Representation Theory

It is a pair \( (V, \rho_V )\) of vector space \( V \) and a homomorphism \( \rho_V\colon A \to \mathrm{GL}(V) \) form an algebra \( A \) to a set invertable linear maps \( \mathrm{GL}(V) \).

• If the homomorphism is clear from the context, the vector space alone might be called the representation.
• The image of the algebra can be called the action of the algebra (or group, etc.).

• The irreducible representation of commutative algebra is one dimensional.

A linear map \( f\colon V\to W \) is called an equivariant map (or \( G \)-linear map), if is commutes with the action of \( A \): \[ f\circ \rho_V = \rho_W\circ f. \]

If two vector space \( V, W \) are not isomorphic, then the equivariant map has to be trivial: \( f\colon v\mapsto 0_W \).

It shows that a non-trivial equivariant map establishes isomorphism between two representations. The vector spaces has to be isomorphic by the theorem and the homomorphism has to be isomorphic by the definition of the equivariant map.

It is the trace of the representation: \( \chi = \mathrm{tr}\circ \rho \).

• Character is not a homomorphism in general.
• It is an element of the group ring \( K^G \) or equivalently \( K[G] \)

• Since the trace is basis independent and the isomorphism is established by the change of basis, isomorphic representations have the same character.
  • Furthermore, over a field of characteristic 0, two representations are isomorphic if and only if they have the same character.
  • linear algebra - 2 complex representations have the same character iff they a…
• \( \chi_{\rho\oplus \sigma} = \chi_\rho + \chi_\sigma \)
• \( \chi_{\rho\otimes \sigma } = \chi_\rho \cdot \chi_\sigma \)
• Character is a class function, because the eigenvalue is preserved under conjugation.
• Every character value \( \chi(g) \) is a sum of \(n\) \(m\)-th root of unity, where \( n\) is the degree, and \( m \) is the order of \( g\).
• If \( \operatorname{char} K \) does not devide \( | G| \), The number of irreducible characters of \( G\) is equal to the number of conjugacy classes of \( G \).

• The number of distint irreducible representations is equal to the number of conjugacy classes.

\[ \sum_i\chi_i(e)^2 = |G|. \] It is because \( \chi_i(e)^2 \) is the dimension of the \( i \)th representation, and the direct sum of all distinct irreducible representation must be isomorphic to the group itself.

• class function is any function that has it constant value on a conjugacy class.

• It is the center of the group ring \( K[G] \).

Set of mapping \( f\colon G\to K \) from a group \( G \) to a field \( K \) of finite support, for which the addition and scalar product are defined canonically, and the multiplication is defined by: \[ f\cdot g: x\mapsto \sum_{uv = x} f(u)g(v). \] This comes from the analogy to the polynomial ring, of the formal linear combinations of elements of \( G \) \[ \sum_{g\in G} f(g) g. \]

• It is a free module and ring at the same time.

On a natural inner product space of complex-valued class functions of a finite group \( G \) with the inner product: \[ \langle \alpha, \beta \rangle := \frac{1}{|G|} \sum_{g\in G} \alpha(g)\overline{\beta{g}} \] the irreducible characters form an orthonormal basis.