2.4 Maschke's theorem

Maschke's theorem is a game-changer for finite group representations. It lets us break down complex representations into simpler, irreducible pieces, making our lives way easier when studying how groups act on vector spaces.

The theorem only works for finite groups and certain fields, but when it does, it's super powerful. It gives us a neat way to organize representation matrices and simplify calculations, which is a big deal in representation theory.

Understanding Maschke's Theorem

Maschke's theorem and implications

Top images from around the web for Maschke's theorem and implications

• 

  Representation Theory [The Physics Travel Guide] View original

  Is this image relevant?

• 

  Representation Theory [The Physics Travel Guide] View original

  Is this image relevant?

1 of 1

Top images from around the web for Maschke's theorem and implications

• 

  Representation Theory [The Physics Travel Guide] View original

  Is this image relevant?

• 

  Representation Theory [The Physics Travel Guide] View original

  Is this image relevant?

1 of 1

• Maschke's theorem states for finite group G and field F with characteristic not dividing |G|, every representation of G over F completely reducible (decomposable into irreducible subrepresentations)
• Enables decomposition of representations into direct sums of irreducible representations simplifying study of group representations
• Guarantees existence of basis where representation matrices are block diagonal facilitating computations (eigenvalues, traces)

Conditions for Maschke's theorem

• Applies to groups with finite order ensures averaging process in proof well-defined
• Field characteristic must not divide group order |G| (complex numbers, real numbers, rational numbers)
• Encompasses all linear representations of group including finite and infinite-dimensional representations
• Fails for infinite groups or fields with characteristic dividing |G| (modular representation theory)

Proof using averaging operator

1. Consider G-module V and G-submodule W, aim to find G-invariant complement U
2. Define projection operator P: V → W
3. Construct averaging operator: $P' = \frac{1}{|G|} \sum_{g \in G} gPg^{-1}$
4. Prove G-equivariance: $P'(gv) = gP'(v)$ for all $g \in G$ and $v \in V$
5. Show projection onto W: $P'(w) = w$ for all $w \in W$
6. Define complement U = ker(P') and prove V = W ⊕ U
7. Demonstrate U is G-invariant completing proof

Decomposition into irreducible components

• Iteratively decompose representation V finding proper subrepresentation W
• Apply Maschke's theorem to obtain V = W ⊕ U
• Repeat process for W and U until reaching irreducible components
• Jordan-Hölder theorem ensures unique decomposition up to isomorphism
• Enables expression of characters as sums of irreducible characters
• Facilitates obtaining block diagonal form for representation matrices
• Examples: regular representation decomposition, tensor product decomposition (SU(2) representations)