Representations of Compact Groups (Part 1)

I've written a previous post on representation theory for finite groups. The representation theory of finite groups is very nice, but many of the groups whose representations we care about are not finite. For example, representations of $SU(2)$ are important for understanding the behavior of particles with nonzero spin. So we want to extend representation theory to more general groups. A nice family of groups to consider are compact groups. In many ways, compactness is a generalization of finiteness. To use an example from that link, every real-valued function on a finite set is bounded and attains its maximum. This is untrue for real-valued functions on infinite sets: consider the functions $f(x) = \tan x$ and $f(x) = x$ respectively on the interval $(0, \frac\pi 2)$. However, continuous real-valued functions on a compact interval must be bounded and must attain their maxima. Similarly, compact groups generalize finite groups, and many of the nice features of the representation theory of finite groups extend to the representation theory of compact groups. This post will mostly follow the notes about representations of compact groups available here

Compact Topological Groups

First, we will start with some nice properties of compact topological groups. Recall that a topological group is a group endowed with a topology so that multiplication and the inverse map are continuous. When studying the representation theory of finite groups, it was often convenient to sum over the elements of the group (e.g. to define our inner product on the space of characters). Clearly we cannot always sum over the elements of an infinite group. But for compact groups, we have a nice theorem that tells us that we can integrate over the group instead, which is just as good.

(Haar measure) Let $G$ be a locally compact topological group. There exists a non-zero left-$G$-invariant measure on $G$. This measure is nonvanishing and unique up to positive scalar multiplication

This theorem is tricky to prove for locally compact topological groups. But for Lie groups, it is fairly easy. So we will just show a version of the theorem for Lie groups.

Let $G$ be a Lie group. Then $G$ has a left-$G$-invariant measure

First, we will show existence. Let $n$ denote the dimension of $G$. Recall that as long as $G$ is oriented, an $n$-form on $G$ induces a measure. Furthermore, we recall that if we can find a nonvanishing $n$-form on $G$, then $G$ must be orientable. So it is sufficient to find a left-invariant nonvanishing $n$-form on $G$. Let $\Lambda^nT_eG$ denote the space of $k$-covectors on the tangent space to the identity of $G$. Pick any nonzero $\omega_e \in \Lambda^nT_eG$. Now, we can extend $\omega_e$ to a differential form on $G$. Let $L_g$ denote the automorphism of $g$ given by left-multiplication by $g$. This is continuous. $L_{g^{-1}}$ sends $g$ to $e$, so we can pull $\omega_e$ back along this map to define $\omega_g = (L_{g^{-1}})^* \in \Lambda^n T_gG$. This defines a differential $n$-form $\omega \in \Omega^n(G)$ on all of $G$. $\omega$ is left-invariant by construction. \[((L_h)^* \omega)_g = (L_h)^* \omega_{hg} = (L_h)^* (L_{(hg)^{-1}})^* \omega_e = (L_{(hg^{-1})h})^* \omega_e =(L_{g^{-1}})^* \omega_e = \omega_g\] Clearly this differential form is nonvanishing. And by negating $\omega$ if necessary, we see that $\omega$ is positive with respect to $G$'s orientation, so it defines a left-invariant measure on $G$.

Now, you might be wondering why it is important that $G$ is compact, because the above theorems don't require compactness. The nice thing about compactness is that measures only let us integrate functions with compact support - but if $G$ is compact, then every function has compact support. So we can integrate any real- (or complex-) valued functions on $G$. We will write the integral of $f$ with respect to the Haar measure as $\int_G f(g)\;dg$.

In particular, we can integrate the constant function $f(g) = 1$ over compact groups. It is convenient to normalize our Haar measure so that $\int_G 1 \;dg = 1$. I will assume that all Haar measures are normalized in this way.

From now on, I'll assume that all groups are compact Lie groups unless I explicitly state otherwise.

Basic Definitions

We'll start with a whole bunch of definitions. They're essentially the same as the analogous definitions for finite groups, except we require that our maps are continuous. To do so, we have to put topologies on the vector spaces involved.

A topological vector space is a vector space endowed with a Hausdorff topology such that addition and scalar multiplication are continuous.
It turns out that a finite dimensional vector space has a unique topology which turns it into a topological vector space. $\R^n$ naturally has a product topology from the standard topology on $\R^n$, and any linear isomorphism $V \cong \R^n$ lets us transfer this topology to $V$.
Suppose $V$ has an inner product $\inrp \cdot \cdot$. This gives us a norm $\|v\| = \sqrt{\inrp vv}$, which in turn gives us a metric $d(v,w) = \|v-w\|$, and thus a topology. This topology makes $V$ a topological vector space. If $V$ is complete with respect to this metric, we call $V$ a Hilbert space.
Given a topological vector space $V$, the automorphism group of $V$, denoted by $\Aut(V)$ (or $GL(V)$ if we are working with a basis) is the group of continuous linear maps $V \to V$ with continuous inverses.
A representation of a topological group $G$ is a pair $(V, \phi)$ where $V$ is a vector space and $\phi$ is a continuous homomorphism $\phi:G \to \Aut(V)$. Frequently, we will write $\phi(g)(v)$ as $g \cdot v$. Also, we will frequently refer to the representation as $V$, leaving the group action implicit. All of the representations I write about today will be assumed to be finite-dimensional unless specified otherwise.
A morphism or $G$-linear map between representations $V$ and $W$ is a continuous linear map $A:V \to W$ which commutes with the group action on $V$ and $W$ (i.e. $g(Av) = A(gv)$). We denote the set of $G$-linear maps between $V$ and $W$ by $\Hom_G(V,W)$. We will sometimes write $\End_G(V)$ for $\Hom_G(V,V)$. The set of finite-dimensional representations of $G$ together with the $G$-linear morphisms form a category.
An isomorphism is a $G$-linear map with a $G$-linear inverse.
Unless specified otherwise, all functions will be assumed to be continuous.

Useful Constructions

A subrepresentation of a representation $(V, \phi)$ is a linear subspace $W \subseteq V$ which is invariant under the action of $G$. This defines a representaiton $(W, \phi|_W)$.

There are several simple subrepresentations we can consider.

  1. For any representation $V$, $\{0\} \subseteq V$ is a subrepresentation because $g \cdot 0 = 0$.
  2. Similarly, $V$ is a subrepresentation of itself.
  3. We also have a subrepresentation $V^G = \{v \in V\;|\; g\cdot v = v\}$, the subspace of $G$-invariants. Note that the action of $G$ on $V^G$ is trivial.
  4. Given any $G$-linear map $A \in \Hom_G(V,W)$, the kernel is a subrepresentation of $A$ and the image is a subrepresentation of $W$.

Given two representations $(V, \phi)$ and $(W, \psi)$, there are several ways we can build new representations out of them.

  1. We can define a representation of $G$ on the dual space $V^* = \Hom(V, k)$ (where $k$ is the base field) by setting $g(A)(v) = A(g^{-1}v)$ for $A \in \Hom(V,k)$.
  2. We can define a representation of $G$ on the conjugate space $\overline V$. We define $\overline V$ as follows: it is the same topological abelian group as $V$, but the scalar multiplication is changed. Let $v$ denote an element of $V$ and $\overline v$ denote the corresponding element of $\overline V$. Then we set $\lambda \overline v = \overline{\overline \lambda v}$. That is to say, we scalar multiply by the conjugate of $\lambda$ instead of by $\lambda$ itself. The action of $G$ on $\overline V$ is the same as the action of $G$ on $V$.
  3. We can define a representation of $G$ on $V \oplus W$ by setting $g(v,w) = (gv, gw)$.
  4. We can define a representation of $G$ on $V \otimes W$ by setting $g(v \otimes w) = (gv) \otimes (gw)$.
  5. We can define a representation of $G$ on $\Hom(V,W)$ by using the isomorphism $\Hom(V,W) \cong W \otimes V^*$ for finite-dimensional $W,V$ and using our constructions for taking tensor products and duals of representations.
We can write down an explicit formula for the action of $G$ on $\Hom(V,W)$. Let $\{e_i\}$ be a basis for $W$ and $\{f_j\}$ be a basis for $V$. Let $\{f^j\}$ be the corresponding dual basis of $V^*$. Using the definition of the dual representation, $G$ acts on $V^*$ by the formula $g\cdot(f^j)(v) = f^j(g^{-1} \cdot v)$. Therefore, \[(g \cdot (e_i \otimes f^j))(v) = (g\cdot e_i) \otimes (f^j(g^{-1} \cdot v))\]

Note that $f^j(g^{-1}v)$ is a scalar and $ge_i$ is a vector in $W$. So this is just $f^j(g^{-1}v)(ge_i)$. Since $g$ acts by a linear map, we can factor out the $g$ to obtain \[(g \cdot (e_i \otimes f^j))(v) = g \cdot (f^j(g^{-1}v) e_i) = g \cdot ((e_i \otimes f^j)(g^{-1}v))\] So given any $A \in \Hom(V,W)$, we have $(g \cdot A)(v) = g\cdot A(g^{-1}v)$.


$\Hom_G(V,W) = \Hom(V,W)^G$ where the left hand side is the space of $G$-linear maps, and the right hand side is the subspace of invariants of the representation $\Hom(V,W)$ as defined above.

First, suppose that $A \in \Hom(V,W)^G$. Then $g \cdot A = A$, so in particular we have $(g \cdot A)v = Av$ for any $v \in V$. Using the formula for $g \cdot A$, we see that $g \cdot A(g^{-1} v) = Av$ for all $g \in G, v \in V$. Multiplying both sides by $g^{-1}$, we find that $A(g^{-1}v) = g^{-1} Av$. Since this is true for all $g \in G$, we conclude that $A$ is $G$-linear. So $A \in \Hom_G(V,W)$.

Conversely, suppose that $A \in \Hom_G(V,W)$. Then $A(gv) = g(Av)$ for all $g \in G, v \in V$. So $g^{-1}A(gv) = Av$. Letting $h = g^{-1}$, we see that $g A (h^{-1}v) = Av$ for all $h \in G, v \in V$. So $A$ is in the subspace of invariants $\Hom(V,W)^G$.

Complete Reducibility and Schur's Lemma

An irreducible representation, or an irrep is a representation $V$ whose only two subrepresentations are $\{0\}$ and $V$ itself.
A representation is called completely reducible if it is a direct sum of irreps.
Some representations are neither irreducible nor completely reducible. Consider the set of upper triangular $2 \times 2$ matrices \[G = \left\{\left.\begin{pmatrix} 1 & x \\ 0 & 1\end{pmatrix}\;\right|\; a\in \R\right\}\]

These all have determinant one, and are thus invertible. Furthermore, the product of two upper-triangular matrices is an upper-triangular matrix, so this is a group. This group has a natural action on $\R^2$ given by the usual matrix-vector product. This defines a representation of $G$ on $\R^2$.

Note that this representation fixes the subspace $V \subseteq \R^2$ given by

\[V = \left\{\left. \begin{pmatrix}\lambda\\0\end{pmatrix}\;\right|\;\lambda\in\R\right\}\]

But it doesn't fix any other nontrivial subspaces. So $\R^2$ is neither an irreducible representation nor a completely reducible representation of $G$.

It's kind of frustrating that not all representations are completely reducible. One of the nice features of finite groups is that all representations of finite groups are completely reducible. We will show that compact groups are nice in this way as well- all representations of compact groups are completely reducible as well.

(Schur's Lemma)
  1. Let $V, W$ be irreps of $G$. Let $A \in \Hom_G(V,W)$. Then $A$ is either 0 or an isomorphism.
  2. Let $V$ be a complex irrep of $G$. Then $\End_G(V) = \C \cdot \Id_V$ (i.e. any $G$-linear endomorphism of $V$ is a scalar multiple of the identity)
  1. Since $A$ is $G$-linear, we know that $\ker A, \im A$ are subrepresentations. Since $V,W$ are irreps, this implies that $\ker A$ is either $0$ or all of $V$, and $\im A$ is either $0$ or all of $W$. Thus, the only way for $A$ to be nonzero is if $\ker A = 0$ and $\im A = W$. This means that if $A$ is nonzero, it must be an isomorphism.
  2. Since $A$ is a complex matrix, it has an eigenvalue $\lambda$. Clearly $\lambda \Id$ is a $G$-linear endomorphism of $V$. Thus, $A - \lambda \Id \in \Hom_G(V,W)$. But $A-\lambda \Id$ cannot be an isomorphism. So it must be $0$. Thus, $A = \lambda \Id$.
Let $(V, \phi)$ be a representation. There exists a unique projection $Av_G \in \End_G(V)$ onto $V^G$.

First, we will construct one such projection. Explicitly, we define \[Av_G(v) := \int_G g \cdot v \;dg\] This operation averages over the group action, which is why we named the projection $Av_G$. To show that $Av_G$ is a projection, we have to show that it restricts to the identity on its image. First, we note that the image of $Av_G$ is simply $V^G$. We see that $\im Av_G$ is contained in $V^G$. Let $v$ be any vector in $V$. Then for any $h \in G$, we have \[h \cdot Av_G(v) = h \cdot \int_G g \cdot v \;dg = \int_G (hg) \cdot v\;dg = \int_G (hg)\cdot v \;d(hg)\] the last equality follows from the left-invariance of our measure. So we see that $h \cdot Av_G(v) = Av_G(v)$, which implies that $\im Av_G \subseteq V^G$.

Furthermore, any vector of $V^G$ is itself fixed by $Av_G$. If $v \in V^G$, then $Av_G(v) = \int_G g \cdot v\;dg = \int_G v\;dg = v$. So in particular, $v \in \im Av_G$. Thus, we see that $\im Av_G = V^G$, and $Av_G$ acts as the identity on its image. So it is a projection.

Now, we will show uniqueness. First, note that $Av_G$ commutes with any other $T \in \End_G(V)$. \[Av_G \circ T(v) = \int_G g \cdot (Tv)\;dg = \int_G T (g\cdot v) \;dg = T \int_G g \cdot v\;dg = T \circ Av_G (v)\] Suppose that $P$ is another projection onto $V^G$. In particular, it is an element of $\End_G(V)$, so it commutes with $Av_G$. Thus, \[P = Av_G \circ P = P \circ Av_G = Av_G\]

Let $V$ be a representation of $G$. Then there exists a $G$-invariant inner product on $V$.

Let $\inrp \cdot \cdot$ be any hermitian inner product on $V$. We can view $\inrp \cdot \cdot$ as an element of $\Hom(\overline V \otimes V, \C) \cong \Hom(\overline V, V^*)$. We can think of the $G$-invariant inner products as elements of $\Hom(\overline V, V^*)^G$. So $Av_G \inrp \cdot \cdot$ gives us a $G$-invariant inner product.

Explicitly, this just means that we can define a $G$-invariant inner product $\inrp \cdot \cdot _G$ by the formula \[\inrp v w_G := \int_G \inrp {gv}{gw}\;dg\]

If $V$ is endowed with a $G$-invariant inner product, then we call the representation unitary, since for every $g \in G$, $\phi(g)$ is a unitary operator (or orthogonal if $V$ is a real vector space). So the above lemma says that for any representation $V$, there is an inner product on $V$ such that our representation is unitary. This perspective will be useful later.
(Maschke) Let $V$ be a representation of $G$ and let $W \subseteq V$ be a subrepresentation. Then there exists a subrepresentation $U \subseteq V$ such that $V = W \oplus U$.

Let $\inrp \cdot \cdot$ be a $G$-invariant inner product on $V$. Let $U = W^\perp$.

We note that $U$ is a subrepresentation of $V$. Let $u \in U$. By definition, $\inrp u w = 0$ for all $w \in W$. Since the inner product is $G$-invariant, $\inrp {gu} {w} = \inrp u {g^{-1}w}$. Since $W$ is a subreresentation, $g^{-1}w \in W$, so $\inrp u {g^{-1}w} = 0$. Thus, $\inrp {gu} w = 0$ for all $w \in W$, so we conclude that $gu \in U$.

Therefore, $V = W \oplus U$.

Any representation of a compact group is completely reducible.

Just reply Maschke's theorem repeatedly. Since our vector space is finite-dimensional, this process must terminate.

Let $V$ be a representation of $G$. Then we can decompose $V$ into irreducible representations as \[V \cong E_1^{\oplus d_1} \oplus \cdots \oplus E_k^{\oplus d_k}\] where the $E_i$ are nonisomorphic irreps, and we have \[d_i = \dim \Hom_G(V, E_i) = \dim \Hom_G(E_i, V)\] We call $d_i$ the multiplicity of $E_i$ in $V$, and sometimes denote it $[V:E_i]$.

This follows from the above corollary and Shur's lemma.

Characters

Let $(V, \phi)$ be a representation of $G$. The character is the function $\chi_V:G \to \C$ defined by $\chi_V(g) = \tr \phi(g)$. Sometimes, we will denote the character by $\chi_\phi$
If $(V, \phi)$ is a trivial representation of $G$ (i.e. $\phi$ sends every $g \in G$ to the identity), then $\chi_\phi$ is the constant function $\dim V$.
Let $(V, \phi)$ and $(W, \psi)$ be isomorphic representations. Then $\chi_V = \chi_W$.

Let $A:V \to W$ be a ($G$-linear) isomorphism. Then $\psi(g)Av = A \phi(g) v$. So $\phi(g) = A^{-1}\psi(g) A$. By the cyclic property of the trace, $\tr(A^{-1} \psi(g) A) = \tr(\psi(g))$. Thus, \[\chi_V(g) = \tr(\phi(g)) = tr(A^{-1}\psi(g)A) = \tr(\psi(g)) = \chi_W(g)\]

Our operations on representations define the following operations on the characers
  1. $\chi_{V^*} = \chi_V^*$ where $\chi_V^*(g) = \chi_V(g^{-1})$
  2. $\chi_{\overline V} = \overline{\chi_V}$ where $\overline{\chi_V}(g) = \overline{\chi_V(g)}$
  3. $\chi_{V \oplus W} = \chi_V + \chi_W$
  4. $\chi_{V \otimes W} = \chi_V \cdot \chi_W$
  5. $\chi_{\Hom(V,W)} = \chi_V^* \cdot \chi_W$
  6. $\chi_{V^G} = av(\chi_V)$ where $av(\chi_V) = \int_G \chi_V(g)\;dg$ considered as a constant function
  1. Since $g$ acts on $V^*$ by $\phi(g^{-1})^T$, and transposing does not change the trace, we see that $\chi_{V^*}(g) = \chi_V(g^{-1})$.
  2. Since scalar multiplication on $\overline V$ is conjugated, we have to take the complex conjugate of the entries in the matrix $\phi(g)$ to get the matrix which acts on $\overline V$. Thus, $\chi_{\overline V} = \overline{\chi_V}$.
  3. $\chi_{V \oplus W}(g) = \tr (\phi(g) \oplus \psi(g)) = \tr(\phi(g)) + \tr(\psi(g)) = \chi_V(g) + \chi_W(g)$.
  4. $\chi_{V \otimes W}(g) = \tr (\phi(g) \otimes \psi(g)) = \tr(\phi_V(g)) \cdot \tr(\psi_W(g)) = \chi_V(g)\chi_W(g)$.
  5. $\chi_{\Hom(V,W)} = \chi_{W \otimes V^*} = \chi_V^* \cdot \chi_W$.
  6. This one is more complicated. We need to compute $\chi_{V^G}$. To do so, we use a trick involving the averaging projection.

    Note that the averaging projection $Av_G:V \to V^G$ acts as the identity on $V^G$ and acts as $0$ on the orthogonal complement to $V^G$. Thus, $\phi(g) \circ Av_G$ acts as $\phi(g)$ on $V^G$ and acts as $0$ on the orthogonal complement to $V^G$. So $\tr_{V^G} \phi(g) = \tr_V (\phi(g) \circ Av_G)$. (Here $\tr_{V^G}$ denotes the trace over $V^G$ and $\tr_V$ denotes the trace over $V$)

    Therefore, \[\chi_{V^G}(g) = \tr_{V^G} \phi(g) = \tr_V (\phi(g) \circ Av_G) = \tr_V \left(\phi(g)\int_G \phi(h)dh\right) = \tr_V\int_G \phi(gh)dh\] Since our measure is left-invariant, this is just \[\chi_{V^G}(g) = \tr_V \int_G \phi(g)dg = \int_G \tr_V \phi(g)dg = \int_G \chi_V(g)dg = av(\chi_V)\]

It turns out that $\chi_V^* = \overline{\chi_V}$. Note that a $G$-invariant inner product gives an isomorphism $V^* \cong \overline V$ as $G$-representations. Since we proved the existence of $G$-invariant inner products, it follows that the representations $\overline V, V^*$ are isomorphic, so they have the same characters.
We can put an inner product on the space of complex-valued functions on $G$. Given $f_1,f_2:G \to \C$, we define \[\inrp{f_1}{f_2}(g) = \int_G f_1(g) \overline{f_2(g)}dg = av(f_1 \cdot \overline {f_2})\]
For representations $V, W$ we have \[\dim \Hom_G(V,W) = \inrp{\chi_W}{\chi_V}\]
The proof is surprisingly simple using all of the operations we have defined on representations and characters. \[\begin{aligned} \dim \Hom_G(V,W) &= \dim \Hom(V,W)^G\\ &= \chi_{\Hom(V,W)^G}\\ &= av(\chi_{\Hom(V,W)})\\ &= av(\chi_{W \otimes V^*})\\ &= av(\chi_W \cdot \chi_V^*)\\ &= av(\chi_W \cdot \overline{\chi_V})\\ &= \inrp {\chi_W}{\chi_V} \end{aligned}\]
A representation $V$ is irreducible if and only if $\inrp {\chi_V}{\chi_V} = 1$.
By the above proposition, $\inrp {\chi_V}{\chi_V} = \dim \Hom_G(V,V)$. We know that we can decompose $V$ into a direct sum of irreducibles $V = \bigoplus_i E_i^{d_i}$ where $\dim \hom_G(E_i, E_j) = \delta_{ij}$. So $\dim \Hom_G(\bigoplus_i E_i^{d_i}, \bigoplus_i E_i^{d_i}) = 1$ if and only if $V = E_i$ for some $i$. Thus, $\inrp {\chi_V}{\chi_V} = 1$ if and only if $V$ is irreducible.
$\chi_V = \chi_W$ if and only if $V$ is isomorphic to $W$.
We saw earlier that if $V \cong W$, then $\chi_V = \chi_W$. So now, we just have to show that if $\chi_V = \chi_W$, then $V \cong W$. Recall that $V \cong \bigoplus_i E_i^{d_i}$ where $d_i = \dim \Hom_G(V,E_i)$. Since $\dim \Hom_G(V,E_i) = \inrp {\chi_{E_i}}{\chi_V}$, we conclude that if $V$ and $W$ have the same characters, then they must be isomorphic.
(Orthogonality of characters) Let $E, F$ be irreps of $G$. Then $\inrp{\chi_E}{\chi_F}$ is $1$ if $E$ and $F$ are isomorphic and $0$ otherwise.
By Schur's lemma, $\dim \Hom_G(E,F)$ is $1$ if $E$ and $F$ are isomorphic and $0$ otherwise.

The previous propositions tell us that characters of $G$ are a decategorification of the category of finite-dimensional representations of $G$. Decategorification is the process of taking a category, identifying isomorphic objects and forgetting all other morphisms. This eliminates a lot of useful information, but often makes the category easier to work with. For example, if we decategorify the category of finite sets, we identify all sets with the same cardinality, and forget about all other functions. This just leaves us with the natural numbers, because sets are classified by their cardinality.

Frequently, there are nice structures in the category that still make sense after decategorification. For example, decategorifying disjoint unions of finite sets gives us addition of natural numbers, and decategorifying cartesian products gives us multiplication of natural numbers.

Above, we saw that characters are a decategorification of finite-dimensional $G$-representations. Two characters are equal if the corresponding representations are isomorpic, and the direct sums, tensor products, etc. of $G$-representations translate nicely into operations on characters. One of the most interesting aspects of this decategorification is that $\Hom$s turn into inner products.

Recall that a pair of adjoint functors are functors $F:\mathcal{C} \to \mathcal{D}, G:\mathcal{D} \to \mathcal{C}$ such that $\Hom_D(F(X), Y) \cong \Hom_C(X, G(Y))$ for all $X \in \Ob(C), Y \in \Ob(D)$. Adjoint functors are so named in analogy with adjoint linear operators (Recall that two operators $T,U$ on a Hilbert space are adjoint if $\inrp {Tx} y = \inrp x {Uy}$ for all vectors $x,y$.) This connection between inner products and Hom sets can be formalized to give a categorification of Hilbert spaces.

For an introduction to (de)categorification, you can look here (for a simpler introduction) or here (for a more complicated introduction).

Application: Irreducible Representations of $SU(2)$

Before proceeding, let's use some of this machinery we have built up so far to find all irreducible representations of $SU(2)$.

$SU(2)$ is the special unitary group of degree $2$. It is the group of $2 \times 2$ complex matrices which are unitary and have determinant $1$.

It will be helpful to use another characterization of $SU(2)$ as well.

$SU(2) \cong S^3$, where $S^3$ is given a multiplicative structure by identifying it with the group of unit quaternions $\H^\times$.

Recall that the quaternions are defined by \[\H = \{a + jb\;|\; a,b \in \C\}\] where $jb = \overline b j$. Then $S^3 = \{a + jb \in \H\;|\; |a|^2 + |b|^2 = 1\}$. We have a natural action of $S^3$ on $\H$ by left-multiplication. This gives us a two-dimensional complex representation of $S^3$. Writing it out explicitly, we see that \[\begin{aligned} (a+jb) : 1 &\mapsto a+jb\\ (a+jb) : j &\mapsto - \overline b + j \overline a\\ \end{aligned}\] Thus, our representation is given by \[ (a+jb) \mapsto \begin{pmatrix} a & -\overline b\\b & \overline a \end{pmatrix} \] This matrix is unitary, and has determinant $|a|^2 + |b|^2 = 1$. So this is clearly a continuous bijection from $S^3$ to $SU(2)$. You can check that this bijection is a group isomorphism.

Since $SU(2)$ acts on $\C^2$, we also get an action of $SU(2)$ on $\C[z_1, z_2]$, the space of complex polynomials in 2 variables. Given $A \in SU(2), p \in \C[z_1, z_2]$, we define \[(A \cdot p)\left(\vvec{z_1}{z_2}\right) = p \left(A^{-1} \vvec {z_1}{z_2}\right)\] We note that this action does not change the degree of monomials. Thus, the space of homogeneous polynomials of degree $k$ is invariant under this action. So it is a subrepresentation. Let $V_k \subseteq \C[z_1, z_2]$ denote the space of homogeneous polynomials of degree $k$. We will show that $\{V_k\}$ are nonisomorphic irreducible representations, and every irreducible representation of $SU(2)$ is isomorphic to some $V_k$. First, we'll start with a lemma about the structure of $S^3$.

  1. Every element of $S^3 \subseteq \H$ can be written $ge^{i\theta}g^{-1}$
  2. For fixed $\theta$, $\{ge^{i\theta}g^{-1}\}$ is a 2D sphere with radius $\sin \theta$ intersecting $\C$ at $e^{i\theta}, e^{-i\theta}$
  1. Using our identification of $S^3$ with $SU(2)$, we can think of points on the sphere as special unitary matrices. Unitary matrices are unitarily-diagonalizable. Clearly we can rescale these matrices so that the matrices we diagonalize with are in $SU(2)$. Finally, note that a diagonal matrix in $SU(2)$ must have the form \[\begin{pmatrix} a & 0 \\ 0 & \overline a\end{pmatrix}\] for $a \in \C$, $|a|^2 = 1$. Thus, diagonal matrices in $SU(2)$ correspond to points $e^{i\theta}$ on the sphere.
  2. Since the quaternion norm is multiplicative, and elements of $S^3$ have norm 1, we see that $|g e^{i\theta}g^{-1}| = |g||e^{i\theta}||g|^{-1} = 1$. Furthermore, \[\begin{aligned} ge^{i\theta}g^{-1} &= g (\cos \theta + i \sin \theta)g^{-1}\\ &= \cos \theta + \sin \theta \; g i g^{-1} \end{aligned}\] Now, we will consider the map $\pi:g \mapsto g i g^{-1}$. Note that for unit quaternions, $g^{-1} = \overline g$, the conjugate of $g$. So we can also write this map $\pi:g \mapsto g i \overline g$. Note that \[\overline{g i \overline g} = g \overline i \overline g = -g i \overline g\] So $gi\overline g$ is purely imaginary. Furthermore, since $g, i$ and $\overline g$ are all unit quaternions, so is their product. Thus, we can think of $\pi$ as a map $\pi : S^3 \to S^2$, where we view $S^3$ as the unit quaternions and $S^2$ as the unit imaginary quaternions.

    Furthermore, $\pi$ is surjective. If we represent vectors in $\R^3$ as imaginary quaternions, then $v \mapsto g v \overline g$ is a representation of $SU(2)$ on $\R^3$ which acts by rotations. Since we can write all rotations in this form, and the rotation group $SO(3)$ acts transitively on the two-sphere, we see that $\{gi \bar g\}_{g \in SU(2)}$ covers all of $S^2$. So $\pi$ is surjective.

    So since $g e^{i\theta} g^{-1} = \cos \theta + \sin \theta \pi(g)$, we see that $\{g e^{i\theta} g^{-1}\}$ is a sphere with radius $\sin \theta$. Now, we just have to check that the intersection of this sphere with $\C$ is $e^{\pm i \theta}$. Note that $\im \pi$ is the imaginary unit quaterions, and the only imaginary unit quaternions that lie in $\C$ are $\pm i$. Thus, the intersection of $\{g e^{i\theta} g^{-1}\}$ with $\C$ is $\cos \theta \pm i \sin \theta$.

The homogeneous subspaces $\{V_k\}$ are nonisomorphic irreducible representations of $SU(e)$ and every irreducible representation of $SU(2)$ is isomorphic to some $V_k$.

To prove this, we will use characters. For convenience, let us write $\chi_{V_k}$ as $\chi_k$. Recall that the image of $e^{i\theta} \in S^3$ in $SU(2)$ is the matrix \[\begin{pmatrix} e^{i\theta}& 0 \\ 0 & e^{-i\theta}\end{pmatrix}\] Note that the eigenspaces of this operator on $V_k$ are $\{\C z_1^\ell z_2^{k-\ell}\}_\ell$ with eigenvalues $\{e^{(2 \ell - k)i \theta}\}_\ell$. Therefore, \[\begin{aligned} \chi_k(e^{i\theta}) &= \sum_{\ell=0}^k e^{(2 \ell - k) i \theta}\\ &= \frac{e^{(k+1)i\theta} - e^{-(k+1)i\theta}}{e^{i\theta} - e^{-i\theta}}\\ &= \frac{\sin[(k+1)\theta]}{\sin\theta} \end{aligned}\] Note that all of these characters are different. This shows that all of the representations $V_k$ are distinct. Now, we will show that the characters are orthonormal.

Recall that the inner product on characters is given by \[\inrp{\chi_k}{\chi_\ell} = \int_{S^3} \chi_k(g) \overline{\chi_\ell(g)}\;dg\] Since the volume of $S^3$ is $2\pi^2$, we can write $dg = \frac 1 {2\pi^2}d\sigma$ where $d\sigma$ is the standard volume element on $S^3$. So we want to compute \[\inrp{\chi_k}{\chi_\ell} = \frac 1 {2\pi^2} \int_{S^3} \chi_k(g) \overline{\chi_\ell(g)}\;d\sigma\] Recall that characters are constant on conjugacy classes. Since every element of $SU(2)$ is conjugate to exactly two unit complex numbers, we have \[\inrp{\chi_k}{\chi_\ell} = \frac 1 {2\pi^2} \int_0^\pi \chi_k(e^{i\theta}) \overline{\chi_\ell(e^{i\theta})}\vol(\text{orbit})\;d\theta\] Above, we showed that these orbits are spheres with radius $\sin \theta$. Therefore, the volume of an orbit is $4 \pi \sin^2 \theta$. Substituting this and our expressions for the characters, we see that our inner product is \[\inrp{\chi_k}{\chi_\ell} = \frac 2 {\pi} \int_0^\pi \sin[(k+1)\theta)\sin[(\ell+1)\theta]\;d\theta\] Because sines with different frequencies are orthogonal, we conclude that \[\inrp{\chi_k}{\chi_\ell} = \delta_{k\ell}\] So our characters are orthonormal.

Finally, we will show that these are all of the irreducible representations. Suppose that $W$ was another irreducible representation. Then \[0 = \inrp{\chi_W}{\chi_k} = \int_G \chi_W(g) \overline{\chi_k(g)}\;dg\] Using the same computational tricks, we see that \[0 = \frac 2 \pi \int_0^\pi \chi_W(e^{i\theta}) \sin[(k+1)\theta]\sin\theta\;d\theta\] Since sinces form an orthonormal basis for the set of square-integrable functions on the circle, we see that $\chi_W(e^{i\theta}) = 0$, which is impossible. Thus, every irreducible representation must be isomorphic to some $V_k$.

Characters made it fairly easy to classify all of the irreducible representations of $SU(2)$. Later on, we will generalize some of the computational techniques we used here to find the Weyl character formula and Weyl Integration Formula, which will be very useful for understanding representations. But that will have to be another post, since this one is already much longer than I realized it would be.

279 comments:

  1. Yes, I am entirely agreed with this article, and I just want say that this article is very helpful and enlightening. I also have some precious piece of concerned info !!!!!!Thanks. kids quads

    ReplyDelete
  2. I was reading some of your content on this website and I conceive this internet site is really informative ! Keep on putting up. polaris ranger

    ReplyDelete
  3. Thank you because you have been willing to share information with us. we will always appreciate all you have done here because I know you are very concerned with our. taxibusje rotterdam

    ReplyDelete
  4. I really like your writing style, great information, thankyou for posting. 토토사이트

    ReplyDelete
  5. Hi there! Nice stuff, do keep me posted when you post again something like this! i99pro

    ReplyDelete
  6. It is my first visit to your blog, and I am very impressed with the articles that you serve. Give adequate knowledge for me. Thank you for sharing useful material. I will be back for the more great post. 스포츠토토

    ReplyDelete
  7. Hi there! Nice stuff, do keep me posted when you post again something like this! esa letter

    ReplyDelete
  8. Thankyou for this wondrous post, I am glad I observed this website on yahoo. 토토사이트

    ReplyDelete
  9. Thankyou for this wondrous post, I am glad I observed this website on yahoo. 메이저사이트

    ReplyDelete
  10. Hello I am so delighted I located your blog, I really located you by mistake, while I was watching on google for something else, Anyways I am here now and could just like to say thank for a tremendous post and a all round entertaining website. Please do keep up the great work. 메이저사이트

    ReplyDelete
  11. Home of the weekend rental, our team is fast, efficient, courteous and passionate about having fun.​i thought about this

    ReplyDelete
  12. This is actually the kind of information I have been trying to find. Thank you for writing this information. 먹튀검증

    ReplyDelete
  13. Thankyou for this wondrous post, I am glad I observed this website on yahoo. 먹튀검증

    ReplyDelete
  14. Apply for LottoVIP Login Lotto VIP Apply online lottery website to join the fun with Thai lottery,pay the best price, 100% confident.หวย lottovip

    ReplyDelete
  15. Radonisasilentkiller.It’sacolorless,odorlesswhichnaturallyoccursinthegroundbelowus.Homepage

    ReplyDelete
  16. I think this is an informative post and it is very beneficial and knowledgeable. Therefore, I would like to thank you for the endeavors that you have made in writing this article. All the content is absolutely well-researched. Thanks... security guard services singapore

    ReplyDelete
  17. It is a fantastic post – immense clear and easy to understand. I am also holding out for the sharks too that made me laugh. BizOp

    ReplyDelete
  18. Pretty nice post. I just stumbled upon your weblog and wanted to say that I have really enjoyed browsing your blog posts. After all I’ll be subscribing to your feed and I hope you write again soon! know more

    ReplyDelete
  19. Wow, What an Outstanding post. I found this too much informatics. It is what I was seeking for. I would like to recommend you that please keep sharing such type of info.If possible, Thanks. 안전놀이터

    ReplyDelete
  20. Good website! I truly love how it is easy on my eyes it is. I am wondering how I might be notified whenever a new post has been made. I have subscribed to your RSS which may do the trick? Have a great day! ssrmovies

    ReplyDelete
  21. Hi there! Nice stuff, do keep me posted when you post again something like this! 7starhd

    ReplyDelete
  22. I can’t imagine focusing long enough to research; much less write this kind of article. You’ve outdone yourself with this material. This is great content. Canninghill Piers

    ReplyDelete
  23. If you are passionate about what you do, you will have endless energy to devote.Declutter for success

    ReplyDelete
  24. I can’t imagine focusing long enough to research; much less write this kind of article. You’ve outdone yourself with this material. This is great content. 토토사이트

    ReplyDelete
  25. Thanks for the blog filled with so many information. Stopping by your blog helped me to get what I was looking for. Now my task has become as easy as ABC. 에볼루션카지노

    ReplyDelete
  26. Joker Gaming is better than ever. Collection of many games from Joker Game, unique, better than ever.joker slot

    ReplyDelete
  27. It is my first visit to your blog, and I am very impressed with the articles that you serve. Give adequate knowledge for me. Thank you for sharing useful material. I will be back for the more great post. 토토사이트

    ReplyDelete
  28. Great job here on _______ I read a lot of blog posts, but I never heard a topic like this. I Love this topic you made about the blogger's bucket list. Very resourceful. 릴게임

    ReplyDelete
  29. Thanks for your post. I’ve been thinking about writing a very comparable post over the last couple of weeks, I’ll probably keep it short and sweet and link to this instead if thats cool. Thanks. 안전놀이터

    ReplyDelete
  30. That is the excellent mindset, nonetheless is just not help to make every sence whatsoever preaching about that mather. Virtually any method many thanks in addition to i had endeavor to promote your own article in to delicius nevertheless it is apparently a dilemma using your information sites can you please recheck the idea. thanks once more. 먹튀검증업체

    ReplyDelete
  31. This particular papers fabulous, and My spouse and i enjoy each of the perform that you have placed into this. I’m sure that you will be making a really useful place. I has been additionally pleased. Good perform! 먹튀검증

    ReplyDelete
  32. From traditional limos to classic rides, all our cars are of first-class quality with certified drivers for our client's safety.Cheap Party Bus Chicago

    ReplyDelete
  33. We are Wisconsin’s best radon mitigation and radon testing specialists.  With over 20 years of combined experience.why not try here

    ReplyDelete
  34. Kaun Banega Crorepati Head office number, now you can feel free to call.kbc head office number

    ReplyDelete
  35. Bridal shops varying in quality and the types of wedding dresses they have available.mother of the bride dresses

    ReplyDelete
  36. Excellent to be visiting your blog again, it has been months for me. Rightly, this article that I've been served for therefore long. I want this article to finish my assignment within the faculty, and it has the same topic together with your article. Thanks for the ton of valuable help, nice share. pii-email

    ReplyDelete
  37. Impressive web site, Distinguished feedback that I can tackle. Im moving forward and may apply to my current job as a pet sitter, which is very enjoyable, but I need to additional expand. Regards. research transcription services

    ReplyDelete
  38. Discovering how 100 % free sporting betting succeeds mandates comprehension of how within the internet activities wagering is effective.검증사이트

    ReplyDelete
  39. The source of the best online slot games It comes in the form of famous anime from Japan such as One Piece, Naruto.สมัคร PG Slot

    ReplyDelete
  40. Hello I am so delighted I located your blog, I really located you by mistake, while I was watching on google for something else, Anyways I am here now and could just like to say thank for a tremendous post and a all round entertaining website. Please do keep up the great work. 먹튀검증

    ReplyDelete
  41. I’ve been searching for some decent stuff on the subject and haven't had any luck up until this point, You just got a new biggest fan!.. 88카

    ReplyDelete
  42. Thank you a bunch for sharing this with all of us you actually realize what you are talking about! Bookmarked. Please also seek advice from my site =). We could have a hyperlink change contract between us! สล็อต

    ReplyDelete
  43. I exactly got what you mean, thanks for posting. And, I am too much happy to find this website on the world of Google. 검증사이트

    ReplyDelete
  44. Merely a smiling visitant here to share the love (:, btw outstanding style. 먹튀검증

    ReplyDelete
  45. At Inovi, we’re committed to achieving the highest success rates while providing personalized care to our patients. Our Houston location offers expert care and is home to our state-of-the-art embryology lab. inovifertility.com

    ReplyDelete
  46. That is the excellent mindset, nonetheless is just not help to make every sence whatsoever preaching about that mather. Virtually any method many thanks in addition to i had endeavor to promote your own article in to delicius nevertheless it is apparently a dilemma using your information sites can you please recheck the idea. thanks once more. sbobet mobile

    ReplyDelete
  47. Attractive, post. I just stumbled upon your weblog and wanted to say that I have liked browsing your blog posts. After all, I will surely subscribe to your feed, and I hope you will write again soon! linear algebra and its applications 5th edition solutions

    ReplyDelete
  48. Good website! I truly love how it is easy on my eyes it is. I am wondering how I might be notified whenever a new post has been made. I have subscribed to your RSS which may do the trick? Have a great day! canvas bag singapore

    ReplyDelete
  49. Kbc Lottery Winner 2021 list for information online like Kaun Banega Crorepati Lucky draw Winners 2021.Kbc lottery winner list 2022

    ReplyDelete
  50. Great job for publishing such a beneficial web site. Your web log isn’t only useful but it is additionally really creative too. singapore casino

    ReplyDelete
  51. I have been impressed after read this because of some quality work and informative thoughts. I just want to say thanks for the writer and wish you all the best for coming! Your exuberance is refreshing. Dallas Manslaughter Defense lawyer

    ReplyDelete
  52. Kbc Sweepstakes Success 2021 listing for facts on line just like Kaun Banega Crorepati Fortuitous bring Those who win 2021.Kbc lottery number

    ReplyDelete
  53. I wanted to thank you for this great read!! I definitely enjoying every little bit of it I have you bookmarked to check out new stuff you post. 파워볼

    ReplyDelete
  54. I was reading some of your content on this website and I conceive this internet site is really informative ! Keep on putting up. 토토사이트

    ReplyDelete
  55. I haven’t any word to appreciate this post.....Really i am impressed from this post....the person who create this post it was a great human..thanks for shared this with us. 파워볼사이트

    ReplyDelete
  56. Eat-and-go commerce is the NO.1 eat-and-see community that listens to the voices of users and always strives to create a fair Toto culture.토토사이트

    ReplyDelete
  57. At Inovi, we’re committed to achieving the highest success rates while providing personalized care to our patients. Our Houston location offers expert care and is home to our state-of-the-art embryology lab. Donor Program

    ReplyDelete
  58. Liposuction should never be considered an alternative to a healthy lifestyle, it is one of the steps that can get you to a better version of yourself through fat reduction. cheeks

    ReplyDelete
  59. We place a great deal of attention on the values, principles, knowledge, relationships and abilities that allow for wealth creation. Our values-based model for wealth transition influences human, intellectual, social and structural capital to support and promote growing financial capital. Go Here

    ReplyDelete
  60. Often referred to as the "natural filler," Sculptra Aesthetic is a unique injection that stimulates your body's own collagen production. The anti-aging results are both long-lasting and natural looking. https://omahacosmeticcenter.com/

    ReplyDelete
  61. Great post, and great website. Thanks for the information! pengeluaran togel hongkong

    ReplyDelete
  62. We provide Kbc lottery winner 2022 Mumbai and Kbc lottery winner season 13 data online like Kaun Banega Crorepati Lucky Winners, Lucky Winners 2022.Kbc lottery winner

    ReplyDelete
  63. I think this is an informative post and it is very beneficial and knowledgeable. Therefore, I would like to thank you for the endeavors that you have made in writing this article. All the content is absolutely well-researched. Thanks... 대전건마

    ReplyDelete
  64. You have a real ability for writing unique content. I like how you think and the way you represent your views in this article. I agree with your way of thinking. Thank you for sharing. nursing test bank

    ReplyDelete
  65. Quality leads and a consistent volume of them are vital to fill your sales pipeline with the right people, the decision-makers your sales team want to reach. We are reaching a high volume of prospects during this time. you could check here

    ReplyDelete
  66. Imagen offers a variety of advanced treatments and procedures, unavailable at any other clinic in Nebraska. Our board-certified physicians are highly trained in the most advanced facial rejuvenation techniques, performing thousands of injections and cosmetic procedures each year. Discover More

    ReplyDelete
  67. Best Plumbers take the listings of plumbers published in our plumbing directory very serious, and try hard to provide accurate.plumber directory

    ReplyDelete
  68. Provides the highest odds in the industry, covering matches from around the world, with multiple guesses such as handicaps.Hypebetcash

    ReplyDelete
  69. I was looking at some of your posts on this website and I conceive this web site is really instructive! Keep putting up.. 현금바둑이

    ReplyDelete
  70. Only strive to mention one's content can be as incredible. This clarity with your post is superb! Thanks a lot, hundreds of along with you should go on the pleasurable get the job done. 온라인바둑이

    ReplyDelete
  71. Kbc Lottery Winner 2022 list for information online like Kaun Banega Crorepati Lucky draw Winners 2022.kbc office number

    ReplyDelete
  72. Positive site, where did u come up with the information on this posting?I have read a few of the articles on your website now, and I really like your style. Thanks a million and please keep up the effective work. 美国作业代写

    ReplyDelete
  73. We provide Kbc lottery winner 2022 Mumbai and Kbc lottery winner season.Kbc online game winner

    ReplyDelete
  74. This is my first time i visit here. I found so many interesting stuff in your blog especially its discussion. From the tons of comments on your articles, I guess I am not the only one having all the enjoyment here keep up the good work http://www.toryburchoutlet.org/

    ReplyDelete
  75. Hello, this weekend is good for me, since this time i am reading this enormous informative article here at my home. http://digitalanalog.in/

    ReplyDelete
  76. You made such an interesting piece to read, giving every subject enlightenment for us to gain knowledge. Thanks for sharing the such information with us to read this... https://dynamichealthstaff.com/nursing-jobs-in-kuwait

    ReplyDelete
  77. Thanks for posting this info. I just want to let you know that I just check out your site and I find it very interesting and informative. I can't wait to read lots of your posts. 먹튀폴리스

    ReplyDelete
  78. Kbc Financial success 2022 showing with respect to truth on the internet like Kaun Banega Crorepati Fortuitous draw People acquire 2022.Kbc official website

    ReplyDelete
  79. It is doing accomplice with, yet look at the information at this space. Eyes specialist in delhi

    ReplyDelete
  80. Repair in UAE provides 24/7 emergency services related to property maintenance services covering plumbing,appliances.Bosch Service Center

    ReplyDelete
  81. A good termites for one'ohydrates dwelling bring about regrettable effects, if he does not dealt with around ideal time.Pest Control Service Lahore

    ReplyDelete
  82. Lavish Men's Salon is now proud to offer complete range of hair stylinghttps://lavishsaloon.com.pk

    ReplyDelete
  83. Today Kbc lottery champ 2022 Mumbai as well as Kbc lottery champ month roughly 13 data files information online. Fortunate enough Winners 2022.Kbc lottery winner

    ReplyDelete
  84. Previously Kbc lottery champ 2022 Mumbai and in addition Kbc lottery champ season and maybe 13 knowledge archives online. Fortunate enough Winners 2022.kbc head office

    ReplyDelete
  85. 바카라사이트
    This is my website and it has been very helpful. Guys, this is a really good post here. Thank you for taking your valuable time to post this valuable information. Quality content is always what attracts visitors. You are so cool! Thanks for getting started.
    Here is my website. Please leave a comment on my site as well.

    ReplyDelete
  86. You know your projects stand out of the herd. There is something special about them. It seems to me all of them are really brilliant! Lamborghini car parts

    ReplyDelete
  87. Kbc official website list for information online like Kaun Banega Crorepati Lucky draw Winners 2022.Kbc winner list 2022

    ReplyDelete
  88. Travel Guides by the Experienced Tourist Consulter.We are a professional and experienced tourism consultant.travelguidline.com

    ReplyDelete
  89. We surely are stuck in a hectic routine with our education and job every day.how to get rid of sugar ants

    ReplyDelete
  90. Our totostation is always safe and recommends only major toto sites certified by the eat-and-run verification.토토사이트

    ReplyDelete
  91. At Simplibuy we specialize in servicing independent healthcare facilities.https://simplibuy.ca/

    ReplyDelete
  92. Emergency Dental Center provides same-day emergency and routine dental care for Houston and the surrounding areas.emergency dental center

    ReplyDelete
  93. Youre thus great! We don’t suppose I have discover anything like this just before. So great to locate an individual with some authentic applying for grants this topic. realy thank you for beginning this particular up. this web site is something that’s needed on the web, somebody with a little originality. beneficial job for bringing one thing new to the internet! 먹튀검증커뮤니티

    ReplyDelete
  94. I am always looking for some free kinds of stuff over the internet. There are also some companies which give free samples. But after visiting your blog, I do not visit too many blogs. Thanks. 토토사이트

    ReplyDelete
  95. Ahdena. Digital Creator. ✨Revitalizing Faith Continuing to open my heart to the beauty of Islam.https://ahdena.com/

    ReplyDelete
  96. When I originally commented I clicked the -Notify me when new comments are added- checkbox and now each time a comment is added I receive four emails with the same comment. Can there be however you possibly can get rid of me from that service? Thanks! สล็อต

    ReplyDelete
  97. We are serving industries like hotels, restaurants, storage houses, construction, interior designers, retail shops, institutions, shopping centers and more.Termite Control service Lahore

    ReplyDelete
  98. You can get custom packaging boxes with logo here. Our company makes packaging with good material at wholesale prices. If you order more than 50 pounds, our company give you free shipping for all over the world.Bath bomb boxes

    ReplyDelete
  99. At Simplibuy we know how busy you are.You have a busy facility to run.You have to watch your bottom line.Revital-Ox

    ReplyDelete
  100. Pretty good post. I have just stumbled upon your blog and enjoyed reading your blog posts very much. I am looking for new posts to get more precious info. Big thanks for the useful info. 오피아트

    ReplyDelete
  101. Thanks so much for this information. I have to let you know I concur on several of the points you make here and others may require some further review, but I can see your viewpoint. 먹튀검증업체

    ReplyDelete
  102. You can get custom packaging boxes with logo here. Our company makes packaging with good material at wholesale prices. If you order more than 50 pounds, our company give you free shipping for all over the world.Custom packaging boxes with logo

    ReplyDelete
  103. You make so many great points here that I read your article a couple of times. Your views are in accordance with my own for the most part. This is great content for your readers. 먹튀폴리스

    ReplyDelete
  104. This site seems to inspire me a lot. Thank you so much for organizing and providing this quality information in an easy to understand way. I think that a healthy era of big data can be maintained only when such high-quality information is continuously produced. And I, too, are working hard to organize and provide such high-quality information. It would be nice to come in once and get information.

    Also visit my site:스포츠토토

    ReplyDelete
  105. Great knowledge, do anyone mind merely reference back to it lottery sambad

    ReplyDelete
  106. One of the best things you can do for yourself and/or your family is to buy quality bedsheets.Online curtains in Pakistan

    ReplyDelete
  107. I think this is an informative post and it is very useful and knowledgeable. therefore, I would like to thank you for the efforts you have made in writing this article. 먹튀검증사이트

    ReplyDelete
  108. Good post but I was wondering if you could write a litte more on this subject? I’d be very thankful if you could elaborate a little bit further. Appreciate it! ufabet เว็บตรง

    ReplyDelete
  109. SwiftRecharge comes in picture with its simple, fast and secure cutting-edge platform to help millions of people.digicel top up

    ReplyDelete
  110. Every Hamptons Airport Car Service vehicles such as sedans, SUVs, Sprinter Vans is latest model.nyc to montauk

    ReplyDelete
  111. Baccarat formula  is another secret that will allow you to play baccarat to make money and profit from online baccarat  games .บาคาร่าออนไลน์

    ReplyDelete
  112. Jio KBC Lottery Winner. Kbc Lottery Winner 2022 list for information online like Kaun Banega Crorepati Lucky draw Winners 2022.Bloody hell

    ReplyDelete
  113. Literally the word Casino means - a small house or villa for summer season, which is built on a larger ground.pgslot

    ReplyDelete
  114. I am glad you take pride in what you write. This makes you stand way out from many other writers that push poorly written content. 토토사이트

    ReplyDelete
  115. Simplibuy has you covered with a huge range of medical products, diagnostic imaging supplies and healthcare products for customers in Toronto.GTA Medical Supplies

    ReplyDelete
  116. we are announcing that Auspicious Labs has added Covid Testing to the list of services we provide to our community.pcr covid test Houston

    ReplyDelete
  117. Thanks for picking out the time to discuss this, I feel great about it and love studying more on this topic. It is extremely helpful for me. Thanks for such a valuable help again. 토토사이트

    ReplyDelete
  118. pimah offer several services to help you manage your health.doctor for internal medicine katy

    ReplyDelete
  119. Deep Cleaning Services has been providing domestic cleaning services for over Two years.SOFA CLEANING SERVICES

    ReplyDelete
  120. National Eye Center has been providing both indoor and outdoor eye care facilities to patients since 2004 at Sanda Road Lahore.pediatric eye specialist in lahore

    ReplyDelete
  121. Simplexmed is a full service medical billing company that helps healthcare providers lower costs and increase revenues.Medical Billing Texas

    ReplyDelete
  122. Baccarat, online casino game, international standard, can be played for both iOS & Android, direct game maker Fast service giving you the feeling of being in a real casino.บาคาร่าออนไลน์

    ReplyDelete
  123. Our Pet Flight Nanny will pickup pet from the Shipper at the closest major airport.flight nanny for dogs

    ReplyDelete
  124. Thank you for visiting Mazinger TV, a free broadcasting community specializing in sports.https://matv04.com/

    ReplyDelete
  125. Download Crackwinz With Setup Download Latest software.AIRSERVER Licesnce Key

    ReplyDelete
  126. Our Head Office has created many branches of our customer care department in many other cities to facilitate customers.Kbc Lottery winner 2022

    ReplyDelete
  127. This minute must be given to Asia99th, which is the hottest online gambling website now, tearing all the rules of playing online games for real money.คาสิโนออนไลน์

    ReplyDelete
  128. Whatever our Golden Time is, if you start it haphazardly without any preparation, damage or damage may occur and you will not feel comfortable. 메이저놀이터

    ReplyDelete
  129. If you sign up with the mpolice code at the safety playground introduced by Eat-Police and experience unfair damage from a major playground.https://www.mt-police07.com/

    ReplyDelete
  130. Sureman is a company that thoroughly verifies scam sites so that you can safely enjoy Sports Toto.https://sureman01.com/

    ReplyDelete
  131. We provide a collection of the latest links from all over the world to a sports community site, a sports relay site, and a real-time address.블랙링크

    ReplyDelete
  132. We only provide various major recommendations, and only Toto sites that are guaranteed to be eaten and run without any occurrences are carefully selected and accurately recommended to you.https:// toriters.com/

    ReplyDelete
  133. Why is a dog even extraordinary is the fact they may easily go along with people.pet portraits

    ReplyDelete
  134. A table game in which winning or losing is determined by the size of the board.gameone

    ReplyDelete
  135. Remote broadband network access is growing in locations outside of the workplace or home.Etisalat wireless internet Dubai

    ReplyDelete
  136. I wish more authors of this type of content would take the time you did to research and write so well. I am very impressed with your vision and insight. mustang start button cover

    ReplyDelete
  137. Flight Nanny Pups on the Move is a professional and experienced pet delivery service.puppy flight nanny

    ReplyDelete
  138. At the Pediatric Pod we offer comprehensive physical exams, assess the growth and development of the children.Houston Pediatrician

    ReplyDelete
  139. Payday loans are unsecured loans that generally get paid back in 30 days or less Payday loans online no credit check instant approval

    ReplyDelete
  140. Regal Marketing Network is among the fastest-growing marketing network around the world having its offices in many cities of Pakistan.kingdom valley

    ReplyDelete
  141. Online, you may purchase award-winning coffee beans, ground coffee, and compostable coffee pods.Coffee beans Uk

    ReplyDelete
  142. Boost Social Media provides the highest quality buy instagram likes at a cheap price in the market. Get our best offers!

    ReplyDelete
  143. We always aim to deliver exceptional quality; we care about our reputation, our clients and staff, our local community and our planet.SOFA CLEANING SERVICES

    ReplyDelete
  144. The main theme of Kaun Banega Crorepati is Airtel lottery winner inspired by a British game show, so if we see it on a global stage its not a unique show by any means even the Kaun Banega Crorepati Winner 2022 Kbc winner list 2022

    ReplyDelete
  145. We offer specialised sameday courier services to firms in a variety of industrial sectors to and from any UK location.Business couriers

    ReplyDelete
  146. Buy Venoms: Online Venom Pharmacy, Scorpion, Snake, Frog, & Toad at https://www.buyvenoms.com snake venom for sale

    ReplyDelete
  147. If Anyone To Buy Bitcoin Miner - Cryptocurrency Mining Hardware Supplier Online at crypto mining rig for sale

    ReplyDelete
  148. PG SLOT is considered one of the best. online slot games That has been very popular PGSLOT from people all over the world that has it all. pg slot

    ReplyDelete
  149. Our goal is to help new entrepreneurs establish themselves successfully and see their online businesses prosper.Bluehost review

    ReplyDelete
  150. Refrigerator Repair Before the time comes that you don't have to search for a solution for an immediate fridge repair in Dubai, give us a call today and let us discuss the options we offer. Our team is not merely willing to provide you with the resolve business, we also provide other expert services also. Repairing your freezer in Dubai is among one of numerous, on the contrary it's an essential one single.

    ReplyDelete
  151. Siemens Service Center Dubai Give us a call today and let us discuss the options we offer before the time comes that you don't have to search for a solution for an immediate fridge repair in Dubai. Our group is not only capable of offer the fix company, we can provide other companies also. Restoring your refrigerator in Dubai is one of the countless, nevertheless it's a crucial an individual.

    ReplyDelete
  152. UFABET is open to everyone 24 hours a day, depositing-withdrawing within 1 minute with an automatic system AUTO on the website.UFABET

    ReplyDelete
  153. Airport Transfer Edinburgh
    We have been industry experts in bringing first-class solutions towards the prospects by designing and consequently are very proud to become Scotland’s primary minicabs taxi agency scotminicabs. We do business without doubt one of Edinburgh’s major personal employ fleets, supplying a easy personalized employ and taxi cab company to our customers. We will offer right cars and trucks for your wants offering the highest regulations of luxury and comfort.

    ReplyDelete
  154. At Simcare PLLC, our goal is to further your health needs through Preventative and Acute Medical Care.Internal Medicine Sugar Land

    ReplyDelete
  155. Airport Taxis Edinburgh
    We have been industry experts in giving you advanced solutions for our shoppers by means of making and they are pleased to always be Scotland’s primary minicabs taxi provider scotminicabs. We manage definitely one of Edinburgh’s primary personal hire fleets, providing a smooth personalized get and taxi cab service to our clientele. We could provide best suited motors for all of your demands supplying the optimum measures of luxury and comfort.

    ReplyDelete
  156. Nature provides us blossoms, and also we provide you the best blossom bouquets - Carmel Flowers is a leading Florist in UAE that provides rejuvenating services to all your flower needs. Our widespread visibility throughout India as well as in the international blossom online market is to guarantee with over 25 years of experience in the floral market. Our unmatched floral gifting solutions offer the very best presents for all occasional and regular flower needs in simply a couple of clicks for individuals across the UAE. From flower arrangements, flower arrangements, boxed flowers, flower baskets, Forever Roses to exceptional flowers online, scroll with a countless variety of flower opportunities that will suit almost every occasion and also relationship online on our website. And also if you're unsure what to buy online, after that visit our physical flower store in Dubai to witness the handpicked blooms right in front of your eyes. With our network across the UAE, you can quickly send out stunning flower bouquets as well as even more to your loved ones in Dubai, Abu Dhabi, Sharjah, Umm al-Qaiwain, Fujairah, Ajman as well as Ra's al-Khaimah. Choose from an impressive variety of flowers on the internet as well as use same-day, midnight and fixed-time delivery choices to enjoy unique fresh flower delivery in Dubai as well as other cities across the UAE. We likewise supply encouraging 24-hours flower distribution in Dubai and 1-hour Abu Dhabi blossom distribution solutions for different celebrations. Carmel Flowers understand the feelings as well as love behind every order, and for this reason we are devoted to offering you the best as well as rejuvenating floral gifts each time, whenever. Flower Delivery Dubai

    ReplyDelete
  157. Eye treatment set up has been inaugurated in new six storied, purpose built Eye Treatment center. In which, Eye consultants are available 7 days a week, except national holidays.DCR Surgery in Lahore

    ReplyDelete
  158. Buy tramadol, Liquid ketamine, and other drugs from best suppliers in USA at
    tramadol red pill 225

    ReplyDelete
  159. Buy Venoms: Online Venom Pharmacy, Scorpion, Snake, Frog, & Toad at toad venom for sale

    ReplyDelete
  160. If you've in no way taken on a tiling venture before, you might be surprised with the aid of the various unique styles of tiles available. Ceramic and porcelain tiles are the maximum famous, however there are also glass tiles, cement tiles, steel tiles, and stone tiles—to call only a few.tile stores lexington

    ReplyDelete
  161. If you want your rest room tiles to remaining a long time, porcelain is the best fabric to pick. It’s strong, long lasting, and surprisingly water resistant, and it isn’t clean to scratch or stain.Porcelain can also be used to create lovely consequences, as it’s to be had in a huge range of colours and also can be made to seem like wood or stone, imparting a extra long lasting alternative to either.bathroom tile

    ReplyDelete
  162. Ceramic is a water-proof, resilient, smooth-to-preserve floors alternative this is suitable for lots under-grade basement installations. However, there are some precautions that want to be taken during and after set up to ensure the integrity of the basement.basement tile

    ReplyDelete
  163. Buy Tramadol Online And Buy Tramadol for Dogs & Cats Online at liquid ketamine for sale

    ReplyDelete
  164. At Teak we pleasure ourselves on being the leaders of producing and presenting the nice a-grade teak furnishings within the out of doors furnishings market. With years of revel in and after learning the exceptional out of doors materials we are pleased to introduce Aluminum Outdoor Furniture.teak table

    ReplyDelete
  165. An outside dining set is a short and smooth way to get a coordinated search for your outdoor space, patio or balcony. Choose from a extensive variety of sizes, substances and styles – from neat, café-like patio sets to sturdy, wood garden eating units. Simply upload some out of doors cushions for max comfort!outdoor dining furniture.

    ReplyDelete
  166. If you're searching out a long lasting, pleasant folding chair, nothing beats the teak folding chair. Teak Closeouts has some of the lowest teak furnishings charges on the net!teakcloseouts.com.

    ReplyDelete
  167. Ciaocustom is founded with one simple goal: to provide amazing custom products at amazing prices.serenity prayer wall art

    ReplyDelete
  168. i never know the use of adobe shadow until i saw this post. thank you for this! this is very helpful. GT350 STYLE MUSTANG FRONT BUMPER

    ReplyDelete
  169. salwar kameez online uk ALL DELIVERIES TO EU ARE EXPERIENCING DELAYS DUE TO COVID 19 - CUSTOMS & DUTY MAY NOW BE APPLICABLE FOR EU CUSTOMERS. SHOP NOW

    ReplyDelete
  170. buy elife ultra and enjoy upto AED in free benefits.http://applywifi.net/

    ReplyDelete
  171. JOKER123 has become Asia's leading live online gaming software development company.皇朝娛樂

    ReplyDelete
  172. wabohk has become Asia's leading live online gaming software development company.網上賭場

    ReplyDelete
  173. It less complicated for betters to pick any website listed on their platform because toto software program is known to verifying the most depended on playing web page from the bunch of faux having a bet web sites.안전놀이터.

    ReplyDelete
  174. Internet playing (a time period in large part interchangeable with interactive far flung and on line gambling) refers to the variety of wagering and gaming activities presented via Internet-enabled gadgets, which include computers, mobile and clever phones, pills and virtual tv.사설토토.

    ReplyDelete
  175. This is but some other benefit of on line playing. Once you sign up and begin playing, for each cent or dollar that you spend, you may earn loyalty factors. These then gather that will help you move up the membership program ranges with more suitable blessings and returns at distinct ranges.안전놀이터.

    ReplyDelete
  176. Indeed, you may be gambling with others truly and in real-time, but there's no physical presence or crowding of strangers while you play in online poker rooms.사설토토.

    ReplyDelete
  177. There are absolutely 3 foremost categories of Muktupolis merchandise that may be purchased on-line. These classes encompass a 먹튀폴리스 in addition to a chain of different products and services.먹튀폴리스.

    ReplyDelete
  178. The HCPL provide outstanding business consulting services through team work.Real Estate

    ReplyDelete
  179. Essentially, sports analytics is the exercise of applying mathematical and statistical principles to sports activities and associated peripheral sports. While there are numerous factors and priorities unique to the industry, sports analysts use the same primary strategies and method as some other type of statistics analyst.먹튀.

    ReplyDelete
  180. We are a diverse team of professionals with laser–sharp focus and passion. We are award-winning marketers, designers, and developers.Facebook ads help with Facebook ads

    ReplyDelete
  181. We as a dedicated and passionate team air cargo to pakistan Cargo work hard to ensure the Best Pakistan Cargo & Shipping services.air cargo to pakistan.

    ReplyDelete
  182. air cargo from dubai to pakistan Tracking made smooth - click on at the airline call and go to the air cargo tracking site of the Airline.air cargo from dubai to pakistan.

    ReplyDelete
  183. I read this article, it is really informative one. Your way of writing and making things clear is very impressive. Thanking you for such an informative article.dunning labs

    ReplyDelete
  184. Sports analytics is the exercise of applying mathematical and statistical standards to sports and associated peripheral activities. It is a field that applies information evaluation techniques to analyze diverse additives of the sports activities.토토사이트

    ReplyDelete
  185. Sports analytics has emerged as a field of research with increasing reputation propelled, in part, via the real-world success illustrated by the satisfactory-promoting book and movement photo, Moneyball.안전놀이터

    ReplyDelete
  186. 현금과 입회비를 지급하는 토토 1차 사이트 콤벳은 사기 검증은 물론 안전한 금전거래가 가능한 토토 대표 사이트를 온라인으로 추천한다꽁머니.

    ReplyDelete
  187. 사설토토는 중죄가 아닌 불법이므로 토토 가입 전 미리 인증을 받아야 합니다. 지금부터 프라이빗토토 가입 방법과 사기 확인 방법을 알려드리겠습니다사설토토.

    ReplyDelete
  188. ShareEcard is the Global Leader in Digital Business Cards.digital visiting card maker

    ReplyDelete
  189. In the event that you want to get acquainted with the diverse Walk Through and TOTO games offered in this website, at that point kindly visit this TOTO stroll through.먹튀폴리스

    ReplyDelete

Powered by Blogger.