Character orthogonality theorem: Difference between revisions

Revision as of 16:53, 21 June 2008

This fact is related to: linear representation theory
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This article describes an orthogonality theorem. View a list of orthogonality theorems

Name

This result is known as the first orthogonality theorem, character orthogonality theorem or row orthogonality theorem.

Statement

Statement over complex numbers

Let $G$ be a finite group and $C$ denote the field of complex numbers. Let $\bar{z}$ denote the complex conjugate of $z$ . Then, if $ρ_{1}$ and $ρ_{2}$ are two inequivalent irreducible linear representations, and $χ_{1}$ and $χ_{2}$ are their characters, we have:

$\sum_{g \in G} χ_{1} (g) \bar{χ_{2} (g)} = 0$

and:

$\sum_{g \in G} χ_{1} (g) χ_{1} (g) = | G |$

Statement over complex numbers in terms of inner product of class functions

Consider the space of complex-valued functions $G \to C$ . This is a $C$ -vector space in a natural way, with basis being the indicator functions of elements of $G$ . Consider the Hermitian inner product on this vector space given by:

$⟨ f_{1}, f_{2} ⟩ = \frac{1}{| G |} \sum_{g \in G} f_{1} (g) \bar{f_{2} (g)}$

Then, the characters form an orthonormal set of functions with respect to this basis.

Statement over general fields

Let $G$ be a finite group and $k$ a field whose characteristic does not divide the order of $G$ . Let $ρ_{1}$ and $ρ_{2}$ be two inequivalent irreducible linear representations of $G$ over $k$ and let $χ_{1}$ and $χ_{2}$ denote their characters. Then, the following are true:

$\sum_{g \in G} χ_{1} (g) χ_{2} (g^{- 1}) = 0$

And:

$\sum_{g \in G} χ_{1} (g) χ_{1} (g^{- 1}) = d | G |$

where $d = 1$ if the field $k$ is a splitting field for $G$ (for instance, if $k$ is sufficiently large for $G$ , viz., contains all the $m^{t h}$ roots of $1$ where $m$ is the exponent of $G$ ).

When $k$ is not sufficiently large, $d$ is the number of irreducible constituents of Failed to parse (unknown function "\chii"): {\displaystyle \chii} when taken over a splitting field containing $k$ .

Statement over general fields in terms of inner product of class functions

For functions $f_{1}, f_{2} : G \to k$ , define the following inner product:

$⟨ f_{1}, f_{2} ⟩ = \frac{1}{| G |} \sum_{g \in G} f_{1} (g) f_{2} (g^{- 1})$

Then, the character orthogonality theorem states that the characters of irreducible linear representations form an orthogonal set of elements, and further, if we are working over a sufficiently large field, they form an orthonormal set.

Note that by Maschke's lemma, the irreducible linear representations are precisely the indecomposable linear representations when the characteristic of $k$ does not divide the order of $G$ , so we can replace irreducible in the above statement with indecomposable.

Consequences

@@ Line 1: / Line 1: @@
-{{factrelatedto|linear representation theory}}
+{{fact related to|linear representation theory}}
+{{orthogonality theorem}}
+==Name==
 This result is known as the '''first orthogonality theorem''', '''character orthogonality theorem''' or '''row orthogonality theorem'''.
 ==Statement==
+===Statement over complex numbers===
+Let <math>G</math> be a [[finite group]] and <math>\mathbb{C}</math> denote the field of complex numbers. Let <math>\overline{z}</math> denote the complex conjugate of <math>z</math>. Then, if <math>\rho_1</math> and <math>\rho_2</math> are two inequivalent [[irreducible linear representation]]s, and <math>\chi_1</math> and <math>\chi_2</math> are their characters, we have:
+<math>\sum_{g \in G} \chi_1(g) \overline{\chi_2(g)} = 0</math>
+and:
+<math>\sum_{g \in G} \chi_1(g)\chi_1(g) = |G|</math>
+===Statement over complex numbers in terms of inner product of class functions===
+Consider the space of complex-valued functions <math>G \to \mathbb{C}</math>. This is a <math>\mathbb{C}</math>-vector space in a natural way, with basis being the indicator functions of elements of <math>G</math>. Consider the Hermitian inner product on this vector space given by:
+<math>\langle f_1, f_2 \rangle = \frac{1}{|G|}\sum_{g \in G} f_1(g) \overline{f_2(g)}</math>
+Then, the characters form an orthonormal set of functions with respect to this basis.
+===Statement over general fields===
 Let <math>G</math> be a [[finite group]] and <math>k</math> a [[field]] whose characteristic does not divide the order of <math>G</math>. Let <math>\rho_1</math> and <math>\rho_2</math> be two inequivalent [[irreducible linear representation]]s of <math>G</math> over <math>k</math> and let <math>\chi_1</math> and <math>\chi_2</math> denote their [[character]]s. Then, the following are true:
@@ Line 13: / Line 34: @@
 <math>\sum_{g \in G} \chi_1(g)\chi_1(g^{-1}) = d|G|</math>
-where <math>d=1</math> if the field <math>k</math> is a [[sufficiently large field]] for <math>G</math> (viz contains all the <math>m^{th}</math> roots of <math>1</math> where <math>m</math> is the [[exponent of a group|exponent]] of <math>G</math>).
+where <math>d=1</math> if the field <math>k</math> is a [[splitting field]] for <math>G</math> (for instance, if <math>k</math> is [[sufficiently large field|sufficiently large]] for <math>G</math>, viz., contains all the <math>m^{th}</math> roots of <math>1</math> where <math>m</math> is the [[exponent of a group|exponent]] of <math>G</math>).
-When <math>k</math> is not sufficiently large, <math>d</math> is the number of irreducible constituents of <math>\phi</math> when taken over a [[sufficiently large field]] containing <math>k</math>.
+When <math>k</math> is not sufficiently large, <math>d</math> is the number of irreducible constituents of <math>\chii</math> when taken over a [[splitting field]] containing <math>k</math>.
-===In terms of inner product of class functions===
+===Statement over general fields in terms of inner product of class functions===
 For functions <math>f_1,f_2: G \to k</math>, define the following inner product:
-<math><f_1,f_2> = \frac{1}{|G|}\sum_{g \in G}f_1(g)f_2(g^{-1})</math>
+<math>\langle f_1,f_2 \rangle = \frac{1}{|G|}\sum_{g \in G}f_1(g)f_2(g^{-1})</math>
 Then, the character orthogonality theorem states that the characters of irreducible linear representations form an orthogonal set of elements, and further, if we are working over a [[sufficiently large field]], they form an orthonormal set.