# Changes

## Lcm of degrees of irreducible representations

, 00:14, 13 April 2011
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==Definition==
Supose ===For a group over a field=== Suppose [itex]G[/itex] is a [[finite group]] and [itex]K[/itex] is a [[splitting field]] for [itex]G[/itex]. The '''lcm of degrees of irreducible representations''' of [itex]G[/itex] is defined as the least common multiple of all the [[defining ingredient::degrees of irreducible representations]] of [itex]G[/itex] over [itex]K[/itex]. ===Typical context: finite group and splitting field=== The typical context is where [itex]G[/itex] is a [[finite group]] and [itex]K[/itex] is a [[splitting field]] for [itex]G[/itex]. In particular, the characteristic of [itex]K[/itex] is either zero or is a prime not dividing the order of [itex]G[/itex], and every irreducible representation of [itex]G[/itex] over any extension field of [itex]K[/itex] can be realized over [itex]K[/itex].
Note that the lcm of degrees of irreducible representations depends (if at all) only on the characteristic of the field [itex]K[/itex]. 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 [itex]K = \mathbb{C}[/itex].