3.11. Lecture 11

We now turn to various ways of constructing new representations out of old ones. The methods will split into two general classes:

(i)

Linear algebraic
(ii)

Group theoretic

We start with linear algebra:

3.11.1. The dual representation

Given a vector space $V$ , recall that $V^{*}$ (the dual space of $V$ ) is the vector space consisting of all linear functionals from $V$ to ${\mathbb{C}}$ , i.e. $V^{*}=\operatorname{Hom}(V,{\mathbb{C}})$ . We define a representation $(\pi^{*},V^{*})$ of $G$ by letting (for every $g\in G$ and $\lambda\in V^{*}$ ) $\pi^{*}(g)\lambda$ be the functional defined via the formula

[\pi^{*}(g)\lambda]({\bf v}):=\lambda\big{(}\pi(g^{-1}){\bf v}\big{)}\qquad% \forall{\bf v}\in V.

Note that this may be seen as a special case of some representations we have seen before:

(i)

$(\pi^{*},V^{*})$ is a subrepresentation of the permutation representation on the space of ${\mathbb{C}}$ -valued functions on $V$ that arises from the linear $\pi$ -action of $G$ on $V$ .
(ii)

Recall the representation $(c_{\pi}^{\rho},\operatorname{Hom}(V,W))$ (cf. Lemma 2.9.1). The dual representation is the special case $(\rho,W)=({\mathrm{triv}},{\mathbb{C}})$ .

Being a special case of $c_{\pi}^{\rho}$ , Lemma 2.9.2 (b) gives $\chi_{\pi^{*}}=\overline{\chi_{\pi}}$ . Problem 3.11.4 below gives another proof of this fact.

3.11.2. Quotient representations

Let $(\pi,V)$ be a representation of a group $G$ and $(\pi|_{W},W)$ a subrepresentation of $(\pi,V)$ . Recall that the quotient space $V/W$ is the vector space $V$ modulo the equivalence relation ${\bf v}\sim{\bf w}\Leftrightarrow{\bf v}-{\bf w}\in W$ . We normally write elements of $V/W$ in the form ${\bf v}+W$ . We define a representation $(\hat{\pi},V/W)$ of $G$ on $V/W$ via the formula $\hat{\pi}(g)({\bf v}+W):=\pi(g){\bf v}+W$ . Note that it is critical that $(\pi,W)$ is a subrepresentation for this to make sense (i.e. $\pi(g)W=W$ ).

The following proposition will let us avoid using quotient representations too often:

Proposition 3.11.1.

If $(\pi,V)=(\pi|_{W},W)\oplus(\pi|_{W^{\prime}},W^{\prime})$ , then

(\hat{\pi},V/W)\cong(\pi|_{W^{\prime}},W^{\prime}).

As a consequence of the above, we have

Proposition 3.11.2.

- Let $(\pi,V)$ be a finite dimensional representation of a finite group $G$ and $(\pi|_{W},W)$ a subrepresentation with corresponding quotient representation $(\hat{\pi},V/W)$ . Then

\chi_{\hat{\pi}}=\chi_{\pi}-\chi_{\pi|_{W}}.

The proofs of both these results are left as exercises below.

We won’t need to use quotient representations too often, but they are required to make what we do next rigorous.

3.11.3. Tensor Products

We now define the tensor product of two representations; this is a way of ‘multiplying’ two or more representations. Throughout this section, let $(\pi,V)$ and $(\rho,W)$ be two representations of a group $G$ .

Firstly, we consider the permutation representation of $G$ on the free vector space ${\mathbb{C}}(V\times W)$ ; we write this as $\pi\otimes\rho$ , i.e. the elements of ${\mathbb{C}}(V\times W)$ are finite formal sums

\sum_{({\bf v},{\bf w})\in V\times W}z_{({\bf v},{\bf w})}({\bf v},{\bf w}),

for $z_{({\bf v},{\bf w})}\in{\mathbb{C}}$ , where all but finitely many $z_{({\bf v},{\bf w})}$ are non-zero. Note that since $V\times W$ is an infinite set, ${\mathbb{C}}(V\times W)$ is an infinite-dimensional vector space, with basis given by all possible pairs $({\bf v},{\bf w})\in V\times W$ . The group $G$ acts via the formula

[\pi\otimes\rho](g)\left(\sum_{({\bf v},{\bf w})\in V\times W}z_{({\bf v},{\bf w% })}({\bf v},{\bf w})\right)=\sum_{({\bf v},{\bf w})\in V\times W}z_{({\bf v},{% \bf w})}\big{(}\pi(g){\bf v},\rho(g){\bf w}\big{)}.

This may seem like a sensibly defined representation however note that in ${\mathbb{C}}(V\times W)$ , the vectors $({\bf v}_{1}+{\bf v}_{2},{\bf w})$ and $({\bf v}_{1},{\bf w})+({\bf v}_{2},{\bf w})$ are different. In fact we don not have any relations that allow us to split up brackets or pull out scalar factors. We therefore define the vector space

	$\displaystyle U=\mathrm{span}\bigg{\{}(\alpha{\bf v}_{1}+\beta{\bf v}_{2}$	$\displaystyle,\lambda{\bf w}_{1}+\mu{\bf w}_{2})-\alpha\lambda({\bf v}_{1},{% \bf w}_{1})-\alpha\mu({\bf v}_{1},{\bf w}_{2})$
	$\displaystyle-$	$\displaystyle\beta\lambda({\bf v}_{2},{\bf w}_{1})-\beta\mu({\bf u}_{2},{\bf w% }_{2})\,\|\,{\bf v}_{1},{\bf v}_{2}\in V,\;{\bf w}_{1},{\bf w}_{2}\in W,\;% \alpha,\beta,\lambda,\mu\in{\mathbb{C}}\bigg{\}}.$

We claim that $(\pi\otimes\rho,U)$ is a subrepresentation of $\big{(}\pi\otimes\rho,{\mathbb{C}}(V\times W)\big{)}$ , and define the tensor product representation as the quotient representation

\big{(}\pi\otimes\rho,V\otimes W\big{)}:=\big{(}\pi\otimes\rho,{\mathbb{C}}(V% \times W)/U\big{)}.

Taking the quotient space by $U$ allows us to identify vectors that differ by elements of $U$ . If we consider our example notice that they differ by $({\bf v}_{1}+{\bf v}_{2},{\bf w})-({\bf v}_{1},{\bf w})-({\bf v}_{2},{\bf w})$ which is one of the spanning elements of $U$ , hence

({\bf v}_{1}+{\bf v}_{2},{\bf w})+U=({\bf v}_{1},{\bf w})+({\bf v}_{2},{\bf w}% )+U

as elements of $V\otimes W$ . The same holds for the other linear relations in both components of ${\mathbb{C}}(V\times W)$ . In other words, modding out the subspace $U$ is a way of forcing desired linear relations on the space ${\mathbb{C}}(V\times W)$ .

If ${\bf v}\in V$ and ${\bf w}\in W$ , then we write ${\bf v}\otimes{\bf w}$ for the element $({\bf v},{\bf w})+U$ in $V\otimes W$ .

Important: Not every element of $V\times W$ may be written in the form ${\bf v}\otimes{\bf w}$ (see Problem 3.11.8 below). However, the vectors ${\bf v}\otimes{\bf w}$ do span $V\otimes W$ , and the following proposition gives a convenient way of finding a basis from this spanning set:

Proposition 3.11.3.

Let $V$ and $W$ be finite-dimensional vector spaces, with ${\bf v}_{1},\ldots,{\bf v}_{n}$ being a basis of $V$ and ${\bf w}_{1},\ldots,{\bf w}_{m}$ being a basis of $W$ . Then

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

is a basis of $V\otimes W$ . In particular, $\operatorname{dim}(V\otimes W)=\operatorname{dim}(V)\,\operatorname{dim}(W)$ .

Example 3.11.4.

The above basically says ${\mathbb{C}}^{n}\otimes{\mathbb{C}}^{m}\cong{\mathbb{C}}^{nm}$ .

Remark 3.11.5.

An element in $V\otimes W$ of the form ${\bf v}\otimes{\bf w}$ ( ${\bf v}\in V$ , ${\bf w}\in W$ ) is called a pure tensor.

The previous proposition can be used to find the characters of tensor products of representations:

Proposition 3.11.6.

$\chi_{\pi\otimes\rho}=\chi_{\pi}\chi_{\rho}$

Proof.

Let ${\bf v}_{1},\ldots,{\bf v}_{n}$ be an eigenbasis of $V$ for $\pi(g)$ ( $\pi(g){\bf v}_{i}=\lambda_{i}{\bf v}_{i}$ ) and ${\bf w}_{1},\ldots,{\bf v}_{m}$ be an eigenbasis of $W$ for $\rho(g)$ $\rho(g){\bf w}_{j}=\mu_{j}{\bf w}_{j}$ ). Then from the Proposition 3.11.3, $\{{\bf v}_{i}\otimes{\bf w}_{j}\}$ is a basis of $V\otimes W$ , and is in fact an eigenbasis for $[\pi\otimes\rho](g)$ , since

[\pi\otimes\rho](g){\bf v}_{i}\otimes{\bf w}_{j}=\big{(}\pi(g){\bf v}_{i}\big{% )}\otimes\big{(}\rho(g){\bf w}_{j}\big{)}=(\lambda_{i}{\bf v}_{i})\otimes(\mu_% {j}{\bf w}_{j})=\lambda_{i}\mu_{j}({\bf v}_{i}\otimes{\bf w}_{j}).

This gives $\chi_{\pi\otimes\rho}(g)=\sum_{ij}\lambda_{i}\mu_{j}=\left(\sum_{i}\lambda_{i}% \right)\left(\sum_{j}\mu_{j}\right)=\chi_{\pi}(g)\chi_{\mu}(g)$ . ∎

Example 3.11.7.

From Lemma 2.9.2 (b), we have that the character of $(c_{\pi}^{\rho},\operatorname{Hom}(V,W))$ is $\chi_{\rho}\overline{\chi_{\pi}}$ . From Proposition 3.11.6 and Problem 3.11.4, we then get

\big{(}c_{\pi}^{\rho},\operatorname{Hom}(V,W)\big{)}\cong(\rho,W)\otimes(\pi^{% *},V^{*}).

3.11.4. Exercises

Problem 64. Let $(\pi,V)$ be a finite-dimensional representation of a finite group $G$ with dual representation $(\pi^{*},V^{*})$ .

(a)

Let ${\mathcal{B}}=\{{\bf v}_{i}\}_{i=1,\ldots,\operatorname{dim}V}$ be a basis of $V$ and ${\mathcal{A}}=\{\lambda_{i}\}_{i=1,\ldots,\operatorname{dim}V^{*}}$ the corresponding dual basis of $V^{*}$ , i.e.

$\lambda_{i}({\bf v}_{j})=\begin{cases}1\quad&\mathrm{if}\;i=j,\\ 0\quad&\mathrm{if}\;i\neq j.\end{cases}$

Show that $[\pi^{*}(g)]_{{\mathcal{A}}}=[\pi(g^{-1})]_{{\mathcal{B}}}^{T}$ and $\chi_{\pi^{*}}=\overline{\chi_{\pi}}$ .
(b)

Show that $(\pi^{*},V^{*})$ is irreducible if and only if $(\pi,V)$ is.

Problem 65. Let $(\pi,V)$ be a representation of a group $G$ and $(\pi,W)$ a subrepresentation of $(\pi,V)$ . Verify that defining

\pi(g)({\bf v}+W):=\pi(g){\bf v}+W\qquad\forall\;{\bf v}+W\in V/W,\;g\in G

gives a $G$ -representation $(\pi,V/W)$ .

Problem 66. Prove Propositions 3.11.1 and 3.11.2.

Problem 67. Consider the representation of $S_{3}$ on $(\rho,V)$ , where $V={\mathbb{C}}^{2}$ , given by

\rho(1\;2)=\begin{pmatrix}0&1\\ 1&0\end{pmatrix},\qquad\rho(1\;2\;3)=\begin{pmatrix}\omega&0\\ 0&\omega^{2}\end{pmatrix}

with $\omega=e^{2\pi i/3}$ .

(a)

Write down the matrices of $(\rho\otimes\rho)(1\;2)$ and $(\rho\otimes\rho)(1\;2\;3)$ with respect to the basis

${\bf e}_{1}\otimes{\bf e}_{1},{\bf e}_{1}\otimes{\bf e}_{2},{\bf e}_{2}\otimes% {\bf e}_{1},{\bf e}_{2}\otimes{\bf e}_{2}$

of $V\otimes V$ (where ${\bf e}_{1}$ and ${\bf e}_{2}$ are the standard basis of $V$ ).
(b)

Write the character of $(\rho,V)\otimes(\rho,V)$ as a sum of irreducible characters.
(c)

For each of the irreducible characters occurring in the expression of $\chi_{\rho\otimes\rho}$ used in the previous part, find a subrepresentation of $V\otimes V$ with that character.
(d)

Find an isomorphism from $(\mathrm{sgn},{\mathbb{C}})\otimes(\rho,V)$ to $(\rho,V)$ .
(e)

Find an isomorphism from $(\rho^{*},V^{*})$ to $(\rho,V)$ .
(f)

Let $(\rho^{\otimes n},V^{\otimes n})=(\rho,V)\otimes\ldots\otimes(\rho,V)$ with $n$ factors. Decompose $(\rho^{\otimes n},V^{\otimes n})$ into irreducible representations.

Problem 68. Let $(\pi,V)$ , $(\rho,U)$ , and $(\sigma,W)$ be three finite-dimensional representations of a finite group $G$ . Show that

(a)

$\big{(}(\pi,V)\otimes(\rho,U)\big{)}\otimes(\sigma,W)\cong(\pi,V)\otimes\big{(% }(\rho,U)\otimes(\sigma,W)\big{)}$
(b)

$\big{(}(\pi,V)\oplus(\rho,U)\big{)}\otimes(\sigma,W)\cong\big{(}(\rho,U)% \otimes(\sigma,W)\big{)}\oplus\big{(}(\pi,V)\otimes(\sigma,W)\big{)}$

Remark 3.11.8.

Note that (a) lets us write $(\pi,V)\otimes(\rho,U)\otimes(\sigma,W)$ unambiguously (and similarly for larger products of representations).

Problem 69. Let ${\bf e},{\bf f}$ form a basis of ${\mathbb{C}}^{2}$ . Show that ${\bf e}\otimes{\bf e}+{\bf f}\otimes{\bf f}\in{\mathbb{C}}^{2}\otimes{\mathbb{% C}}^{2}$ cannot be written in the form ${\bf v}\otimes{\bf w}\in{\mathbb{C}}^{2}\otimes{\mathbb{C}}^{2}$ .

Problem 70. For two finite-dimensional representations $(\pi,V)$ and $(\rho,W)$ of a finite group $G$ , construct an isomorphism to show that

\big{(}c_{\pi}^{\rho},\operatorname{Hom}(V,W)\big{)}\cong(\pi^{*},V^{*})% \otimes(\rho,W).

Problem 71. Given a finite group $G$ , let $\big{(}\pi,{\mathbb{C}}(G)\big{)}$ denote the permutation representation of $G$ on ${\mathbb{C}}(G)$ associated to the conjugation action of $G$ on itself, i.e.

\pi(g)\left(\sum_{h\in G}z_{h}h\right)=\sum_{h\in G}z_{h}ghg^{-1}=\sum_{h\in G% }z_{g^{-1}hg}h\qquad\forall g\in G,\;\sum_{h\in G}z_{h}h\in{\mathbb{C}}(G).

(a)

Show that $\chi_{\pi}(g)=\frac{|G|}{|{\mathcal{C}}_{g}|}$ , where ${\mathcal{C}}_{g}$ is the conjugacy class $g$ belongs to. Hint: use Problem 2.7.2.
(b)

Then show that

$\big{(}\pi,{\mathbb{C}}(G)\big{)}\cong\bigoplus_{\rho\in\mathrm{Irr}(G)}(\rho,% W_{\rho})\otimes(\rho^{*},W^{*}_{\rho}).$