Exploration 6: Groups as matrices

November 17, 2023.

Questions:

• TODO

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Group representation examples

Recall that we can represent the complex numbers $\CC$ with matrices:

$$\begin{aligned} 1&=\left[\begin{array}{ccc}1&0\\0&1\end{array}\right]\\ i&=\left[\begin{array}{ccc}0&1\\-1&0\end{array}\right]\\ a+bi&=\left[\begin{array}{ccc}a&b\\-b&a\end{array}\right]\\ \end{aligned}$$

This works because $i^2=-1$, and the matrices for $1$ and $i$ form an independent basis of the $\RR^2$ plane. It is one of many unique representations of the complex numbers using matrices.

In more formal terms, this defines a homomorphism $R:\CC\to GL_2(\RR)$ which we call the matrix representation of $\CC$ on $\RR^2$. In this case the dimension of the representation is $2$ – it takes $2$ by $2$ matrices for this representation to represent the complex numbers

Another example: the defining representation $R:D_n\to GL_2(\CC)$ of the dihedral group $D_n$ says

$$\begin{aligned} \left[\begin{array}{ccc}\zeta&0\\0&\zeta^{-1}\end{array}\right]&\text{ is rotation}\\ \left[\begin{array}{ccc}0&1\\1&0\end{array}\right]&\text{ is reflection} \end{aligned}$$

where $\zeta=\exp(\frac{2\pi i}{n})$.

Another: the sign representation sign : Sₙ → GL₁(F) of the symmetric group Sₙ takes the sign of each permutation. Since the dimension is 1, it is a linear representation.

Another: All complex linear representations Cₙ → GL₁(ℂ) = ℂˣ of the cyclic group Cₙ can be given as some mapping from Cₙ to one of {ζ⁰,ζ¹,…,ζⁿ⁻¹}. In fact, there are n of them.

Another: One way to represent ℤ is as matrices [1 n; 0 1], so the product of two matrices is equal to addition on the top right. It’s a representation ℤ → GL₂(F).

⎡1 n⎤⎡1 m⎤ = ⎡1 n+m⎤
⎣  1⎦⎣  1⎦   ⎣    1⎦

Last one:

The obvious action of a group G on a vector space V is the permutation action, where we view G as a bunch of permutations (see group actions exploration).

This is exactly related to how the obvious representation of a group G is the representation of G as a bunch of permutations.

In fact, the operation of G on V is equivalent to the representation of G on V.

This is the permutation representation of G.

Matrix representations are invariant under conjugation

It is a fact that change-of-basis of a linear transformation gives the same linear transformation. What if you do that for representations? If two matrices are similar, you can always find an isomorphism from a matrix A to a matrix B via change-of-basis: B = PAP⁻¹. Since you always get some matrix P, it should still have all the structure it needs to represent the group G, just in a different basis.

So matrix representations are invariant under conjugation.

Representations in general

The general construction uses ρ : G → GL(V) to denote a representation of G on V. This generalizes ℂ to some group G, and ℝⁿ to some arbitrary vector space V. Given a representation ρ, let ρ_g = ρ(g) be the image of g over ρ (so ρ₁ = I always, as homomorphisms preserve identity).

A matrix representation is just one where V is the space of column vectors Eⁿ. All representations of G on finite dimensional V can be reduced to matrix representations by choosing a basis for the space. However, all representations are equivalent up to conjugation (change-of-basis), so we need to study representations without regard to a specific basis. That’s why the other definition is used.

Isomorphic representations

The map χ: G → F given by χ(g) = tr(Rg) just sums the diagonal of the matrix for Rg.

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Properties of representations

Every matrix representation R of a finite group G is conjugate to a unitary representation (find it via conjugation).

It’s true that for every matrix representation, there is a conjugate unitary representation. This is because each Rg generates a subgroup of GLₙ, which is conjugate to a subgroup of Uₙ.

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Exploration 5: Permutation groups Exploration 7: Sylow theory

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