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Lcm of degrees of irreducible representations

484 bytes added, 00:14, 13 April 2011
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==Definition==
Supose ===For a group over a field=== Suppose <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>. ===Typical context: finite group and splitting field=== 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.
 
===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>.
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