Representation theory of compact groups

The representation theory of compact groups is a branch of mathematics that examines where you how groups operate on existing structures.

Representation theory is used in many areas of mathematics, as well as in quantum chemistry and physics. Representation theory is used, among other things, in algebra to examine the structure of groups, in harmonic analysis and in number theory . For example, representation theory is used in the modern approach to better understand automorphic forms.

The theory of representations of compact groups can be in some measure on locally compact expand groups. In this context, representation theory is of great importance for harmonic analysis and the study of automorphic forms. For more detailed insights, evidence and more extensive information than given in this brief overview, can be used.

history

Characters of finite Abelian groups have been a tool used in number theory since the 18th century, but Frobenius was the first to extend this concept to non-Abelian groups in 1896. The theory of the characters of symmetrical and alternating groups was worked out by Frobenius and Young around 1900.

Burnside and Schur formulated Frobenius' character theory on the basis of matrix representations instead of characters. Burnside proved that every finite-dimensional representation of a finite group leaves a scalar product invariant and thus obtained a simpler proof of the (already known) decomposability into irreducible representations. Schur proved the lemma named after him and the orthogonality relations.

It was Emmy Noether who gave the customary definition of representations using linear mappings of a vector space.

Schur observed in 1924 that one can use invariant integration to extend the representation theory of finite groups to compact groups . Weyl then developed the representation theory of compact, connected Lie groups.

Definition and characteristics

A topological group is a group with a topology with respect to which the group linkage and the inverse formation are continuous . Such a group is called compact if every open coverage of in the topology has a finite partial coverage . Completed subgroups of a compact group are compact again. ${\displaystyle G}$

Let be a compact group and be a finite-dimensional vector space. A linear representation from to is a continuous group homomorphism d. h., is a continuous function in the two variables A linear representation of in a Banach space is defined as a continuous group homomorphism from the set of all bijective, limited linear operators to with continuous inverse. As can be dispensed with the last requirement. From now on we will deal particularly with representations of compact groups in Hilbert spaces . ${\displaystyle G}$ ${\displaystyle V}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle G}$ ${\displaystyle V}$ ${\displaystyle \rho \colon G\to {\text{GL}}(V),}$ ${\displaystyle \rho (s)v}$ ${\displaystyle s\in G,v\in V.}$
${\displaystyle G}$ ${\displaystyle V}$ ${\displaystyle G}$ ${\displaystyle V}$ ${\displaystyle \pi (s)^{-1}=\pi (s^{-1})}$

As with finite groups, one can define group algebra and convolution algebra . However, in the case of non-finite groups, the group algebra does not provide any helpful information, since the continuity condition is lost during formation. Instead, convolutional algebra takes its place. ${\displaystyle L^{1}(G)}$

Most of the properties of representations of finite groups can be transferred to compact groups with appropriate changes. For this we need an equivalent for the summation over a finite group:

Existence and uniqueness of the hair measure ${\displaystyle G}$

There is exactly one measure on a compact group so that: ${\displaystyle G}$ ${\displaystyle \mathrm {d} t,}$

• ${\displaystyle \int _{G}^{}f(t)\,\mathrm {d} t=\int _{G}f(st)\,\mathrm {d} t}$for all d. i.e. the measure is left invariant. ${\displaystyle s\in G,}$
• ${\displaystyle \int _{G}\,\mathrm {d} t=1,}$ so the whole group has moderation ${\displaystyle 1.}$

Such a left-invariant, normalized measure is called the hair measure of the group Da is compact, one can show that this measure is also right-invariant, i.e. i.e., it also applies ${\displaystyle G.}$
${\displaystyle G}$

• ${\displaystyle \int _{G}^{}f(t)\,\mathrm {d} t=\int _{G}f(ts)\,\mathrm {d} t}$ for all ${\displaystyle s\in G.}$

On a finite group, the hair measure with the normalization property from above is given by for all ${\displaystyle \mathrm {d} t(s)={\frac {1}{|G|}}}$ ${\displaystyle s\in G.}$

All definitions of finite group representations given in the Definition and Properties section also apply to compact group representations. There are a few modifications:
For a sub-representation you now need a closed sub-space. In the case of finitely dimensional presentation spaces, this is not required, since in this case each subspace is closed. Furthermore, two representations of a compact group are called equivalent if there is a linear operator between the respective representation spaces, which is continuous and invertible, and whose inverse is also continuous and which satisfies for all . If unitary, then the two representations are called unitary equivalent. In order to get an -invariant scalar product from a non-invariant one, one does not use the sum over but the integral. If a scalar product on a Hilbert space is not invariant with respect to the representation of , then forms ${\displaystyle \rho ,\pi }$ ${\displaystyle G}$ ${\displaystyle T}$ ${\displaystyle T\circ \rho (s)=\pi (s)\circ T}$ ${\displaystyle s\in G}$
${\displaystyle T}$
${\displaystyle G}$ ${\displaystyle G,}$ ${\displaystyle (\cdot |\cdot )}$ ${\displaystyle V,}$ ${\displaystyle \rho }$ ${\displaystyle G}$

${\displaystyle (v|u)_{\rho }=\int _{G}(\rho (t)v|\rho (t)u)\,\mathrm {d} t}$

an -invariant scalar product due to the hair dimension properties of With this, representations on Hilbert spaces can be viewed as unitary without restriction. ${\displaystyle G}$ ${\displaystyle V,}$ ${\displaystyle \mathrm {d} t.}$

Let be a compact group and let on the Hilbert space of quadratically integrable functions the operator is defined by where The mapping is a unitary representation of it is called the left-regular representation. You can also define the regular representation . Since the hair measure on is additionally right-invariant, the operator on is given by The right-regular representation is then the unitary representation given by the two representations and are dual to one another. If is not finite, these representations have no finite degree. For a finite group, the left and right regular representation, as defined at the beginning, are isomorphic to the right and left regular representation just defined, as in this case ${\displaystyle G}$ ${\displaystyle s\in G.}$ ${\displaystyle L^{2}(G)}$ ${\displaystyle G}$ ${\displaystyle L_{s}}$ ${\displaystyle L_{s}\Phi (t)=\Phi (s^{-1}t),}$ ${\displaystyle \Phi \in L^{2}(G),t\in G.}$
${\displaystyle s\mapsto L_{s}}$ ${\displaystyle G.}$ ${\displaystyle G}$ ${\displaystyle R_{s}}$ ${\displaystyle L^{2}(G)}$ ${\displaystyle R_{s}\Phi (t)=\Phi (ts).}$ ${\displaystyle s\mapsto R_{s}.}$ ${\displaystyle s\mapsto L_{s}}$ ${\displaystyle s\mapsto R_{s}}$
${\displaystyle G}$ ${\displaystyle L^{2}(G)\cong L^{1}(G)\cong \mathbb {C} [G].}$

Constructions and dismantling

The various construction possibilities of new representations from given works for compact groups as well as for finite groups, with the exception of the dual representation, which will be discussed later. The direct sum and the tensor product with a finite number of summands / factors are defined in exactly the same way as with finite groups. This also applies to the symmetrical and alternating square. In order to get the result for compact groups as well, that the irreducible representations of the product of two groups are exactly the tensor products of the irreducible representations of the individual groups up to isomorphism, we need a hair measure on the direct product . With the product topology, the direct product of two compact groups provides a compact group again. The hair measurement on this group is given by the product of the hair measurement on the individual groups. ${\displaystyle G_{1}\times G_{2}}$

For the dual representation on compact groups we need the topological dual of the vector space.This is the vector space of all continuous linear forms on Be a representation of the compact group in The dual representation is then defined by the property for all It follows that the dual representation is given is through for all This is again a continuous group homomorphism and thus a representation. The following applies to Hilbert spaces: is irreducible if and only if is irreducible. ${\displaystyle V'}$ ${\displaystyle V.}$ ${\displaystyle V.}$ ${\displaystyle \pi }$ ${\displaystyle G}$ ${\displaystyle V.}$ ${\displaystyle \pi '\colon G\to {\text{GL}}(V')}$ ${\displaystyle \langle \pi '(s)v',\pi (s)v\rangle =\langle v',v\rangle :=v'(v)}$ ${\displaystyle v\in V,v'\in V',s\in G.}$ ${\displaystyle \pi '(s)v'=v'\circ \pi (s^{-1})}$ ${\displaystyle v'\in V',s\in G.}$
${\displaystyle \pi }$ ${\displaystyle \pi '}$

By transferring the results from the section Representation Theory of Finite Groups # Decomposition of Representations we get the following theorems:

sentence

Each irreducible representation of a compact group in a Hilbert space is finite dimensional and there is a dot on so is unitary. This scalar product is unambiguous due to the normalization of the hair measure. Every representation of a compact group is isomorphic to a direct Hilbert sum of irreducible representations. ${\displaystyle (\tau ,V_{\tau })}$ ${\displaystyle G}$ ${\displaystyle V_{\tau },}$ ${\displaystyle \tau }$

Let be a unitary representation of the compact group For an irreducible representation we define the isotype of or the isotypic component in as the subspace, as with finite groups ${\displaystyle (\rho ,V_{\rho })}$ ${\displaystyle G.}$ ${\displaystyle (\tau ,V_{\tau })}$ ${\displaystyle \tau }$ ${\displaystyle \rho }$

${\displaystyle V_{\rho }(\tau )=\sum _{U\subset V_{\rho } \atop U\cong V_{\tau }}U.}$

This is the sum of all invariant closed subspaces that are -isomorphic to . Note that the isotypes of non-equivalent, irreducible representations are pairwise orthogonal. ${\displaystyle U,}$ ${\displaystyle G}$ ${\displaystyle V_{\tau }}$

sentence

• ${\displaystyle V_{\rho }(\tau )}$ is a closed invariant subspace of ${\displaystyle V_{\rho }.}$
• ${\displaystyle V_{\rho }(\tau )}$is -isomorphic to a direct sum of copies of ${\displaystyle G}$ ${\displaystyle V_{\tau }.}$
• ${\displaystyle V_{\rho }}$is the direct Hilbert sum of the isotypes where all isomorphism classes of the irreducible representations run through. This division is the canonical division. ${\displaystyle V_{\rho }(\tau ),}$ ${\displaystyle \tau }$

The projection belonging to the canonical decomposition where is an isotype of is given by for compact groups ${\displaystyle p_{\tau }\colon V\to V(\tau ),}$ ${\displaystyle V(\tau )}$ ${\displaystyle V}$

${\displaystyle p_{\tau }(v)=n_{\tau }\int _{G}{\overline {\chi _{\tau }(t)}}\rho (t)(v)\,\mathrm {d} t,}$

where and is the character belonging to the irreducible representation . ${\displaystyle n_{\tau }={\text{dim}}(V(\tau ))}$ ${\displaystyle \chi _{\tau }}$ ${\displaystyle \tau }$

Projection formula

For each representation of a compact group, define ${\displaystyle (\rho ,V)}$ ${\displaystyle G}$

${\displaystyle V^{G}=\{v\in V\mid \rho (s)v=v\;\;\forall s\in G\}.}$

In general it is non- linear. Set the mapping is defined as endomorphism on by the property ${\displaystyle \rho (s)\colon V\to V}$ ${\displaystyle G}$ ${\displaystyle \textstyle Pv:=\int _{G}\rho (s)v\,\mathrm {d} s.}$ ${\displaystyle P}$ ${\displaystyle V}$

${\displaystyle (\int _{G}\rho (s)v\,\mathrm {d} s|w)=\int _{G}(\rho (s)v|w)\,\mathrm {d} s,}$

which applies to the scalar product of the Hilbert space . ${\displaystyle V}$

Then there is a -linear mapping because it holds ${\displaystyle P}$ ${\displaystyle G}$

${\displaystyle {\begin{aligned}(\int _{G}\rho (s)(\rho (t)v)\,\mathrm {d} s|w)&=\int _{G}(\rho (tst^{-1})(\rho (t)v)|w)\,\mathrm {d} s\\&=\int _{G}(\rho (ts)v|w)\,\mathrm {d} s\\&=\int (\rho (t)\rho (s)v|w)\,\mathrm {d} s\\&=(\rho (t)\int _{G}\rho (s)v\,\mathrm {d} s|w),\end{aligned}}}$

taking advantage of the invariance of the hair measure.

Proposition

The figure is a projection from to ${\displaystyle P}$ ${\displaystyle V}$ ${\displaystyle V^{G}.}$

If the representation is finite dimensional one can determine the direct sum of the trivial partial representations, as with finite groups.

Characters, Schur's Lemma and the dot product

The representations of compact groups are generally viewed on Hilbert or Banach spaces . These are usually not finite-dimensional. It does not make sense to speak of characters for any representations of compact groups . However, one can usually restrict oneself to the finite-dimensional case:

Since irreducible representations of compact groups are finite-dimensional and, with the results from the first sub-chapter, are unitary without restriction, irreducible characters can be defined in the same way as for finite groups.
As with finite groups, the characters get along with the constructions as long as the constructed representations remain finite dimensional.

Also for compact groups that applies Schur's Lemma:
Be an irreducible unitary representation of a compact group then each bounded operator with the property of a scalar multiple of identity, d. i.e., there is such a thing ${\displaystyle (\pi ,V)}$ ${\displaystyle G.}$ ${\displaystyle T\colon V\to V}$ ${\displaystyle T\circ \pi (s)=\pi (s)\circ T,}$ ${\displaystyle \lambda \in \mathbb {C} ,}$ ${\displaystyle T=\lambda {\text{Id}}.}$

Definitions

On the set of all quadratically integrable functions of a compact group one can define a scalar product by ${\displaystyle L^{2}(G)}$

${\displaystyle (\Phi |\Psi )=\int _{G}\Phi (t){\overline {\Psi (t)}}\,\mathrm {d} t.}$

A bilinear form is also defined for a compact group ${\displaystyle L^{2}(G)}$ ${\displaystyle G}$

${\displaystyle \langle \Phi ,\Psi \rangle =\int _{G}\Phi (t)\Psi (t^{-1})\,\mathrm {d} t.}$

The bilinear form on the display spaces is defined in the same way as with finite groups.
Analogous to finite groups, the following results apply:

Theorem (Schur's orthogonality relations)

Are the characters of two nonisomorphic irreducible representations shall apply ${\displaystyle \chi ,\chi '}$ ${\displaystyle V,V',}$

• ${\displaystyle (\chi |\chi ')=0.}$
• ${\displaystyle (\chi |\chi )=1,}$d. h., has "norm" ${\displaystyle \chi }$ ${\displaystyle 1.}$

sentence

Let be a representation of It applies where the are irreducible. Since the direct sum is finite, a character can be defined for by the sum of the irreducible characters . Let now be an irreducible representation of with character Then the following applies: The number of partial representations that are too equivalent does not depend on the given decomposition and corresponds to the scalar product , i.e. the isotype of is independent of the choice of decomposition and it applies : ${\displaystyle V}$ ${\displaystyle G.}$ ${\displaystyle V=W_{1}\oplus \cdots \oplus W_{k},}$ ${\displaystyle W_{i}}$ ${\displaystyle V}$ ${\displaystyle \xi }$ ${\displaystyle (\tau ,W)}$ ${\displaystyle G}$ ${\displaystyle \chi .}$
${\displaystyle W_{i}}$ ${\displaystyle W}$ ${\displaystyle (\xi |\chi ).}$
${\displaystyle \tau }$ ${\displaystyle V(\tau )}$ ${\displaystyle V}$

${\displaystyle {\text{dim}}(W)(\xi |\chi )={\text{dim}}(V(\tau ))={\text{dim}}(W)\langle V,W\rangle .}$

sentence

Two irreducible representations with the same character are isomorphic.

Irreducibility criterion

Let the character of a representation then be and it applies if and only if is irreducible. ${\displaystyle \chi }$ ${\displaystyle V,}$ ${\displaystyle (\chi |\chi )\in \mathbb {N} _{0}}$ ${\displaystyle (\chi |\chi )=1}$ ${\displaystyle V}$

Together with the first set thus form the characters of irreducible representations with respect to this scalar product an orthonormal system on ${\displaystyle G}$ ${\displaystyle L^{2}(G).}$

Corollary

Every irreducible representation of is contained -mal in the left-regular representation. ${\displaystyle V}$ ${\displaystyle G}$ ${\displaystyle {\text{dim}}(V)}$

lemma

Be a compact group. Then are equivalent: ${\displaystyle G}$

• ${\displaystyle G}$ is abelian.
• All irreducible representations of have degrees ${\displaystyle G}$ ${\displaystyle 1.}$

Orthonormal property

Be a compact group. The pairwise non-isomorphic irreducible characters of form an orthonormal basis of ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle L^{2}(G).}$

This is shown in the same way as with finite groups by showing that there is no quadratically integrable function other than that which is orthogonal to the irreducible characters. ${\displaystyle 0}$

As with finite groups, the following also applies: The number of irreducible representations of a group except for isomorphism corresponds to the number of conjugation classes of However, a compact group generally has an infinite number of conjugation classes. ${\displaystyle G}$ ${\displaystyle G.}$

Induced representations

If there is a closed subgroup of finite index in the compact group , the definition of the induced representation can be adopted as for finite groups. However, the induced representation can also be defined more generally, so that the definition is also valid if the index of in is not finite. Be to a unitary representation of the completed sub-group The steadily induced representation is defined as follows: With we denote the Hilbert space of all measurable, square-integrable functions with the property that for all is the standard and the presentation is given by law Translation: The induced representation then again a unitary representation. Since is compact, the induced representation is broken down into the direct sum of irreducible representations of. It is true that all irreducible representations belonging to the same isotype occur with the multiplicity that corresponds. If there is a representation of then there is a canonical isomorphism ${\displaystyle H}$ ${\displaystyle G}$
${\displaystyle H}$ ${\displaystyle G}$
${\displaystyle (\eta ,V_{\eta })}$ ${\displaystyle H.}$ ${\displaystyle {\text{Ind}}_{H}^{G}(\eta )=(I,V_{I})}$
${\displaystyle V_{I}}$ ${\displaystyle \Phi \colon G\to V_{\eta }}$ ${\displaystyle \Phi (ls)=\eta (l)\Phi (s)}$ ${\displaystyle l\in H,s\in G.}$ ${\displaystyle ||\Phi ||_{G}={\text{sup}}_{s\in G}||\Phi (s)||}$ ${\displaystyle I}$ ${\displaystyle I(s)\Phi (k)=\Phi (ks).}$

${\displaystyle G}$ ${\displaystyle G.}$ ${\displaystyle {\text{dim}}({\text{Hom}}_{G}(V_{\eta },V_{I}))}$
${\displaystyle (\rho ,V_{\rho })}$ ${\displaystyle G,}$

${\displaystyle T\colon {\text{Hom}}_{G}(V_{\rho },I_{H}^{G}(\eta ))\to {\text{Hom}}_{H}(V_{\rho }|_{H},V_{\eta })=\langle V_{\eta },V_{I}\rangle _{G}.}$

The Frobenius reciprocity is carried over to compact groups with the modified definition of the scalar product and the bilinear form, whereby the theorem applies here to quadratically integrable functions instead of to class functions and the subgroup must be closed. ${\displaystyle G}$ ${\displaystyle H}$

Peter-Weyl's theorem

→ Main article : Theorem of Peter-Weyl

Another important result of the representation theory of compact groups is the Peter-Weyl theorem. This is usually proven in the harmonic analysis , in which it takes a central place.

Peter-Weyl's theorem

Be a compact group. For any irreducible representation of let be an orthonormal basis of Define the matrix coefficients for Then is ${\displaystyle G}$ ${\displaystyle (\tau ,V_{\tau })}$ ${\displaystyle G}$ ${\displaystyle e_{1},\dotsc ,e_{{\text{dim}}(\tau )}}$ ${\displaystyle V_{\tau }.}$
${\displaystyle \tau _{k,l}(s)=\langle \tau (s)e_{k},e_{l}\rangle }$ ${\displaystyle 1\leq k,l\leq {\text{dim}}(\tau ),s\in G.}$

${\displaystyle ({\sqrt {{\text{dim}}(\tau )}}\tau _{k,l})_{k,l}}$

an orthonormal basis of ${\displaystyle L^{2}(G).}$

Second version of Peter-Weyl's theorem

There is a natural isomorphism ${\displaystyle G\times G}$

${\displaystyle L^{2}(G)\cong _{G\times G}{\widehat {\bigoplus }}_{\tau \in {\hat {G}}}{\text{End}}(V_{\tau })\cong _{G\times G}{\widehat {\bigoplus }}_{\tau \in {\hat {G}}}\tau \otimes \tau ^{*},}$

where denotes the set of all irreducible representations from up to isomorphism and the representation space belonging to the representation. ${\displaystyle {\hat {G}}}$ ${\displaystyle G}$ ${\displaystyle V_{\tau }}$ ${\displaystyle \tau }$

This isomorphism maps a given to where ${\displaystyle \Phi \in L^{2}(G)}$ ${\displaystyle \sum _{\tau \in {\hat {G}}}\tau (\Phi ),}$

${\displaystyle \tau (\Phi )=\int _{G}\Phi (t)\tau (t)\,\mathrm {d} t\in {\text{End}}(V_{\tau }).}$

In this way we get a generalization of the Fourier series for functions on compact groups.
This sentence is just a reformulation of the first version.

A proof of this theorem and more information on representation theory of compact groups can be found in.

literature

1. ^ Anton Deitmar: Automorphic forms. Springer-Verlag, 2010, ISBN 978-3-642-12389-4 , pp. 89-93, 185-189.
2. ^ ^{A b} Siegfried Echterhoff, Anton Deitmar: Principles of harmonic analysis. Springer-Verlag, 2009, ISBN 978-0-387-85468-7 , pp. 127-150.
3. ↑ Frobenius: About group characters. Meeting reports of the Royal Prussian Academy of Sciences in Berlin (1896), 985-1021; in Gesammelte Abhandlungen, Volume III Springer-Verlag, New York, 1968, 1-37.
4. ^ Anthony W. Knapp: Group representations and harmonic analysis from Euler to Langlands. Notices of the American Mathematical Society 43 (1996).