Orthogonality relations

In the mathematical branch of group theory , orthogonality relations are certain relationships between characters in representations of a group. The name comes from the fact that one can define an inner product on a suitable function space that contains the characters , with regard to the different characters of which are actually orthogonal.

Definitions

In the following we assume a finite group . For a body we consider the set of all functions . Since you can use the definition ${\displaystyle G}$ ${\displaystyle K}$ ${\displaystyle S(G,K)}$ ${\displaystyle \alpha \colon G\rightarrow K}$

${\displaystyle (\alpha +k\beta )(g):=\alpha (g)+k\beta (g)}$   For   ${\displaystyle \alpha ,\beta \in S(G,K),\,k\in K,\,g\in G}$

can add and multiply with elements from the body, there is obviously a K -vector space . You can even multiply two such functions, that is, it is even a K -algebra .

Finite-dimensional representations of the group over a body are homomorphisms into the general linear group over a finite-dimensional vector space . If the trace denotes , the composition is called the character of the representation. The characters of representations are evidently elements of space . The character of an irreducible representation is also called irreducible. ${\displaystyle G}$ ${\displaystyle K}$ ${\displaystyle \rho \colon G\rightarrow \mathrm {GL} (V)}$ ${\displaystyle K}$ ${\displaystyle V}$ ${\displaystyle \mathrm {tr} }$ ${\displaystyle \mathrm {GL} (V)\rightarrow K}$ ${\displaystyle \mathrm {tr} \circ \rho \colon G\rightarrow K}$ ${\displaystyle S(G,K)}$

From now on we consider the case that the characteristics of the body are not part of the group order . This is the case for bodies with characteristic 0 and thus for the important bodies or always. In particular, we can divide in the body by the group order and thus ${\displaystyle \mathbb {Q} ,\mathbb {R} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle |G|}$

${\displaystyle \langle \alpha ,\beta \rangle _{G}:={\frac {1}{|G|}}\sum _{x\in G}\alpha (x)\beta (x^{-1})}$

define. It is easy to show that a symmetrical, non-degenerate K -bilinear form is on . One speaks therefore of an inner product, even if this term is reserved for the body or for many authors . is by far the most important use case because of the additional algebraic closure . ${\displaystyle \langle \,,\,\rangle _{G}}$ ${\displaystyle S(G,K)}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle K=\mathbb {C} }$

The orthogonality relations

Let it be a finite group, a body, the characteristics of which are not part of the group order, and let us be two different, irreducible characters of the group over . Then: ${\displaystyle G}$ ${\displaystyle K}$ ${\displaystyle \chi }$ ${\displaystyle \psi }$ ${\displaystyle K}$

• ${\displaystyle \langle \chi ,\psi \rangle _{G}=0}$, that is, different irreducible characters are orthogonal.
• If algebraically closed, then the following applies , that is, the irreducible characters are orthonormal . ${\displaystyle K}$ ${\displaystyle \langle \chi ,\chi \rangle _{G}=1}$

Character board

→ Main article : Character board

We look at the body . As is well known, a finite group has as many irreducible characters as there are conjugation classes . Furthermore, the characters on conjugation classes are constant, so that it is sufficient to know the values for any chosen element . If you determine that the one-element conjugation class of the neutral element and always the character of the trivial representation should always be, then you can easily overlook the entirety of the characters in the following square scheme called the character table, with the entries corresponding to the dimensions of the characters are irreducible representations. ${\displaystyle \mathbb {C} }$ ${\displaystyle \chi _{1},\ldots ,\chi _{r}}$ ${\displaystyle C_{1},\ldots ,C_{r}}$ ${\displaystyle \chi _{i}(c_{j})}$ ${\displaystyle c_{j}\in C_{j}}$ ${\displaystyle C_{1}}$ ${\displaystyle \chi _{1}}$ ${\displaystyle d_{i}=\chi _{i}(1)}$

${\displaystyle G}$	${\displaystyle 1}$	${\displaystyle |C_{2}|}$	${\displaystyle \ldots }$	${\displaystyle |C_{r}|}$
	${\displaystyle 1}$	${\displaystyle c_{2}}$	${\displaystyle \ldots }$	${\displaystyle c_{r}}$
${\displaystyle \chi _{1}}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle \ldots }$	${\displaystyle 1}$
${\displaystyle \chi _{2}}$	${\displaystyle d_{2}}$	${\displaystyle \chi _{2}(c_{2})}$	${\displaystyle \ldots }$	${\displaystyle \chi _{2}(c_{r})}$
${\displaystyle \vdots }$	${\displaystyle \vdots }$	${\displaystyle \vdots }$	${\displaystyle \ddots }$	${\displaystyle \vdots }$
${\displaystyle \chi _{r}}$	${\displaystyle d_{r}}$	${\displaystyle \chi _{r}(c_{2})}$	${\displaystyle \ldots }$	${\displaystyle \chi _{r}(c_{r})}$

The orthogonality relations are reflected in the character table as follows.

Orthogonality of the lines

The orthogonality relations are written using the Kronecker delta as compact

${\displaystyle |G|\delta _{i,j}=|G|\langle \chi _{i},\chi _{j}\rangle _{G}=\sum _{x\in G}\chi _{i}(x)\chi _{j}(x^{-1})=\sum _{k=0}^{r}|C_{k}|\chi _{i}(c_{k}){\overline {\chi _{j}(c_{k})}}}$,

because the characters are constant on conjugation classes and for characters . Despite the factors that arise , this relationship is called the orthogonality of the lines of the character table. You can also read these equations as matrix multiplication . If one defines and , then the above equation is nothing other than ${\displaystyle \chi (x^{-1})={\overline {\chi (x)}}}$ ${\displaystyle \chi }$ ${\displaystyle |C_{k}|}$ ${\displaystyle X=(\chi _{i}(c_{k}))_{i,k=1,\ldots ,n}}$ ${\displaystyle Y=(|C_{k}|{\overline {\chi _{j}(c_{k})}})_{k,j=1,\ldots ,n}}$

${\displaystyle |G|\cdot 1_{r}=XY}$,

where is the identity matrix . In particular, and are invertible . ${\displaystyle 1_{r}}$ ${\displaystyle X}$ ${\displaystyle Y}$

Orthogonality of the columns

If you multiply the above matrix equation from the left by and from the right by , you get: ${\displaystyle X^{-1}}$ ${\displaystyle X}$

${\displaystyle |G|\cdot 1_{r}=YX}$

In component notation, that means

${\displaystyle |G|\delta _{k,l}=\sum _{j=1}^{r}|C_{k}|{\overline {\chi _{j}(c_{k})}}\chi _{j}(c_{l})}$

or, since that is constant under the sum: ${\displaystyle |C_{k}|}$

${\displaystyle {\frac {|G|}{|C_{k}|}}\delta _{k,l}=\sum _{j=1}^{r}{\overline {\chi _{j}(c_{k})}}\chi _{j}(c_{l})}$

This relationship is called in an obvious way the orthogonality of the columns.

Orthogonality relations for representations

Since characters are the traces of representations, one would expect similar orthogonality relations for representations, in fact these are used for the proof of the above orthogonality relations. Since representations do not assume their values ​​in the body , but in general linear groups over vector spaces, the formulation is somewhat more complex. As above, we limit ourselves to finite-dimensional representations and choose the coordinate space as the vector space of a -dimensional representation , which ultimately corresponds to the ambiguous choice of a base. A representation thus has values ​​in the- row square matrices above and one can use the component functions ${\displaystyle K}$ ${\displaystyle d}$ ${\displaystyle K^{d}}$ ${\displaystyle \rho \colon G\rightarrow \mathrm {GL} (K^{d})\cong \mathrm {Mat} (d,K)}$ ${\displaystyle d}$ ${\displaystyle K}$

${\displaystyle \rho _{i,j}\colon G\rightarrow K,\quad \rho _{i,j}(x)={\text{(i,j)-te Komponente von }}\rho (x)}$

consider. With these definitions the following sentence consists:

Let it be a finite group, a body, the characteristics of which are not part of the group order; and let two irreducible representations of the group over . Then: ${\displaystyle G}$ ${\displaystyle K}$ ${\displaystyle \rho }$ ${\displaystyle \sigma }$ ${\displaystyle K}$

• Are and are not equivalent, so is ${\displaystyle \rho }$ ${\displaystyle \sigma }$

${\displaystyle \sum _{x\in G}\rho _{i,j}(x)\sigma _{r,s}(x^{-1})=0}$   for all component functions of and . ${\displaystyle \rho }$ ${\displaystyle \sigma }$

• If algebraically closed, then applies ${\displaystyle K}$

${\displaystyle \sum _{x\in G}\rho _{i,j}(x)\rho _{r,s}(x^{-1})={\frac {|G|}{\mathrm {dim} (\rho )}}\delta _{i,s}\delta _{j,r}}$   for all component functions of . ${\displaystyle \rho }$

Applications

The orthogonality relations form a cornerstone of the very extensive representation theory of groups . In the following we restrict ourselves to the case and only bring a few very elementary applications to illustrate the use of the orthogonality relations. ${\displaystyle K=\mathbb {C} }$

Hum of irreducible characters

The different, irreducible characters of a group are not only orthonormal, for dimensional reasons they also create the space of so-called class functions, i.e. of functions that are constant on conjugation classes. The irreducible characters therefore form an orthonormal basis in the space of the class functions. In particular, every class function is a unique linear combination of irreducible characters. ${\displaystyle \chi _{1},\ldots ,\chi _{r}}$

According to Maschke's theorem , every finite-dimensional representation of a finite group is a direct sum of irreducible representations. By creating a trace one obtains that every character is the sum of irreducible characters, that is: ${\displaystyle \chi }$

${\displaystyle \chi =\sum _{i=1}^{r}n_{i}\chi _{i}}$   With ${\displaystyle n_{i}\in \mathbb {N} _{0}}$

The coefficients can be determined immediately using orthogonality: ${\displaystyle n_{i}}$

${\displaystyle n_{i}=\sum _{j=1}^{r}\delta _{i,j}n_{j}=\sum _{j=1}^{r}n_{j}\langle \chi _{i},\chi _{j}\rangle _{G}=\langle \chi _{i},\sum _{j=1}^{r}n_{j}\chi _{j}\rangle _{G}=\langle \chi _{i},\chi \rangle _{G}}$

Irreducibility criterion

If there is a finite-dimensional representation with character , then is irreducible if and only if . ${\displaystyle \rho }$ ${\displaystyle \chi }$ ${\displaystyle \rho }$ ${\displaystyle \langle \chi ,\chi \rangle _{G}=1}$

Proof: That characters of irreducible representations have this property is the second point of the above orthogonality relations. Conversely, every character is the sum of irreducible characters and this follows because of the orthonormality with natural numbers . If this is equal to 1, there is only the option for one and for all other coefficients. It follows from this that it is irreducible and therefore also . ${\displaystyle \textstyle \chi =\sum _{i=1}^{r}n_{i}\chi _{i}}$ ${\displaystyle \textstyle \langle \chi ,\chi \rangle _{G}=\sum _{i=1}^{r}n_{i}^{2}}$ ${\displaystyle n_{i}}$ ${\displaystyle n_{i}=1}$ ${\displaystyle i}$ ${\displaystyle n_{j}=0}$ ${\displaystyle \chi =\chi _{i}}$ ${\displaystyle \chi }$ ${\displaystyle \rho }$

Completion of character boards

By means of the orthogonality relations, parts of character tables can be made accessible. As an example we consider the symmetric group S ₃ . In addition to the trivial conjugation class , we have the conjugation class of the three transpositions and the two elements of order  3. As obvious one-dimensional representations, we have the trivial representation and the sign function . Since there are just as many characters as there are conjugation classes, there is still one character missing whose values ​​we do not yet know. So the character board has the shape ${\displaystyle C_{1}}$ ${\displaystyle C_{2}}$ ${\displaystyle C_{3}}$ ${\displaystyle \chi _{1}}$ ${\displaystyle \chi _{2}}$ ${\displaystyle \chi _{3}}$

${\displaystyle S_{3}}$	${\displaystyle 1}$	${\displaystyle 3}$	${\displaystyle 2}$
	${\displaystyle c_{1}=1}$	${\displaystyle c_{2}=(1,2)}$	${\displaystyle c_{3}=(1,2,3)}$
${\displaystyle \chi _{1}}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 1}$
${\displaystyle \chi _{2}}$	${\displaystyle 1}$	${\displaystyle -1}$	${\displaystyle 1}$
${\displaystyle \chi _{3}}$	${\displaystyle x}$	${\displaystyle y}$	${\displaystyle z}$

with as yet unknown . These can be determined using the orthogonality relations without knowing the missing irreducible representation; no further details of the group are even required. ${\displaystyle x,y,z}$

From the orthogonality for columns it follows for the first column

${\displaystyle 6=|S_{3}|=1^{2}+1^{2}+x^{2}}$,

so . Since the dimensions (traces of standard matrices) are in the first column, it must be, that is . ${\displaystyle x=\pm 2}$ ${\displaystyle x>0}$ ${\displaystyle x=2}$

For the second column follows

${\displaystyle {\frac {6}{3}}={\frac {|S_{3}|}{|C_{2}|}}=1^{2}+1^{2}+y^{2}}$,

and there just remains . ${\displaystyle y=0}$

Since the third column is orthogonal to the first, it follows

${\displaystyle 0=1\cdot 1+1\cdot 1+2\cdot z}$,

so . The character board of group S ₃ is thus completely determined. ${\displaystyle z=-1}$

Remarks

The orthogonality relations go back to a work by Ferdinand Georg Frobenius from 1896, where character tables are also discussed. A revision of this theory was undertaken by Issai Schur , which is why one also finds the term Schur's orthogonality relations.

Finite groups are compact groups whose hairy measure assigns the measure to every single-element set . Analogous results are obtained for infinite compact groups if one replaces summations of the form with integrals according to Haar's measure. John von Neumann had achieved the first results in this direction in 1934, but still using almost periodic functions . A more modern representation that uses the hairy measure can be found, for example, in the textbook "Representations of Finite and Compact Groups" by Barry Simon mentioned below . ${\displaystyle \textstyle {\frac {1}{|G|}}}$ ${\displaystyle \textstyle {\frac {1}{|G|}}\sum _{x\in G}}$

Individual evidence

1. ^ DJS Robinson : A Course in the Theory of Groups. Springer-Verlag, 1996, ISBN 0-387-94461-3 , sentence 8.3.5.
2. ↑ Kurt Meyberg: Algebra II. Carl Hanser Verlag, 1976, ISBN 3-446-12172-2 , sentence 9.5.6 (orthogonality relation).
3. ^ DJS Robinson: A Course in the Theory of Groups. Springer-Verlag, 1996, ISBN 0-387-94461-3 , Chapter 8, Page 232: The Character Table.
4. ^ DJS Robinson: A Course in the Theory of Groups. Springer-Verlag, 1996, ISBN 0-387-94461-3 , sentence 8.3.4.
5. ^ Kurt Meyberg: Algebra II. Carl Hanser Verlag, 1976, ISBN 3-446-12172-2 , sentence 9.6.6.
6. ^ DJS Robinson: A Course in the Theory of Groups. Springer-Verlag, 1996, ISBN 0-387-94461-3 , sentence 8.3.12.
7. ^ Kurt Meyberg: Algebra II. Carl Hanser Verlag, 1976, ISBN 3-446-12172-2 , sentence 9.6.4.
8. ^ Kurt Meyberg: Algebra II. Carl Hanser Verlag, 1976, ISBN 3-446-12172-2 , example 9.7.1.b.
9. ^ FG Frobenius: About group characters. Prussian Academy of Sciences Berlin: Meeting reports of the Prussian Academy of Sciences in Berlin 1896, available here as a pdf.
10. ^ I. Schur: New justification of the theory of group characters. Meeting reports of the Prussian Academy of Sciences in Berlin 1905, page 406.
11. ↑ J. v. Neumann: Almost periodic functions in groups. Trans. Amer. Math. Soc. (1934), Vol. 36, No. 3, pp. 445-492.
12. ↑ Barry Simon: Representations of Finite and Compact Groups. American Mathematical Society 1996, Volume 10.