Representation Theory

It is a pair \( (V, \rho )\) of vector space \( V \) and a homomorphism \( \rho\colon A \to \mathrm{GL}(V) \) 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.).

A representation is called faithful, if \( \rho \) is injective.

• irrep

A representation is called irreducible, if there does not exists a subspace of \( V \) fixed by the action of \( A \) other than the trivial space and \( V\) itself.

If such an subspace exists it is called the subrepresentation, with the homomorphism clearly being the homomorphism restricted to the subspace.

One way to find the irreducible representation is to find the basis that diagonalizes all the matrices.

A representation \( V \) is called indecomposable if \( V \) cannot be written as a direct sum of two subrepresentations.

Irreducible representation is automatically indecomposable.

Form a vector space \( V_{\rm reg} \) from the elements \( g \) of a group \( G \) as a basis vector \( b_g \). The representation \( \rho_{\rm reg} \) of \( G \) is the permutation operation of those basis vector according to the group operation: \[ \rho_{\rm reg}\colon G \to \operatorname{Hom}(V_{\rm reg}, V_{\rm reg})\colon g \mapsto (b_h \mapsto b_{gh}). \]

It is said to be the "representation on itself".

Each basis vector has its coefficient, and its the same for a function \( G \to \mathbb{C} \): \[ \mathbb{C}^{G} \cong V_{\rm reg}. \]

Given two representation \( \rho_V, \rho_W \), \[ \rho_{\operatorname{Hom}(V,W)}(g) \colon \operatorname{Hom}(V, W) \to \operatorname{Hom}(V, W)\colon f\mapsto \rho_W(g) \circ f \circ \rho_V(g^{-1}) \] is also a representation.

It conjugate the linear map \( f \) with \( g \) in a sense.

The dual representation \( (\rho_{V})^{*}\colon \operatorname{Hom}(V, K)\to \operatorname{Hom}(V, K) \) of a representation \( \rho_V \) is defined as: \[ (\rho_{V})^{*}(g)\colon f\mapsto f\circ \rho_V(g^{-1}). \]

Dual representations transform opposite to regular ones.

Given two representation \( \rho_V, \rho_W \), their tensor product representation is defined to map each basis tensor as follows: \[ \rho_{V\otimes W}(g)\colon a_i\otimes b_j \mapsto \rho_V(g)(a_i)\otimes \rho_W(g)(b_t) \] where \( \{a_i\} \) and \( \{b_j\} \) are the basis of \( V \) and \( W \) resepctively.

\( V\otimes W \cong \operatorname{Hom}(V^{*}, W) \) by noticing the isomorphism: \[ (f_{ji}\colon \alpha_i\mapsto b_j) \leftrightarrow a_i\otimes b_j. \]

A left module of \( A \) is a \( K \)-vector space with an action \( \cdot \colon A\times V \to V \) such that:

• Unitality: \( \exists 1 \in A, \forall v\in V, 1\cdot v = v \)
• Associativity: \( \forall g,h \in A, h\cdot (g\cdot v) = hg\cdot v \)
• Linearity: \( \forall g \in A, \forall v, w\in V, \forall a, b \in K, g\cdot (av+ bw) = ag\cdot v + bg\cdot w\)

Right module is similarly defined as well.

Given a representation \( \rho_V \), the invariant subrepresentation \( V^G \) is defined as: \[ V^G := \{ x\in V \mid \rho_V(g)(x) = (x), \forall g\in G\}. \]

The set \( \operatorname{Hom}_G(V,W) \) of G-linear maps is equal to the invariant subrepresentation \( \operatorname{Hom}(V, W)^G \) of the representation \( \operatorname{Hom}(V, W) \)

Any representation \( V \) of \( G \) can be decomposed into its irreducible representations (irreps). In particular, the regular representation decomposes into all possible irreps \( \{ U_i \} \), each raised to their dimension \( d_i \): \[ V_{\rm reg} = \bigoplus_i U_i^{\oplus d_i}. \] If \( G \) is abelian, then \( d_i = 1 \) and there exists \( |G| \) distinct irreps.

If two vector space \( V, W \) are not isomorphic, then a \( G \)-linear 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 irreducible representation of commutative algebra is one dimensional.
• \( \dim \operatorname{Hom}_G(V, W) = 1_{V\cong W} \) for two irreps \( V, W \).
• \( \dim \operatorname{Hom}_G(V,W) = \text{\#number of irreps isomorphic to $W$} \) for any \( V \) and an irrep \( W \), or an irrep \( V \) and any \( W \).

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

• Simple Character

Character of irreducible representation.

Inner product is defined for the space \( \mathbb{C}^G \): \[ \langle \xi | \zeta \rangle := \frac{1}{|G|}\sum_{g\in G} \xi(g) \overline{\zeta(g)}. \]

Given any representation \( V, W \), let \( \Psi \) be the following projection of \( \operatorname{Hom}(V,W) \) onto \( \operatorname{Hom}_G(V,W) \). \( \Psi \) is given in terms of the representation \( \rho_{\operatorname{Hom}(V,W)} \): \[ \Psi\colon f\mapsto \frac{1}{|G|}\sum_{g\in G} \rho_{\operatorname{Hom}(V,W)}(g)(f). \] The trace of \( \Psi \) is equal to the inner product:

\begin{align*} \operatorname{tr}(\Psi) &= \frac{1}{|G|}\sum_{g\in G} \operatorname{tr}\left(\rho_{\operatorname{Hom}(V,W)}(g)\right) \\ &= \frac{1}{|G|}\sum_{g\in G} \chi_{\operatorname{Hom}(V,W)}(g) = \frac{1}{|G|} \sum_{g\in G} \overline{\chi_V(g)}\chi_W(g) \\ &= \langle \chi_W | \chi_V\rangle. \end{align*}

The third equality is from the property of character of tensor product representation.

The same trace is given by: \[ \langle \chi_W|\chi_V\rangle = \operatorname{tr}(\Psi) = \dim \operatorname{Hom}_{G}(V, W). \]

Since character is a class function, the inner product can be written equivalently: \[ \langle \chi_V | \chi_W \rangle := \frac{1}{|G|}\sum_{[g]} |[g]|\chi_V(g) \overline{\chi_W(g)} \] where \( [g] \) are the conjugacy classes.

• Character is not a homomorphism in general.
• It is an element of the group ring \( K^G \) or equivalently \( K[G] \). Often the field is complex numbers: \( \chi \in \mathbb{C}^G \).
• Any character is a linear combination of irreducible characters.
• \( \chi_{\rho\oplus \sigma} = \chi_\rho + \chi_\sigma \)
• \( \chi_{\rho\otimes \sigma } = \chi_\rho \cdot \chi_\sigma \)
• Trace is basis independent, so 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.
  • The inner product with irreducible characters are the same, and the irrep decomposition are the same. Therefore, two representations are isomorphic.
  • linear algebra - 2 complex representations have the same character iff they a…
• \( \langle \chi|\chi\rangle = 1 \iff \chi\text{ irreducible} \)
• Character is a class function, because trace is invariant 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 \( G \).

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 defiintion is another way of describing the coefficients of multiplication between two formal linear combinations of the elements of \( G \): \[ (f(x_1)x_1 + \cdots + f(x_n)x_n)(g(x_1)x_1 + \cdots + g(x_n)x_n). \]

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

Let \( V \) be a vector space. \( A_V := \operatorname{Hom}(V,V) \) equipped with pointwise addition and scalar multiplication by \( K \), and function composition is called the matrix algebra.

\( A_V \)-module \( V \) is simple module.

\( A_V \)-module \( A_V \) is isomorphic to \( V^{\oplus n} \), considering each column constitutes a submodule isomorphic to \( V \).

For any \( A_V \)-module \( M \), \[ M\cong \operatorname{Hom}_{A_V}(A_V, M) \cong \operatorname{Hom}_{A_V}(V, M)^{\oplus n} \] and \[ \dim M = nk,\quad k := \dim \operatorname{Hom}_{A_V}(V, M). \] Furthermore, \( M \cong V^{\oplus k} \)

Semi-simple algebra is isomorphic to a direct sum of matrix algebras. \( A \)-module \( A \) is isomorphic to \( M_1^{\oplus d_1}\oplus \cdots \oplus M_r^{\oplus d_r} \), and algebra \( A \) is isomorphic to \( A_{M_1}\oplus \cdots \oplus A_{M_r} \).

The irreducible characters of a finite group \( G \) form an orthonormal basis in the inner product space of class functions of \( G \).

There exists an isomorphism of algebras: \[ \tilde{\rho}\colon \mathbb{C}[G] \to \bigoplus_{i=1}^r A_{U_i} \] where \( A_{U_i} \) are the matrix algebra over irreps \( U_i \) of \( G \).

In particular, the center \( Z_{\mathbb{C}[G]} \) of group algebra is isomorphic to the center \( \mathbb{C}^{\oplus r} \) of the direct sum of matrix algebras, and the number of conjugacy classes \( \dim Z_{\mathbb{C}[G]} \) is equal to the number of irreps \( \dim \mathbb{C}^{\oplus r} = r \).

The character table is the covariant transformation matrix from the basis \( \{\delta_{[g_j]} \} \) to \( \{ \chi_i\} \). Due to the orthogonality of the \( \{ \chi_i \} \) basis, the modified character table \( B \): \[ B_{ij} = \chi_i(g_j) \sqrt{\frac{|[g_j]|}{|G|}} \] is unitary, and the columns are also orthogonal: \[ \sum_{k} \overline{C_{ki}} C_{kj} = |C_{g_i}| \delta_{ij} \] where \( C_{g_i} \) is the centralizer of \( g_i \).

In particular, \[ \sum_i\chi_i(e)^2 = |G|. \] Two reasons why this is the case. The first one is the fact that the centralizer of \( e \) is the entire group. And the second is that the regular representation consists of \( d_i \) copies of each irreps with dimension \( d_i \).

\[ \frac{1}{h} \sum_{g\in G}\Gamma_{ab}^{(i)}(g) \overline{\Gamma_{cd}^{(j)}(g)} = \frac{1}{l_i}\delta_{ij}\delta_{ac}\delta_{bd} \] where \( \Gamma_{ab}^{(i)} \) is the \( a,b \) entry of \( i \)th irrep, \( h \) is the order of the group \( G \), and \( l_i \) is the dimension of the irrep.

\begin{align*} \sum_{g\in G} \chi_i(g) \chi_i(g^{-1} u) &= \sum_{g\in G}\sum_{a,b,c} \Gamma_{aa}^{(i)}(g)\Gamma_{bc}^{(i)}(g^{-1}) \Gamma_{cb}^{(i)}(u) \\ &= \sum_{g\in G}\sum_{a,b,c} \Gamma_{aa}^{(i)}(g)\overline{\Gamma_{cb}^{(i)}(g)} \Gamma_{cb}^{(i)}(u) \\ &= \sum_{a,b,c}\frac{h}{l_i}\delta_{ac}\delta_{ab} \Gamma_{cb}^{(i)}(u) = \frac{h}{l_i} \sum_a \Gamma_{aa}^{(i)}(u) \\ &= \frac{h}{l_i} \chi_i(u). \end{align*}

\[ P^{(i)} := \frac{l_i}{h}\sum_{g\in G}\overline{\chi_i(g)}\Gamma(g) \] is a projection operator onto the irreducible representation \( U_i \).