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Ma424 Group Representations

Lecturer: Dr. Timothy Murphy

Website: Link

A page of theorems and proofs.

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Finite groups

Examples of finite groups:

The cyclic group $C_n = \mathbb{Z}/n = \lbrace e,a,a^2, \dots , a^{n-1} \rbrace$ such that $a^n=e$ .
The symmetric group $S_n$ consisting of permutations of $n$ elements. This group is generated by the transpositions.
The alternating group $A_n$ consisting of the even permutations in $S_n$ (a permutation is said to be even if it can be written as an even number of transpositions).
The dihedral group $D_n$ consisting of the symmetries of the regular $n$ -gon. This consists of $n$ rotations (forming a subgroup isomorphic to $C_n$ ) and $n$ reflections. The group can be written as

$D_n = \left \langle s,t : s^n = t^2 = 1, (st)^2 =1 \right \rangle$

Definition (Group representation): A representation of a group $G$ in a vector space $V$ over a field $k$ is defined by a homomorphism

$\alpha: G \rightarrow GL(V)$

from $G$ to the general linear group on $V$ - this is the group of all invertible linear maps on $V$ . Given a basis for $V$ we can express each operator in $GL(V)$ as a matrix, and the representation can be thought of as a homomorphism from $G$ to the group $GL(n,k)$ of invertible $n \times n$ matrices over the field $k$ , in which we give matrices $A(g)$ for all $g \in G$ , such that

$A(gh) = A(g) A(h) \qquad A(e) = \mathbb{I}$

Obviously this works best in practice for small matrices, as everyone knows it's impossible to multiply $4 \times 4$ matrices.

The one dimensional representation of $G$ is a homomorphism from $G$ to the field $k$ without the zero element (here $k = \mathbb{R}$ or $\mathbb{C}$ ). Note that is constant on conjugacy classes (one-dimensional multiplication commutes). The trivial representation is the representation defined by mapping every element of $G$ to $1$ .

Definition (Equivalent representations): Two representations $\alpha, \beta$ of a group $G$ in vector spaces $U,V$ are equivalent if we can find a linear map $t$ from $U$ to $V$ preserving the action of $G$ :

$t(gu) = g(tu) \forall g \in G, u \in U$

In matrix terms, suppose that

$\alpha: g \mapsto A(g) \quad \beta: g \mapsto B(g)$

Then the two representations are equivalent if there is an invertible matrix $P$ such that

$B(g) = PA(g)P^{-1} \quad \forall g \in G$

Definition (Stable): A subspace $U$ of a vector space $V$ is said to be stable under the action of a group $G$ if

$g \in G, u \in U \Rightarrow gu \in U$

Definition (Simple representation): A representation $\alpha$ of a group $G$ in a vector space $V$ over $k$ is said to be simple if no proper subspace of $V$ is stable under $G$ , i.e. the only stable subspaces are $0$ and $G$ itself.

All one dimensional representations are simple (as a one-dimensional space has no proper subspaces). A simple representation of a finite abelian group over $\mathbb{C}$ is one dimensional.

Representations can be added, multiplied (using the tensor product), and have a conjugacy operation.

Definition (Semisimple representation): A representation $\alpha$ of $G$ is said to be semisimple if it is expressible as a sum of simple representations.

If a vector space $V$ is the sum of simple subspaces then $V$ is semisimple. We also have that $V$ is semisimple if and only if each stable subspace $U$ of $V$ has at least one complementary stable subspace $W$ , meaning that $V = U \oplus W$ . (Proofs use sums of simple subspaces, relying on fact that $S_i \cap S_j = 0$ or $S_j$ for $S_j$ simple).

Maschke's Theorem: Every representation of a finite group $G$ over $\mathbb{R}$ or $\mathbb{C}$ is semisimple. (Proof uses an averaged positive definite form on the vector space to define a complement for a stable subspace $U$ ).

Definition (Intertwining number): The intertwining number $I(\alpha, \beta)$ of two representations $\alpha,\beta$ of $G$ over $k$ in vector spaces $U, V$ respectively is the dimension of space of $G$ -maps from $U$ to $V$ ,

$I(\alpha, \beta) = \mbox{dim}\,\mbox{hom}^G\,(U,V)$

If $\alpha$ and $\beta$ are simple representations, then their intertwining number is 0 if $\alpha \neq \beta$ , and $\geq 1$ if $\alpha = \beta$ . If $k=\mathbb{C}$ then $I(\alpha, \alpha)=1$ .

Definition (Character): The character $\chi = \chi_{\alpha}$ of a representation $\alpha$ of $G$ over $k$ is the function $\chi: G \rightarrow k$ defined by

$\chi(g) = \mbox{tr}\,\Big(\alpha(g)\Big)$

We have the following formula for the intertwining number of two representations:

$I(\alpha, \beta) = \frac{1}{||G||} \sum_{g\in G} \chi_{\alpha}(g^{-1}) \chi_{\beta} (g)$

Note that $\chi_{\alpha^*}(g) = \chi_{\alpha}(g^{-1}) = \overline{\chi_{\alpha}(g)}$ when $k=\mathbb{C}$ or $\mathbb{R}$ (obviously then the conjugation does nothing), and that $\alpha = \beta \Leftrightarrow \chi_{\alpha}(g) = \chi_{\beta}(g) \, \forall g \in G$ . Characters are also constant on conjugacy classes.

Definition (Regular representation): The regular representation of a group is the permutation representation defined by the action of the group on itself, $(g,x) \mapsto gx$ . The character of the regular representation is $||G||$ if $g=e$ and 0 otherwise. Every simple representation $\sigma$ occurs in the regular representation; when $k=\mathbb{C}$ they occur $\mbox{dim}\,\sigma$ times in the regular representation. From this we can derive that (when $k=\mathbb{C}$ ) the dimensions of the simple representations $\sigma_i$ satisfy

$\mbox{dim}^2\,\sigma_1 + \dots + \mbox{dim}^2\,\sigma_r = ||G||$

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Compact groups

Much of the theory of finite groups can be passed over to compact groups once we have established the existence of a unique measure on the group.

Definition (Topological group): A topological space $G$ with a group structure defined on it, such that the group operations $(x,y) \mapsto xy$ and $x \mapsto x^{-1}$ are continuous.

Definition (Hausdorff): A topological space $X$ is said to be Hausdorff if given any two points $x,y \in X$ there exist open sets $U,V$ in $X$ such that

$x \in U, y \in V, U \cap V = \emptyset$

Definition (Compact): A space $X$ is said to be compact if it is Hausdorff and every open covering

$X = \bigcup_{i\in I} U_i$

has a finite subcovering, $X=U_{i_1} \cup \dots \cup U_{i_r}$

Note that a subspace $X$ of Euclidean space $E^n$ is compact iff $X$ is closed and bounded.

Examples of compact groups:

The orthogonal group $O(n)$ of real $n\times n$ matrices $T$ such that $T^tT=\mathbb{I}$
The special orthogonal group of $SO(n)$ of real $n\times n$ matrices $T$ such that $T^tT=\mathbb{I}$ and $\det T = 1$
The unitary and special unitary groups, whose definitions mirror the above, except for complex matrices such that $T^{\dagger}T = \mathbb{I}$ .
The symplectic group of quaternionic matrices whose conjugate transposes are their inverses.

We can define integration on a compact group using a measure.

Definition (Measure): A measure $\mu$ on $X$ is a continuous linear functional $\mu: C(X,k) \rightarrow k$ , where $C(X,k)$ is the vector space of functions $f:X\rightarrow k$ . We write

$\mu(f) = \int_X f d \mu$

Haar's theorem (for compact groups): Let $G$ be a compact group. Then there exists a unique real measure $\mu$ on $G$ such that

$\mu$ is invariant under $G$ , $\int_G (gf) d \mu = \int_G f d \mu$ , for all $g \in G, f \in C(G,\mathbb{R})$ .
$\mu$ is normalised such that $G$ has volume 1, i.e. $\int_G 1 d \mu = 1$ .

and moreover

the measure is strictly positive: $f(x) \geq 0 \, \forall x \Rightarrow \int f d \mu \geq 0$ with equality iff $f(g) = 0 \, \forall g$
$\left| \int_G f d \mu \right| \leq | f |$