2.9. Lecture 9

2.9.1. Standard constructions for representations

We give a list of various constructions with representations of Lie groups, and the analogous constructions for their derivatives.

Definition 2.9.1.

The standard representation of a linear Lie group $G\subseteq\operatorname{GL}_{n}(\mathbb{C})$ comes from its action on $\mathbb{C}^{n}$ so for ${\bf v}\in\mathbb{C}^{n}$

	$\displaystyle\rho(g){\bf v}$	$\displaystyle=g{\bf v}$
	$\displaystyle D\rho(X){\bf v}$	$\displaystyle=X{\bf v}.$

Definition 2.9.2.

The direct sum of representations $(\rho_{1},V_{1})$ , $(\rho_{2},V_{2})$ of $G$ is $(\rho_{1}\oplus\rho_{2},V_{1}\oplus V_{2})$ with derivative

D(\rho_{1}\oplus\rho_{2})=D\rho_{1}\oplus D\rho_{2}.

Definition 2.9.3.

The determinant representation of $G\subseteq\operatorname{GL}_{n}(\mathbb{C})$ is $\det:G\to\mathbb{C}^{*}$ which sends $g$ to $\det(g)$ . We have

D\det(X)=\operatorname{tr}(X),

which follows from $\det\exp(tX)=e^{t\operatorname{tr}(X)}$ .

Definition 2.9.4.

If $(\rho,V)$ is a representation of $G$ , the dual representation $(\rho^{*},V^{*})$ of $(\rho,V)$ is defined by

\left(\rho^{*}(g)(\lambda)\right)({\bf v})=\lambda\left(\rho(g^{-1})({\bf v})% \right),

for $\lambda\in V^{*}$ a linear functional on $V$ and $g\in G$ . It has derivative

D\rho^{*}(X)(\lambda)({\bf v})=-\lambda(D\rho(X){\bf v}).

Given a basis of $V$ , then the matrix of $\rho^{*}$ with respect to the dual basis is

\rho^{*}(g)=\rho(g)^{-T},

which differentiates to

(D\rho^{*})(X)=-D\rho(X)^{T}.

We can take tensor/symmetric/alternating products of more than one factor.

Definition 2.9.5.

Suppose $(\rho_{i},V_{i})$ are representations of $G$ . We form the tensor product

V_{1}\otimes V_{2}\otimes\ldots\otimes V_{l}.

It is generated by symbols ${\bf v}_{1}\otimes\ldots\otimes{\bf v}_{l}$ subject to the multilinear relations, i.e.

{\bf v}_{1}\otimes\ldots\otimes(\lambda{\bf v}_{i}+\mu{\bf v}_{i}^{\prime})% \otimes\ldots\otimes{\bf v}_{l}=\lambda({\bf v}_{1}\otimes\ldots\otimes{\bf v}% _{i}\otimes\ldots\otimes{\bf v}_{l})+\mu({\bf v}_{1}\otimes\ldots\otimes{\bf v% }_{i}^{\prime}\otimes\ldots\otimes{\bf v}_{l}),

for each $V_{i}$ . One has

\dim\left(V_{1}\otimes V_{2}\otimes\ldots\otimes V_{l}\right)=\prod_{i=1}^{l}% \dim V_{i}.

The action of $G$ is

(\rho_{1}\otimes\ldots\otimes\rho_{l})(g)({\bf v}_{1}\otimes\ldots\otimes{\bf v% }_{l})=\rho_{1}(g){\bf v}_{1}\otimes\ldots\otimes\rho_{l}(g){\bf v}_{l}.

for $g\in G$ , ${\bf v}_{i}\in V_{i}$ . We also write

V^{\otimes l}=V\otimes\ldots\otimes V.

Lemma 2.9.6.

Suppose $(\rho_{1}\oplus\ldots\oplus\rho_{n},V_{1}\oplus\ldots\oplus V_{n})$ is a tensor product of representations of $G$ . Then the derivative of this representation is given by

	$\displaystyle D(\rho_{1}\otimes\ldots\otimes\rho_{l})(X)=$	$\displaystyle D\rho_{1}(X)\otimes\mathrm{Id}_{V_{2}}\otimes\ldots\otimes% \mathrm{Id}_{V_{l}}$
		$\displaystyle+\mathrm{Id}_{V_{1}}\otimes D\rho_{2}(X)\ldots\otimes\mathrm{Id}_% {V_{l}}$
		$\displaystyle+\ldots$
		$\displaystyle+\mathrm{Id}_{V_{1}}\otimes\mathrm{Id}_{V_{2}}\ldots\otimes D\rho% _{l},$

for all $X\in\mathfrak{g}$ .

Proof.

Suppose $(\pi,V)$ , $(\rho,W)$ are representations of $G$ . Let $X\in\mathfrak{g}$ , ${\bf v}\in V$ , ${\bf w}\in W$ ,then

	$\displaystyle D(\pi\otimes\rho)(X)({\bf v}\otimes{\bf w})$	$\displaystyle=\left.\frac{d}{dt}\pi(\exp(tX)){\bf v}\otimes\rho(\exp(tX)){\bf w% }\right\|_{t=0}.$
		$\displaystyle=\left.\frac{d}{dt}({\bf v}+t\pi(X){\bf v}+O(t^{2}))\otimes({\bf w% }+t\rho(X){\bf w}+O(t^{2}))\right\|_{t=0}$
		$\displaystyle=\left.\frac{d}{dt}{\bf v}\otimes{\bf w}+t(\pi(X){\bf v}\otimes{% \bf w}+{\bf v}\otimes\rho(X){\bf w})+O(t^{2})\right\|_{t=0}$
		$\displaystyle=\pi(X){\bf v}\otimes{\bf w}+{\bf v}\otimes\rho(X){\bf w}$

as required. We can then apply an inductive argument to show this hold for all tensor products. ∎

Definition 2.9.7.

Let $(\rho,V)$ be a representation of $G$ . The $l$ th symmetric product is the space $\operatorname{Sym}^{l}(V)$ generated by symbols ${\bf v}_{1}\ldots{\bf v}_{l}$ with linearity in each of the ${\bf v}_{i}$ s and any permutation of the vectors giving the same element.

We have

\dim\operatorname{Sym}^{l}(V)=\binom{n+l-1}{l}=\binom{n+l-1}{n-1}

where $\dim V=n$ . Indeed, if ${\bf e}_{1},\ldots,{\bf e}_{n}$ is a basis for $V$ then a basis for $\operatorname{Sym}^{l}(V)$ is

\{{\bf e}_{i_{1}}\ldots{\bf e}_{i_{l}}\,|\,1\leq i_{1}\leq i_{2}\leq\ldots\leq i% _{l}\leq n\}

from which finding the dimension is a simple counting problem.

The action of $G$ is

\operatorname{Sym}^{l}\rho(g)({\bf v}_{1}\ldots{\bf v}_{l})=(\rho(g){\bf v}_{1% })\ldots(\rho(g){\bf v}_{l}),

for all $g\in G$ , and all ${\bf v}_{1}\ldots{\bf v}_{l}\in\operatorname{Sym}^{l}(V)$ .

Lemma 2.9.8.

Suppose $(\operatorname{Sym}^{l}\rho,\operatorname{Sym}^{l}(V))$ is the $l$ th symmetric product of a representation of $G$ . Then the derivative of this representation is given by

D\operatorname{Sym}^{l}\rho(X)({\bf v}_{1}\ldots{\bf v}_{l})=(D\rho(X){\bf v}_% {1}){\bf v}_{2}\ldots{\bf v}_{l}+\ldots+{\bf v}_{1}\ldots{\bf v}_{l-1}(D\rho(X% ){\bf v}_{l}),

for all $X\in\mathfrak{g}$ , and all ${\bf v}_{1}\ldots{\bf v}_{l}\in\operatorname{Sym}^{l}(V)$ .

Proof.

Exercise. ∎

Definition 2.9.9.

The $l$ th alternating product is the space $\bigwedge^{l}(V)$ generated by symbols ${\bf v}_{1}\wedge\ldots\wedge{\bf v}_{l}$ with linearity in each entry and having the alternating property: for any permutation $\sigma\in S_{l}$ , we have

{\bf v}_{\sigma(1)}\wedge\ldots\wedge{\bf v}_{\sigma(l)}=\epsilon(\sigma)({\bf v% }_{1}\wedge\ldots\wedge{\bf v}_{l}).

In particular, switching the places of two components reverses the sign, while ${\bf v}_{1}\wedge\ldots\wedge{\bf v}_{l}=0$ if two of the vectors coincide (more generally, if they are linearly dependent). Thus if $l>\dim V$ we have $\bigwedge^{l}V=0$ .

We have

\dim\textstyle\bigwedge^{l}V=\binom{n}{l}

where $\dim V=n$ . Indeed, if ${\bf e}_{1},\ldots,{\bf e}_{n}$ are a basis for $V$ then a basis for $\bigwedge^{l}(V)$ is

\{{\bf e}_{i_{1}}\wedge\ldots\wedge{\bf e}_{i_{l}}\,|\,1\leq i_{1}<i_{2}<% \ldots<i_{l}\leq n\}

from which finding the dimension is a simple counting problem. In particular, $\bigwedge^{n}\mathbb{C}^{n}$ is one-dimensional generated by ${\bf e}_{1}\wedge\ldots\wedge{\bf e}_{n}$ .

The representation on $\bigwedge^{l}(V)$ is

\textstyle\bigwedge^{l}\rho(g)({\bf v}_{1}\wedge\ldots\wedge{\bf v}_{l})=\rho(% g){\bf v}_{1}\wedge\ldots\wedge\rho(g){\bf v}_{l},

for all $g\in G$ , and all ${\bf v}_{1}\wedge\ldots\wedge{\bf v}_{l}\in\bigwedge^{l}(V)$ .

Lemma 2.9.10.

Suppose $(\bigwedge^{l}\rho,\bigwedge^{l}(V))$ is the $l$ th alternating product of a representation of $G$ . Then the derivative of this representation is given by

D\textstyle\bigwedge^{l}\rho(X)({\bf v}_{1}\ldots{\bf v}_{l})=D\rho(X){\bf v}_% {1}\wedge{\bf v}_{2}\wedge\ldots\wedge{\bf v}_{l}+\ldots+{\bf v}_{1}\wedge% \ldots\wedge{\bf v}_{l-1}\wedge D\rho(X){\bf v}_{l}.

for all $g\in G$ , and all ${\bf v}_{1}\wedge\ldots\wedge{\bf v}_{l}\in\bigwedge^{l}(V)$ .

Proof.

Exercise. ∎

Remark 2.9.11.

We defined tensor products (and so on) of representations of Lie groups and then differentiated them. We could also directly make these definitions with Lie algebras. For instance, if $\mathfrak{g}$ is a Lie algebra and $(\pi,V)$ is a representation of $\mathfrak{g}$ , we define the symmetric square representation on $\operatorname{Sym}^{2}(V)$ by

\pi(X)({\bf v}{\bf w})=(\pi(X){\bf v}){\bf w}+{\bf v}(\pi(X){\bf w}).

Functional constructions

We can construct representations as vector spaces of functions on topological spaces with actions of $G$ . If $G$ acts on a set $X$ , then it also acts on the vector space of functions $X\rightarrow\mathbb{C}$ by $(g\cdot f)(x)=f(g^{-1}x)$ . Usually this will be infinite dimensional, and so out of the scope of our course, but sometimes we can impose conditions allowing us to handle it. For example, $\operatorname{GL}_{n}(\mathbb{C})$ acts on $\mathbb{C}^{n}$ , and hence on the space of polynomial functions in $n$ variables. Imposing a further restriction — to homogeneous polynomials of some fixed degree — gives a finite-dimensional representation. The derivative must be calculated on a case-by-case basis. We will see examples of this on the problem sheet.

2.9.2. Exercises

Problem 23. Prove Lemmas 2.9.8 and 2.9.10.

Problem 24.

(a)

If $(\rho,V)$ is an irreducible finite-dimensional complex representation of $\mathfrak{g}$ and $\mathfrak{z}$ is the centre of $\mathfrak{g}$ (see problem 1.7.3), show that there is a linear map $\alpha:\mathfrak{z}\rightarrow\mathbb{C}$ such that $\rho(Z){\bf v}=\alpha(Z){\bf v}$ for all $Z\in\mathfrak{z}$ .
(b)

For $\mathfrak{g}=\operatorname{GL}_{n,\mathbb{C}}$ , find $\mathfrak{z}$ . Find $\alpha$ when $V=\bigwedge^{k}\mathbb{C}^{n}$ , where $\mathbb{C}^{n}$ is the standard representation and $1\leq k\leq n$ . These representations are in fact irreducible, though we haven’t proved that yet; you can just directly show that $\alpha$ exists.

Problem 25. Let $V=\mathbb{C}^{2}$ be the standard representation of $\operatorname{GL}_{2}(\mathbb{C})$ .

(a)

Show that $\bigwedge^{2}(V)\cong\det$ as Lie group representations.
(b)

Show that $\bigwedge^{2}(V)\cong\operatorname{tr}$ as representations of $\mathfrak{gl}_{2}(\mathbb{C})$ . (You could just ‘take the derivative’ of part (a), but please do it directly instead.)
(c)

Find an explicit homomorphism $\rho:\operatorname{GL}_{2}(\mathbb{C})\rightarrow\operatorname{GL}_{3}(\mathbb% {C})$ corresponding to $\operatorname{Sym}^{2}(V)$ .

	$\displaystyle D(\pi\otimes\rho)(X)({\bf v}\otimes{\bf w})$	$\displaystyle=\left.\frac{d}{dt}\pi(\exp(tX)){\bf v}\otimes\rho(\exp(tX)){\bf w% }\right\|_{t=0}.$
		$\displaystyle=\left.\frac{d}{dt}({\bf v}+t\pi(X){\bf v}+O(t^{2}))\otimes({\bf w% }+t\rho(X){\bf w}+O(t^{2}))\right\|_{t=0}$
		$\displaystyle=\left.\frac{d}{dt}{\bf v}\otimes{\bf w}+t(\pi(X){\bf v}\otimes{% \bf w}+{\bf v}\otimes\rho(X){\bf w})+O(t^{2})\right\|_{t=0}$
		$\displaystyle=\pi(X){\bf v}\otimes{\bf w}+{\bf v}\otimes\rho(X){\bf w}$