Representation theory studies groups via their actions on vector spaces, i.e., homomorphisms . This converts abstract group theory into linear algebra, where we have powerful tools (eigenvalues, traces, diagonalization). We work over throughout (or more generally, any algebraically closed field of characteristic 0).
Definition 13.1.1 (Representation). A representation of a group is a vector space (usually finite-dimensional over ) together with a group homomorphism . Equivalently, a -action on by linear operators: ,, with each linear.
Definition 13.1.2 (Subrepresentation and irreducibility).
A subrepresentation is a subspace that is -invariant: for all .
A representation is irreducible if it has no nontrivial subrepresentations (i.e., the only -invariant subspaces are and itself).
Example 13.1.1 (Representations of cyclic groups). If , a representation is a vector space with a linear operator satisfying . Since the minimal polynomial of divides where , and this polynomial has distinct roots, is diagonalizable. The eigenspaces are subrepresentations, and .
Each 1-dimensional subspace with is an irreducible representation, corresponding to the homomorphism sending . There are exactly such irreducible representations, and every representation of decomposes as a direct sum of copies of these.
Proposition 13.1.1 (Representations of finite abelian groups). If is a finite abelian group and is a representation of over , then splits into a direct sum of 1-dimensional (irreducible) subrepresentations.
Proof. The -action is equivalent to commuting operators with . Each is diagonalizable (as above). Commuting diagonalizable operators are simultaneously diagonalizable: the eigenspaces of are invariant under all (since they commute), and the restriction of to an eigenspace of is again of finite order, hence diagonalizable. Proceed by induction on . The simultaneous eigenspaces are 1-dimensional subrepresentations. ∎
Definition 13.1.3 (Dual group). For a finite abelian group , the dual group is , the group of all 1-dimensional representations (characters in the abelian sense) under pointwise multiplication. For , (but not canonically). More generally, for any finite abelian group (non-canonically). This completes the classification of complex representations of finite abelian groups.
13.1.1 Homomorphisms and Constructions
Definition 13.1.4 (Homomorphism of representations). Given representations of , a homomorphism (or -equivariant map, or intertwining operator) is a linear map satisfying for all ,. Equivalently, as operators. We write for the space of such maps.
Note. If , then and are subrepresentations of and respectively (since , so ).
Proposition 13.1.2 (Constructions with representations). Given representations of :
is a representation: .
is a representation: , extended by linearity.
The dual is a representation: . This is the contragredient representation.
is a representation: , i.e., . Note (the -invariant maps are exactly the -equivariant ones).
Under the isomorphism , the -action is , which is consistent with the above.
If is a subrepresentation, then is a representation: .