Difference between revisions of "Size-degree-weighted characters are algebraic integers"

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(New page: ==Statement== Suppose <math>k</math> is an algebraically closed field of characteristic zero, and <math>G</math> is a finite group. Let <math>\rho</math> be an irreducible linear repr...)
 
(Related facts)
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* [[Characters are cyclotomic integers]]: This statement holds in much greater generality. In particular, it holds over any field.
 
* [[Characters are cyclotomic integers]]: This statement holds in much greater generality. In particular, it holds over any field.
 
* [[Characters are algebraic integers]]
 
* [[Characters are algebraic integers]]
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===Applications and further results===
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* [[Degree of irreducible representation divides group order]]
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* [[Zero-or-scalar lemma]]
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* [[Conjugacy class of prime power order implies not simple]]
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* [[Order has only two prime factors implies solvable]]
  
 
===Breakdown for a field that is not algebraically closed===
 
===Breakdown for a field that is not algebraically closed===

Revision as of 21:58, 9 October 2008

Statement

Suppose k is an algebraically closed field of characteristic zero, and G is a finite group. Let \rho be an irreducible linear representation of G over k, and \chi be the character corresponding to \rho. Let c be a conjugacy class in G and g \in c be an element. Then:

\frac{|c|\chi(g)}{\chi(1)}

is an algebraic integer.

Related facts

Applications and further results

Breakdown for a field that is not algebraically closed

Further information: cyclic group:Z3

Let G be the cyclic group of order three and \R be the field. G has an irreducible two-dimensional linear representation over \R given by rotation by multiples of 2\pi/3. For a non-identity element g of G, \chi(g) = -1 for the corresponding character, while \chi(1) = 2. Thus, the expression works out to -1/2, which is not an algebraic integer.

Proof

The proof is based on the idea of the convolution algebra on conjugacy classes.

Description of the convolution algebra on conjugacy classes

Let C(G,\mathbb{Z}) be a \mathbb{Z}-subalgebra of the group ring \mathbb{Z}(G) defined as follows: as a group, it is the free Abelian group on all indicator class functions for conjugacy classes. In other words, for each conjugacy class, we have a free generator that corresponds to the sum of elements of that conjugacy class.

The structure constant for multiplication of elements of C(G,\mathbb{Z}) is defined as follows: given conjugacy classes c_1, c_2, c_3, the coefficient of the c_3-indicator function in the product of the c_1-indicator function and the c_2-indicator function is the number of ways of writing g_1g_2 = g_3 where g_i \in c_i.

Note that all the structure constants are integers.

A homomorphism from this convolution algebra to the matrix ring

The representation \rho gives rise to a homomorphism from C(G,\mathbb{Z}) to the matrix ring M_n(k). The indicator function for a conjugacy class c goes to the matrix given by:

\sum_{g \in c} \rho(g).

This sum commutes with \rho(h) for all h, and thus, by Schur's lemma, the sum is a scalar matrix. The trace of the sum is |c|\chi(g), so the sum must be a scalar matrix with scalar entry:

\frac{|c|\chi(g)}{\chi(1)}.

Thus, the set of scalar matrices with entries described as above additively generate a group that is a ring under multiplication. The structure constants for this ring are the same as the structure constants for the convolution algebra. A result from algebraic number theory now tells us that this forces the entire ring to be a ring of algebraic integers, and in particular, the generating elements are algebraic integers.