Group representation theory MOC

Maschke’s theorem

Let be a finite group Then the group ring is semisimple iff [[Characteristic|]] does not divide . rep2

Proof of semisimplicity for coprime characteristic

Let be a nonzero proper (left) submodule. Since is a -vector space, we have , and we can find a complement subspace , such that . We can define the -linear projection

with . Let

which is -linear. Note . We claim it is also -linear: Indeed, for all , we have

If , then

so is a projection operator onto . Letting we have that is a -submodule, hence

By induction, it follows is semisimple.

The above proof gives a construction of a complementary submodule for any submodule, which we call Maschke’s algorithm.

In terms of unitary irreps

Every unitary representation is the direct sum of unitary irreps, and thus any representation of a compact group is the direct sum of unitary irreps. rep

Proof

Follows from the fact that The orthogonal complement of an invariant subspace under a unitary operator is invariant and Every finite complex representation of a compact group is equivalent to a unitary representation.

This core statement of group representation theory allows for the Decomposition of a representation, and therefore reduces the task of classifying representations to classifying finite ones.

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