3.12. Lecture 12

3.12.1. Symmetric and alternating products

We defined the tensor product $V\otimes V$ as a quotient ${\mathbb{C}}(V\times V)/U$ , where the subspace $U$ is chosen to give the desired linear relations in $V\otimes V$ . We will now do something similar to $V\otimes V$ , to obtain vector spaces that have even more symmetries.

Definition 3.12.1.

Let $U_{1}\subseteq V\otimes V$ be the subspace

U_{1}=\mathrm{span}\{{\bf v}_{1}\otimes{\bf v}_{2}-{\bf v}_{2}\otimes{\bf v}_{% 1}\,|\,{\bf v}_{1},{\bf v}_{2}\in V\},

and $\mathrm{Sym}^{2}(V):=(V\otimes V)/U_{1}$ . The vector space $\mathrm{Sym}^{2}(V)$ is called the (second) symmetric product of $V$ .

If ${\bf v}_{1}\otimes{\bf v}_{2}\in V\otimes V$ , then

{\bf v}_{1}\otimes{\bf v}_{2}+U_{1}={\bf v}_{2}\otimes{\bf v}_{1}+\big{(}{\bf v% }_{1}\otimes{\bf v}_{2}-{\bf v}_{2}\otimes{\bf v}_{1}\big{)}+U_{1}={\bf v}_{2}% \otimes{\bf v}_{1}+U_{1}.

(3.12.2)

Thus, modding out the subspace $U_{1}$ enforces the relation “ ${\bf v}_{1}\otimes{\bf v}_{2}={\bf v}_{2}\otimes{\bf v}_{1}$ ” on $V\otimes V$ . We normally write ${\bf v}_{1}{\bf v}_{2}$ for the element ${\bf v}_{1}\otimes{\bf v}_{2}+U_{1}$ of $\mathrm{Sym}^{2}(V)$ . We have ${\bf v}_{1}{\bf v}_{2}={\bf v}_{2}{\bf v}_{1}$ , as well as linear relations in both arguments. As a consequence we have that, if ${\bf v}_{1},\ldots,{\bf v}_{n}$ is a basis of $V$ then

\{{\bf v}_{i}{\bf v}_{j}\,|\,1\leq i\leq n,\;1\leq j\leq i\}

is a basis of $\mathrm{Sym}^{2}(V)$ . Thus $\operatorname{dim}\big{(}\mathrm{Sym}^{2}(V)\big{)}=\frac{n(n+1)}{2}$ .

In a similar fashion, we will also define a subspace $U_{2}$ of $V\otimes V$ that enforces the relation “ ${\bf v}_{1}\otimes{\bf v}_{2}=-{\bf v}_{2}\otimes{\bf v}_{1}$ ” on $V\otimes V$ :

Definition 3.12.3.

Let $U_{2}\subseteq V\otimes V$ be the subspace

U_{2}:=\mathrm{span}\{{\bf v}_{1}\otimes{\bf v}_{2}+{\bf v}_{2}\otimes{\bf v}_% {1}\,|\,{\bf v}_{1},{\bf v}_{2}\in V\},

and $\bigwedge^{2}V:=(V\otimes V)/U_{2}$ . The vector space $\bigwedge^{2}(V)$ is called the (second) alternating product of $V$ .

By the same method as the computation 3.12.2 above, we have ${\bf v}_{1}\otimes{\bf v}_{2}+U_{2}=-{\bf v}_{2}\otimes{\bf v}_{1}+U_{2}$ . We normally write ${\bf v}_{1}\wedge{\bf v}_{2}$ for ${\bf v}_{1}\otimes{\bf v}_{2}+U_{2}$ ; and ${\bf v}_{1}\wedge{\bf v}_{2}$ is linear both arguments, and ${\bf v}_{1}\wedge{\bf v}_{2}=-{\bf v}_{2}\wedge{\bf v}_{1}$ . Thus, if we let ${\bf v}_{1},\ldots,{\bf v}_{n}$ be a basis of $V$ , we have

\{{\bf v}_{i}\wedge{\bf v}_{j}\,|\,1\leq i\leq n,\;1\leq j<i\}

is a basis of $\bigwedge^{2}(V)$ . Thus $\operatorname{dim}\big{(}\bigwedge^{2}(V)\big{)}=\frac{n(n-1)}{2}$ .

We now look at representations on these spaces:

Proposition 3.12.4.

Let $(\pi,V)$ be a representation of a group $G$ . Then $(\pi\otimes\pi,U_{1})$ and $(\pi\otimes\pi,U_{2})$ are subrepresentations of $(\pi\otimes\pi,V\otimes V)$ , and

(\pi\otimes\pi,V\otimes V)\cong\big{(}\mathrm{Sym}^{2}\pi,\mathrm{Sym}^{2}(V)% \big{)}\oplus\big{(}\pi\wedge\pi,\bigwedge\!\!^{2}(V)\big{)}.

Proof.

See Problem 3.12.3 below. ∎

Proposition 3.12.5.

Let $(\pi,V)$ be a finite-dimensional representation of a finite group $G$ . Then for all $g\in G$ ,

\chi_{\mathrm{Sym}^{2}\pi}(g)=\frac{1}{2}\big{(}\chi_{\pi}(g)^{2}+\chi_{\pi}(g% ^{2})\big{)},\qquad\chi_{\pi\wedge\pi}(g)=\frac{1}{2}\big{(}\chi_{\pi}(g)^{2}-% \chi_{\pi}(g^{2})\big{)}.

Proof.

We only need to prove the first of these identities, the other follows from Proposition 3.12.4 and Proposition 2.7.9 (iii).

Let ${\bf v}_{1},\ldots,{\bf v}_{n}$ be an eigenbasis of $V$ for $\pi(g)$ , i.e. $\pi(g){\bf v}_{i}=\lambda_{i}{\bf v}_{i}$ . Then

\{{\bf v}_{i}{\bf v}_{j}\,|\,1\leq i\leq n,\;1\leq j\leq i\}

is a basis of $\mathrm{Sym}^{2}(V)$ . This is an eigenbasis for $\mathrm{Sym}^{2}\pi(g)$ as

\mathrm{Sym}^{2}\pi(g){\bf v}_{i}{\bf v}_{j}=(\lambda_{i}{\bf v}_{i})(\lambda_% {j}{\bf v}_{j})=(\lambda_{i}\lambda_{j}){\bf v}_{i}{\bf v}_{j}.

The definition of $\chi_{\mathrm{Sym}^{2}\pi}(g)$ gives

\chi_{\mathrm{Sym}^{2}\pi}(g)=\sum_{i=1}^{n}\sum_{j=1}^{i}\lambda_{i}\lambda_{% j}.

On the other hand,

	$\displaystyle\chi_{\pi}(g)^{2}=\left(\sum_{k=1}^{n}\lambda_{k}\right)^{2}$	$\displaystyle=\sum_{i=1}^{n}\lambda_{i}^{2}+2\sum_{i=1}^{n}\sum_{1\leq j<i}% \lambda_{i}\lambda_{j}$
		$\displaystyle=\chi_{\pi}(g^{2})+2\left(\sum_{i=1}^{n}\sum_{1\leq j\leq i}% \lambda_{i}\lambda_{j}-\sum_{i=1}^{n}\lambda_{i}^{2}\right)$
		$\displaystyle=2\chi_{\mathrm{Sym}^{2}\pi}(g)-\chi_{\pi}(g^{2}).$

∎

We conclude this section by noting that we can generalise these constructions to higher symmetric and alternating powers $\mathrm{Sym}^{k}(V)$ , $\bigwedge^{k}V$ for $k=3,4,\ldots$ .

3.12.2. Example: the character table of $S_{5}$

We will now complete the character table of $S_{5}$ . The irreducibles we already know are the trivial representation ${\mathrm{triv}}$ , the sign representation $\operatorname{sgn}$ , the irreducible permutation representation $(\pi,W_{0})$ , and its twist $(\operatorname{sgn}\pi,W_{0})$ , as before. So we can put these entries into the table:

\begin{array}[h]{c|rrrrrrr}&e&(12)&(12)(34)&(123)&(123)(45)&(1234)&(12345)\\ &1&10&15&20&20&30&24\\ \hline\cr{\mathrm{triv}}&1&1&1&1&1&1&1\\ \operatorname{sgn}&1&-1&1&1&-1&-1&1\\ \pi&4&2&0&1&-1&0&-1\\ \operatorname{sgn}\pi&4&-2&0&1&1&0&-1\end{array}

We then try $\pi\wedge\pi$ , which has character as shown below. This is an irreducible character (as it has norm one).

\begin{array}[h]{c|rrrrrrr}&e&(12)&(12)(34)&(123)&(123)(45)&(1234)&(12345)\\ &1&10&15&20&20&30&24\\ \hline\cr\pi\wedge\pi&6&0&-2&0&0&0&1\end{array}

A natural next step could be to try $\operatorname{sgn}(\pi\wedge\pi)$ , but it turns out this is isomorphic to $\pi\wedge\pi$ . We can also try $\operatorname{Sym}^{2}\pi$ , which has character below; it isn’t irreducible.

\begin{array}[h]{c|rrrrrrr}&e&(12)&(12)(34)&(123)&(123)(45)&(1234)&(12345)\\ &1&10&15&20&20&30&24\\ \hline\cr\operatorname{Sym}^{2}\pi&10&4&2&1&1&0&0\end{array}

By taking inner products with the characters we’ve already found, we see that

\mathrm{Sym}^{2}\pi\cong{\mathrm{triv}}\oplus\pi\oplus\rho,

where $\rho$ is another irreducible representation. Finally, we get one more from twisting $\rho$ by $\operatorname{sgn}$ .

\begin{array}[h]{c|rrrrrrr}&e&(12)&(12)(34)&(123)&(123)(45)&(1234)&(12345)\\ &1&10&15&20&20&30&24\\ \hline\cr\rho&5&1&1&-1&1&-1&0\\ \operatorname{sgn}\rho&5&-1&1&-1&-1&1&0\end{array}

This gives all of the irreducible characters, which we assemble into Table 3.1.

\begin{array}[]{c|ccccccc|}&e&(12)&(12)(34)&(123)&(123)(45)&(1234)&(12345)\\ &1&10&15&20&20&30&24\\ \hline\cr{\mathrm{triv}}&1&1&1&1&1&1&1\\ \operatorname{sgn}&1&-1&1&1&-1&-1&1\\ \pi&4&2&0&1&-1&0&-1\\ \operatorname{sgn}\pi&4&-2&0&1&1&0&-1\\ \pi\wedge\pi&6&0&-2&0&0&0&1\\ \rho&5&1&1&-1&1&-1&0\\ \operatorname{sgn}\rho&5&-1&1&-1&-1&1&0\\ \end{array}

Table 3.1. Character table of

S_{5}

3.12.3. Exercises

Problem 72. Let $(\pi,V)$ be a finite-dimensional representation of a finite group $G$ . Define $U_{1},U_{2}\subseteq V\otimes V$ by

U_{1}=\mathrm{span}\{{\bf v}_{1}\otimes{\bf v}_{2}-{\bf v}_{2}\otimes{\bf v}_{% 1}\,|\,{\bf v}_{1},{\bf v}_{2}\in V\},\quad U_{2}=\mathrm{span}\{{\bf v}_{1}% \otimes{\bf v}_{2}+{\bf v}_{2}\otimes{\bf v}_{1}\,|\,{\bf v}_{1},{\bf v}_{2}% \in V\}

Show that $(\pi\otimes\pi,U_{1})$ and $(\pi\otimes\pi,U_{2})$ are subrepresentations of $(\pi\otimes\pi,V\otimes V)$ and thus show that

(\pi\otimes\pi,V\otimes V)\cong\big{(}\mathrm{Sym}^{2}\pi,\mathrm{Sym}^{2}(V)% \big{)}\oplus\big{(}\pi\wedge\pi,\textstyle\bigwedge^{2}V\big{)}.

Problem 73. Let $(\pi,{\mathbb{C}}^{2})$ be a representation of a group $G$ . Suppose that the matrix of $\pi(g)$ with respect to the basis $\{{\bf b}_{1},{\bf b}_{2}\}$ is given by

\pi(g)=\left(\begin{matrix}a&b\\ c&d\end{matrix}\right).

(a)

Compute the matrix of $\mathrm{Sym}^{2}\pi(g)$ with respect to the basis $\{{\bf b}_{1}^{2},{\bf b}_{1}{\bf b}_{2},{\bf b}_{2}^{2}\}$ .
(b)

Compute the matrix of $\pi\wedge\pi(g)$ with respect to the basis $\{{\bf b}_{1}\wedge{\bf b}_{2}\}$ .

Problem 74. Let $(\sigma,V)$ be any irreducible 5-dimensional representation of $S_{5}$ . Decompose $(\operatorname{Sym}^{2}\sigma,\operatorname{Sym}^{2}V)$ and $(\sigma\wedge\sigma,\Lambda^{2}V)$ into irreducible representations.

Problem 75. Let $(\pi,{\mathbb{C}}^{4})$ be the permutation representation of $S_{4}$ , and let $(\rho,V)$ be the 2-dimensional irreducible representation of $S_{4}$ .

(a)

Show that $(\operatorname{Sym}^{2}\pi,\operatorname{Sym}^{2}{\mathbb{C}}^{4})$ has a unique subrepresentation isomorphic to $(\rho,V)$ .
(b)

Use the $\rho$ -isotopic projector to find that subrepresentation.

Problem 76. A group $G$ of order $168$ has conjugacy classes $C_{1}$ , $C_{2}$ , $C_{3}$ , $C_{4}$ , $C_{7A}$ and $C_{7B}$ , where each conjugacy class is labelled by the order of any element in that class (so, for example, any element of $C_{7A}$ or $C_{7B}$ has order $7$ ). The following shows one of the rows of the character table of $G$ .

\begin{array}[]{c|cccccc}\text{size:}&1&21&56&42&24&24\\ \text{class}&C_{1}&C_{2}&C_{3}&C_{4}&C_{7A}&C_{7B}\\ \hline\cr(\pi_{1},V)&3&-1&0&1&\frac{-1+\sqrt{-7}}{2}&\frac{-1-\sqrt{-7}}{2}\\ \end{array}

(a)

Show that if $x$ is an element of $C_{7A}$ or $C_{7B}$ , then $x$ is conjugate to $x^{2}$ .
(b)

Find the character table of $G$ .

3.12. Lecture 12

3.12.1. Symmetric and alternating products

Definition 3.12.1.

Definition 3.12.3.

Proposition 3.12.4.

Proof.

Proposition 3.12.5.

Proof.

3.12.2. Example: the character table of S5S_{5}

3.12.3. Exercises

3.12.2. Example: the character table of $S_{5}$