Difference between revisions of "Lcm of degrees of irreducible representations"

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(Created page with "{{arithmetic function on groups}} ==Definition== Supose <math>G</math> is a finite group and <math>K</math> is a splitting field for <math>G</math>. The '''lcm of degre...")
 
 
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{{arithmetic function on groups}}
 
{{arithmetic function on groups}}
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{{term related to|linear representation theory}}
  
 
==Definition==
 
==Definition==
  
Supose <math>G</math> is a [[finite group]] and <math>K</math> is a [[splitting field]] for <math>G</math>. The '''lcm of degrees of irreducible representations''' of <math>G</math> is defined as the least common multiple of all the [[defining ingredient::degrees of irreducible representations]] of <math>G</math> over <math>K</math>.
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===For a group over a field===
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Suppose <math>G</math> is a [[group]] and <math>K</math> is a [[field]]. The '''lcm of degrees of irreducible representations''' of <math>G</math> is defined as the least common multiple of all the [[defining ingredient::degrees of irreducible representations]] of <math>G</math> over <math>K</math>.
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===Typical context: finite group and splitting field===
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The typical context is where <math>G</math> is a [[finite group]] and <math>K</math> is a [[splitting field]] for <math>G</math>. In particular, the characteristic of <math>K</math> is either zero or is a prime not dividing the order of <math>G</math>, and every irreducible representation of <math>G</math> over any extension field of <math>K</math> can be realized over <math>K</math>.
  
 
Note that the lcm of degrees of irreducible representations depends (if at all) only on the characteristic of the field <math>K</math>. This is because the [[degrees of irreducible representations]] over a splitting field depend only on the characteristic of the field.
 
Note that the lcm of degrees of irreducible representations depends (if at all) only on the characteristic of the field <math>K</math>. This is because the [[degrees of irreducible representations]] over a splitting field depend only on the characteristic of the field.
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===Default case: characteristic zero===
  
 
By default, when referring to the lcm of degrees of irreducible representations, we refer to the case of characteristic zero, and we can in particular take <math>K = \mathbb{C}</math>.
 
By default, when referring to the lcm of degrees of irreducible representations, we refer to the case of characteristic zero, and we can in particular take <math>K = \mathbb{C}</math>.

Latest revision as of 00:15, 13 April 2011

This article defines an arithmetic function on groups
View other such arithmetic functions
This term is related to: linear representation theory
View other terms related to linear representation theory | View facts related to linear representation theory

Definition

For a group over a field

Suppose G is a group and K is a field. The lcm of degrees of irreducible representations of G is defined as the least common multiple of all the degrees of irreducible representations of G over K.

Typical context: finite group and splitting field

The typical context is where G is a finite group and K is a splitting field for G. In particular, the characteristic of K is either zero or is a prime not dividing the order of G, and every irreducible representation of G over any extension field of K can be realized over K.

Note that the lcm of degrees of irreducible representations depends (if at all) only on the characteristic of the field K. This is because the degrees of irreducible representations over a splitting field depend only on the characteristic of the field.

Default case: characteristic zero

By default, when referring to the lcm of degrees of irreducible representations, we refer to the case of characteristic zero, and we can in particular take K = \mathbb{C}.

Related facts

What it divides

Any divisibility fact stating that the degree of every irreducible representation over a splitting field must divide some fixed number implies that the lcm also divides that fixed number. Some of these are listed below:

Subgroups, quotients, direct products