Module VI·Article III·~1 min read
Group Representations and Maschke's Theorem
Tensor Algebra
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Group Representations
What is a Representation
A representation of a group G is a homomorphism $\rho: G \to GL(V)$, that is, each element of the group is assigned an invertible linear operator on $V$ such that $\rho(gh) = \rho(g)\rho(h)$.
The dimension of the representation = $\dim V$. Matrix coefficients: $\rho_{ij}(g)$ are numbers.
Irreducible Representations
A subspace $U \subseteq V$ is invariant if $\rho(g)U \subseteq U$ for all $g \in G$. A representation is irreducible if there are no nontrivial invariant subspaces.
Maschke's Theorem: Any finite-dimensional representation of a finite group over a field of characteristic 0 (or not dividing $|G|$) is completely reducible: it decomposes into a direct sum of irreducible ones.
Proof: average the projector onto the invariant subspace over the group (the averaging method). We obtain a $G$-invariant projector.
Characters
The character $\chi_\rho(g) = \mathrm{tr}(\rho(g))$ — the trace of the operator. The character function does not change under conjugation: $\chi(hgh^{-1}) = \chi(g)$ — a class function.
Irreducible representations are determined by their characters: $\chi_\rho = \chi_\sigma \iff \rho \cong \sigma$.
Orthogonality relation: $(\chi_\rho, \chi_\sigma) = (1/|G|) \sum_g \chi_\rho(g)\chi_\sigma(g^{-1}) = \delta_{\rho\sigma}$.
Character Table
A finite group $G$ has exactly as many irreducible representations as there are conjugacy classes of elements. The character table encodes the entire structure of the group.
For $S_3$ (the symmetry group of the triangle): 3 classes $\to$ 3 irreducibles: trivial (degree 1), sign (degree 1), standard (degree 2).
Applications: in physics, representations of symmetry groups determine the allowed states of a system (Wigner's theorem, multiplets of elementary particles).
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