\lambda }_{\ell },\cdots ,n\}$. Then $\phi$ is an isomorphism of $S_n$-modules. \end{proof}

the Row/Column Stabilizer and the Associated Polytabloid

Definition

Let $\lambda ⊢n$ be a partition, and let $t$ be a tableau in $\lambda$-shape. Suppose $t$ has rows $R_1,\cdots ,R_{\ell }$ and columns $C_1,\cdots ,C_m$. Define subgroups of $S_n$ as $R_t=S_{R_1}\times \cdots \times S_{R_{\ell }}$ and $C_t=S_{C_1}\times \cdots \times S_{C_m}$, and they are called the row-stabilizer and the column-stabilizer of $t$.

Remark. With this definition, we have that $\{t\}=R_t\cdot t$.

associated polytabloid

For any subset $H\subseteq S_n$, define the following elements of $\mathbb{C}{S}_n$

$$H^{+}=\sum _{\pi \in H}\pi ,H^{-}=\sum _{\pi \in H}\mathrm{sgn}\pi \cdot \pi =\sum _{\pi \in H}{\pi }^{-},$$

where ${\pi }^{-}=\mathrm{sgn}\pi \cdot \pi$.

For any tableau $t$ with $C_1,\cdots ,C_m$, define

$$K_{C_i}=C_i^{-}=\sum _{\pi \in S_{C_i}}\mathrm{sgn}\pi \cdot \pi ,K_t=K_{c_1}\times \cdots \times K_{C_m}\text{mbox}and{e}_t=K_t\{t\},$$

where $e_t$ is called the associated polytabloid of $\lambda$-tableau $t$.

Remark. For any $\lambda$-tableau $t$, the associated polytabloid $e_t=K_t\{t\}$ is an element of permutation module $M^{\lambda }$.

Example. Here is an example of $S_5$.

Lemma

Let $t$ be a tableau and $\sigma \in S_n$. Then:

• $R_{\sigma t}=\sigma R_t{\sigma }^{-1}$;
• $C_{\sigma t}=\sigma C_t{\sigma }^{-1}$;
• $K_{\sigma t}=\sigma K_t{\sigma }^{-1}$;
• $e_{\sigma t}=\sigma e_t$, where $e$ is the associated polytabloid of $t$.

\begin{proof} Note that $\pi \in R_{\sigma t}$ iff $\pi \{\sigma t\}=\{\sigma t\}$ iff $\pi \sigma \{t\}=\sigma \{t\}$ iff ${\sigma }^{-1}t\sigma \{t\}=\{t\}$ iff ${\sigma }^{-1}t\sigma \in R_t$, and the proof of ii) and iii) are similar. Furthermore, $e_{\sigma t}=K_{\sigma t}\{\sigma t\}=\sigma K_t{\sigma }^{-1}\{\sigma t\}=\sigma e_t$. \end{proof}

Specht Module

Definition

Let $\lambda ⊢n$. The Specht module $S^{\lambda }$ is the submodule of $M^{\lambda }$ spanned by polytabloids $e_t$ where $t$ is of shape $\lambda$.

Remark. Since $e_{\sigma t}=\sigma e_t$, any $S^{\lambda }$ is a cyclic module (a module that is generated by a single element), i.e., it is generated by any tabloid $t$.

Examples.

• Let $\lambda =(n)$. Then $M^{(n)}=\mathbb{C}[\{t\}]$, $C_t=\{e\}$ and $K_t=ee\cdots e$. Note that $S^{\lambda }$ is spanned by $e_t=\{t\}$ and $S^{\lambda }=M^{\lambda }$.
• Let $\lambda =(1,1,\cdots ,1)$. Then $M^{\lambda }=\mathbb{C}[\{t_1\},\cdots ,\{t_{n!}\}]$ and $K_t={\sum }_{\pi \in S_n}\mathrm{sgn}\pi \cdot \pi$. It follows that $e_t=K_t\{t\}$ and $\sigma e_t=\mathrm{sgn}\sigma \cdot e_t$. Thus, $S^{\lambda }$ is isomorphic to the sign representation.
• Let $\lambda =(n-1,1)$. Then $M^{\lambda }=\mathbb{C}[\{t_1\},\cdots ,\{t_n\}]$ and $K_t=e-(1n)$. It follows that $e_t=K_t\{t\}=\{t\}-(1n)\{t\}$ and $\sigma e_t=\overline{\sigma (\mathbf{n})}-\overline{\sigma (1)}$ where $\overline{\mathbf{k}}$ is the tabloid whose second line is $k$. Thus, $S^{\lambda }=\{c_1\overline{1}+\cdots +c_n\overline{\mathbf{n}}:c_1+\cdots +c_n=0\}$ is the standard $S_n$ module.