Difference between revisions of "Characters are cyclotomic integers"

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* [[Element of finite order is semisimple and eigenvalues are roots of unity]] (the semisimplicity requires that the order be relatively prime to the characteristic of the field, something we don't need here).
 
* [[Element of finite order is semisimple and eigenvalues are roots of unity]] (the semisimplicity requires that the order be relatively prime to the characteristic of the field, something we don't need here).
  
==Consequences==
+
===Applications===
  
* Characters are algebraic integers: Any cyclotomic integer is an algebraic integer, so every character takes values in algebraic integers.
+
* [[Characters are algebraic integers]]: Any cyclotomic integer is an algebraic integer, so every character takes values in algebraic integers.
 
* If, over a field of characteristic zero, a character takes a rational value, that value must be an integer.
 
* If, over a field of characteristic zero, a character takes a rational value, that value must be an integer.
 +
 
==Proof==
 
==Proof==
  
''Given'': A finite group <math>G</math>, field <math>k</math>, a finite-dimensional linear representation <math>(V,\rho)</math> of <math>G</math> over <math>k</math>. <math>\chi</math> is the character of <math>\rho</math>.
+
'''Given''': A finite group <math>G</math>, field <math>k</math>, a finite-dimensional linear representation <math>(V,\rho)</math> of <math>G</math> over <math>k</math>. <math>\chi</math> is the character of <math>\rho</math>.
  
''To prove'': For any <math>g</math>, <math>\chi(g)</math> is a cyclotomic integer over the prime subfield of <math>k</math>.
+
'''To prove''': For any <math>g</math>, <math>\chi(g)</math> is a cyclotomic integer over the prime subfield of <math>k</math>.
  
''Proof'': <math>\chi(g)</math> is the sum of eigenvalues of <math>\rho(g)</math>, counted with multiplicity, in the algebraic closure of <math>k</math>. Since <math>G</math> is finite, <math>g</math> has finite order, so <math>\rho(g)</math> has finite order, and hence <math>\rho(g)</math> satisfies a polynomial of the form <math>x^n - 1</math>. Hence, every eigenvalue of <math>\rho(g)</math> is a <math>n^{th}</math> root of unity, so <math>\chi(g)</math> is a sum of <math>n^{th}</math> roots of unity. Hence, <math>\chi(g)</math> is a cyclotomic integer.
+
'''Proof''': <math>\chi(g)</math> is the sum of eigenvalues of <math>\rho(g)</math>, counted with multiplicity, in the algebraic closure of <math>k</math>. Since <math>G</math> is finite, <math>g</math> has finite order, so <math>\rho(g)</math> has finite order, and hence <math>\rho(g)</math> satisfies a polynomial of the form <math>x^n - 1</math>. Hence, every eigenvalue of <math>\rho(g)</math> is a <math>n^{th}</math> root of unity, so <math>\chi(g)</math> is a sum of <math>n^{th}</math> roots of unity. Hence, <math>\chi(g)</math> is a cyclotomic integer.

Revision as of 17:47, 9 October 2008

This page describes a useful fact in character theory/linear representation theory arising from rudimentary linear algebra
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Statement

Let G be a finite group, and k be any field. Then, the character of any finite-dimensional linear representation of G over k takes, at every element of the group, a value that is a cyclotomic integer over the prime subfield of k: in other words, it is in the subring generated by all the roots of unity.

(The statement, does not require the characteristic of k to not divide the order of the group).

Related facts

Applications

  • Characters are algebraic integers: Any cyclotomic integer is an algebraic integer, so every character takes values in algebraic integers.
  • If, over a field of characteristic zero, a character takes a rational value, that value must be an integer.

Proof

Given: A finite group G, field k, a finite-dimensional linear representation (V,\rho) of G over k. \chi is the character of \rho.

To prove: For any g, \chi(g) is a cyclotomic integer over the prime subfield of k.

Proof: \chi(g) is the sum of eigenvalues of \rho(g), counted with multiplicity, in the algebraic closure of k. Since G is finite, g has finite order, so \rho(g) has finite order, and hence \rho(g) satisfies a polynomial of the form x^n - 1. Hence, every eigenvalue of \rho(g) is a n^{th} root of unity, so \chi(g) is a sum of n^{th} roots of unity. Hence, \chi(g) is a cyclotomic integer.