4.17. Lecture 17

4.17.1. Young diagrams, tableaux, and tabloids

By Theorem 2.8.7, we know that there is a bijection between the irreducible representations and conjugacy classes of any finite group $G$ . However, for most groups there is no canonical way of giving an explicit construction of such a bijection without first finding all the irreducibles.

In the important case $G=S_{n}$ , we can in fact construct every irreducible representation from the conjugacy classes. More exactly, given a conjugacy class ${\mathcal{C}}$ , we will construct an irreducible representation $\pi_{{\mathcal{C}}}$ and show that these representations are all non-isomorphic for different choices of ${\mathcal{C}}$ .

Recall that the conjugacy classes of $S_{n}$ are given by all the permutations with the same cycle type.

Definition 4.17.1.

Let $n\in{\mathbb{N}}$ . A partition $\lambda$ of $n$ is an ordered tuple $\lambda=(\lambda_{1},\lambda_{2},\ldots,\lambda_{\ell})$ , where

\lambda_{1}\geq\lambda_{2}\geq\ldots\geq\lambda_{\ell}>0\quad\mathrm{and}\quad% \lambda_{1}+\lambda_{2}+\ldots+\lambda_{\ell}=n.

We write $\lambda\vdash n$ and $|\lambda|=\ell$ is called the length of $\lambda$ .

Definition 4.17.2 (Young Diagram, 1900).

Given a partition $\lambda=(\lambda_{1},\lambda_{2},\ldots,\lambda_{\ell})$ of $n$ we can define a corresponding Young diagram as $n$ boxes arranged into $\ell$ rows with $\lambda_{i}$ in the $i$ th row (from the top). A partition $\lambda\vdash n$ is also called the shape of the corresponding Young diagram.

We will choose to have the row corresponding to $\lambda_{1}$ at the top and to draw our boxes left adjusted, so the left most column always has $\ell$ entries, but note that there are other conventions.

Notice that we can associate a cycle type in $S_{n}$ to a unique partition, obtained by ordering the cycles by length. We can therefore also associate a Young diagram.

Example 4.17.3.

The conjugacy class in $S_{5}$ of $(123)=(123)(4)(5)$ is illustrated by the Young diagram \ytableausetupsmalltableaux \ydiagram3,1,1.

Example 4.17.4.

The conjugacy class in $S_{7}$ of $(12)(356)=(356)(12)(4)(7)$ is illustrated by the Young diagram \ytableausetupsmalltableaux \ydiagram3,2,1,1.

Definition 4.17.5.

Given a partition $\lambda\vdash n$ , a Young tableau is a Young diagram of shape $\lambda$ with one of the numbers $1,2,\ldots,n$ in each box such that no number occurs twice in the whole diagram. The set of all Young tableaux of shape $\lambda$ is denoted $\mathbf{YT}^{\lambda}$

Example 4.17.6.

The Young tableaux \ytableausetupsmalltableaux {ytableau} 1 & 3 2
6 7 5
4 and \ytableausetupsmalltableaux {ytableau} 4 & 7 1
6 2 3
5 are both elements of $\mathbf{YT}^{(3,3,1)}$ .

Observe that $|\mathbf{YT}^{\lambda}|=n!$ . There is also a natural $S_{n}$ -action on $\mathbf{YT}^{\lambda}$ inherited from the defining action on $1,\ldots,n$ :

(354)(67)\cdot\ytableausetup{smalltableaux}\ytableau 4&71\\ 623\\ 5=\ytableausetup{smalltableaux}\ytableau 3&61\\ 725\\ 4.

In order to create an $S_{n}$ -action that is different to the standard $S_{n}$ action, we identify all Young tableaux with the same entries in each rows. This defines an equivalence relation on $\mathbf{YT}^{\lambda}$ , and each equivalence class is called a Young tabloid. The set of Young tabloids of shape $\lambda$ is denoted $\mathbf{YTD}^{\lambda}$ . We illustrate a Young tabloid by removing the vertical lines in the corresponding Young tableau, as well as not caring about the ordering of the numbers in each row (though we normally write the numbers in each row in ascending order).

Example 4.17.7.

(23)(56)\cdot\ytableausetup{tabloids}\ytableaushort{123,45,6}=\ytableaushort{1% 32,46,5}=\ytableaushort{123,46,5}.

As an exercise, you should verify that

|\mathbf{YTD}^{\lambda}|=\frac{n!}{\lambda_{1}!\lambda_{2}!\cdots\lambda_{\ell% }!}

(4.17.8)

4.17.2. Specht modules

Since we have an $S_{n}$ -action on $\mathbf{YTD}^{\lambda}$ , there is the corresponding permutation representation $\big{(}\pi,{\mathbb{C}}(\mathbf{YTD}^{\lambda})\big{)}$ as in Definition 1.1.9. We write ${\mathcal{M}}^{\lambda}$ for the free vector space ${\mathbb{C}}(\mathbf{YTD}^{\lambda})$ , and call the vectors in this vector space polytabloids.

Example 4.17.9.

Let’s look at the representation $(\pi,{\mathcal{M}}^{(2,2)})$ of $S_{4}$ . From (4.17.8), we have $\operatorname{dim}({\mathcal{M}}^{(2,2)})=6$ , with a basis given by the elements of $\mathbf{YTD}^{(2,2)}$ :

\ytableausetup{tabloids}\ytableaushort{12,34},\quad\ytableaushort{13,24},\quad% \ytableaushort{14,23},\quad\ytableaushort{34,12},\quad\ytableaushort{24,13},% \quad\ytableaushort{23,14}.

We compute the action of $(12)$ on a polytabloid:

\pi(12)\left(\ytableausetup{tabloids}\ytableaushort{12,34}-2\,\ytableaushort{1% 4,23}+3\ytableaushort{23,14}\right)=\ytableaushort{12,34}-2\,\ytableaushort{24% ,13}+3\,\ytableaushort{13,24}.

The representation $(\pi,{\mathcal{M}}^{\lambda})$ is not irreducible (for example, it has the trivial representation as a subrepresentation), it does have a special subrepresentation $(\pi,{\mathcal{S}}^{\lambda})$ that is irreducible and the map $\lambda\mapsto(\pi,{\mathcal{S}}^{\lambda})$ is a bijection between conjugacy classes of $S_{n}$ and irreducible representations.

Before defining this representation, we need to introduce some more notation. Given a Young tableau $t\in\mathbf{YT}^{\lambda}$ , we write $[t]\in\mathbf{YTD}^{\lambda}$ for the corresponding tabloid. We also let $C(t)$ be the set of elements that preserve the numbers in each of the columns of $t$ (i.e. elements of $C(t)$ just permute the entries of each column individually).

Example 4.17.10.

We compute $C\left(\ytableausetup{notabloids,smalltableaux}\ytableau 2&3\\ 14\\ \right)$ : elements have to leave each column of the tableau invariant, so $C(t)$ is generated by the elements $(12)$ and $(34)$ . We therefore have that

C\left(\ytableausetup{smalltableaux}\ytableau 2&3\\ 14\\ \right)=\left\{e,(12),(34),(12)(34)\right\}.

Definition 4.17.11.

For each $t\in\mathbf{YT}^{\lambda}$ , let ${\bf e}_{t}\in{\mathcal{M}}^{\lambda}$ be the polytabloid

{\bf e}_{t}:=\sum_{\sigma\in C(t)}\operatorname{sgn}(\sigma)[\sigma\cdot t].

Theorem 4.17.12.

Let ${\mathcal{S}}^{\lambda}=\mathrm{span}\{{\bf e}_{t}\,|\,t\in\mathbf{YT}^{% \lambda}\}$ . The subspace ${\mathcal{S}}^{\lambda}$ is an irreducible $S_{n}$ -invariant subspace of ${\mathcal{M}}^{\lambda}$ . If $\lambda\vdash n$ and $\mu\vdash n$ are two different partitions of $n$ , then $(\pi,{\mathcal{S}}^{\lambda})\not\cong(\pi,{\mathcal{S}}^{\mu})$ .

The representation $(\pi,{\mathcal{S}}^{\lambda})$ is called the Specht module for $\lambda$ . These are named after Wilhelm Specht who studied them first in 1935. We will prove this theorem next lecture.

Remark 4.17.13.

Note that while the vectors ${\bf e}_{t}$ span ${\mathcal{S}}^{\lambda}$ , they do not form a basis, as they are not linearly independent. A basis for ${\mathcal{S}}^{\lambda}$ is given by $\{{\bf e}_{t}\,|\,t\>$ is a standard Young tableau $\}$ (we will define these in a few lectures). Observe also that even if two tableaux $t_{1}$ , $t_{2}$ give rise to the same tabloid (i.e. $[t_{1}]=[t_{2}]$ ; that is to say, they have the same entries in each row), in general, one has ${\bf e}_{t_{1}}\neq{\bf e}_{t_{2}}$ , since $C(t_{1})$ will be different from $C(t_{2})$ .

4.17.3. Exercises

Problem 89. Decompose $(\pi,{\mathcal{M}}^{(3,2)})$ into irreducible representations of $S_{5}$ .

Problem 90. Show that $(\pi,{\mathcal{S}}^{(n)})$ is the trivial representation of $S_{n}$ and that $(\pi,{\mathcal{S}}^{(1,\ldots,1)})$ is the sign representation.

Problem 91. Show that $(\pi,{\mathcal{M}}^{(n-1,1)})$ is isomorphic to the standard permutation representation of $S_{n}$ on ${\mathbb{C}}^{n}$ and that $(\pi,{\mathcal{S}}^{(n-1,1)})$ is isomorphic to the usual irreducible $n-1$ -dimensional subrepresentation $W_{0}$ of ${\mathbb{C}}^{n}$ .

Problem 92. Show that $(\pi,{\mathcal{M}}^{\lambda})$ is a unitary representation of $S_{n}$ with respect to the inner product

\left\langle\sum_{[t]\in\mathbf{YTD}^{\lambda}}z_{[t]}[t],\sum_{[s]\in\mathbf{% YTD}^{\lambda}}w_{[s]}[s]\right\rangle_{\lambda}:=\sum_{[t]\in\mathbf{YTD}^{% \lambda}}z_{[t]}\overline{w_{[t]}}.