Character orthogonality theorem: Difference between revisions

Revision as of 02:59, 13 July 2011

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:

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

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. In other words, if $χ_{1}, χ_{2}$ are the characters of inequivalent irreducible representations, we get:

$⟨ χ_{1}, χ_{1} ⟩ = 1$

and

$⟨ χ_{1}, χ_{2} ⟩ = 0$

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 a splitting field, $d$ is the number of irreducible constituents (with multiplicities) of $χ_{1}$ 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, if $χ_{1}, χ_{2}$ are the characters of inequivalent irreducible representations, we get:

$⟨ χ_{1}, χ_{1} ⟩ = d$

where $d = 1$ if $k$ is a splitting field and in general $d$ is the number of irreducible constituents (with multiplicites) of $χ_{1}$ when taken over a splitting field). Also:

$⟨ χ_{1}, χ_{2} ⟩ = 0$

Interpretation in characteristic zero and prime characteristic

In characteristic zero, both sides are being viewed as elements in a field of characteristic zero.

In prime characteristic, however, the inner product is taking values modulo the prime characteristic, hence is not actually an integer, whereas the right side (1, 0, or $d$ ) is an integer, which needs to be reduced modulo the prime to be interpreted on the other side.

Consequences

Orthogonal projection formula: This is a formula that uses the character of a finite-dimensional representation to determine the irreducible constituents of the representation and their multiplicities.
Column orthogonality theorem
Character determines representation in characteristic zero

@@ Line 22: / Line 22: @@
 <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.
+Then, the characters form an orthonormal set of functions with respect to this basis. In other words, if <math>\chi_1, \chi_2</math> are the characters of inequivalent irreducible representations, we get:
+<math>\langle \chi_1, \chi_1 \rangle = 1</math>
+and
+<math>\langle \chi_1, \chi_2 \rangle = 0</math>
 ===Statement over general fields===
@@ Line 36: / Line 42: @@
 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>\chi_1</math> when taken over a [[splitting field]] containing <math>k</math>.
+When <math>k</math> is not a splitting field, <math>d</math> is the number of irreducible constituents (with multiplicities) of <math>\chi_1</math> when taken over a [[splitting field]] containing <math>k</math>.
 ===Statement over general fields in terms of inner product of class functions===
@@ Line 44: / Line 50: @@
 <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.
+Then, if <math>\chi_1, \chi_2</math> are the characters of inequivalent irreducible representations, we get:
+<math>\langle \chi_1, \chi_1 \rangle = d</math>
+where <math>d = 1</math> if <math>k</math> is a splitting field and in general <math>d</math> is the number of irreducible constituents (with multiplicites) of <math>\chi_1</math> when taken over a splitting field). Also:
+<math>\langle \chi_1, \chi_2 \rangle = 0</math>
+===Interpretation in characteristic zero and prime characteristic===
+In characteristic zero, both sides are being viewed as elements in a field of characteristic zero.
-Note that by [[Maschke's lemma]], the irreducible linear representations are precisely the [[indecomposable linear representation]]s when the characteristic of <math>k</math> does not divide the order of <math>G</math>, so we can replace irreducible in the above statement with indecomposable.
+In prime characteristic, however, the inner product is taking values modulo the prime characteristic, hence is not actually an integer, whereas the right side (1, 0, or <math>d</math>) is an integer, which needs to be reduced modulo the prime to be interpreted on the other side.
 ==Consequences==