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

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==Definition== | ==Definition== | ||

− | + | ===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>. |

## Revision as of 00:14, 13 April 2011

This article defines an arithmetic function on groups

View other such arithmetic functions

## Contents

## Definition

### For a group over a field

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

### Typical context: finite group and splitting field

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

Note that the lcm of degrees of irreducible representations depends (if at all) only on the characteristic of the field . 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 .

## 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:

- Degree of irreducible representation divides order of group: Hence, the lcm of degrees of irreducible representations divides the order of the whole group.
- Degree of irreducible representation divides index of center: Hence, the lcm of degrees of irreducible representations divides the index of the center, which is also the order of the inner automorphism group.
- Degree of irreducible representation divides index of abelian normal subgroup: Hence, the lcm of degrees of irreducible representations divides the index of any abelian normal subgroup.

### Subgroups, quotients, direct products

- lcm of degrees of irreducible representations of subgroup divides lcm of degrees of irreducible representations of group
- lcm of degrees of irreducible representations of quotient group divides lcm of degrees of irreducible representations of group
- lcm of degrees of irreducible representations of direct product is lcm of lcms of degrees of irreducible representations of each direct factor