## Representation Theory: Basics and Heisenberg Representation (1) May 17, 2010

Posted by Sarah in Uncategorized.
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Here’s a new installment of lecture notes from Shamgar Gurevich’s seminar on representation theory. (This was a small group of applied math people, professors and grad students and me, looking at applications of representation theory to the problem of cryo-electron microscopy. It’s very pretty stuff and well taught.)

The Discrete Fourier Transform (DFT) is defined as
$1/\sqrt{P'} (e^{\frac{2 \pi i}{p} \omega}) : \mathbb{C}(\mathbb{F}_p \to \mathbb{C}(\mathbb{F}_p$
We want to find a natural basis $B_{DFT}$ of eigenvectors, and explain how to compute the DFT fast. Related is the following theorem:

Fix $1 \neq \psi: \mathbb{F}_p \to \mathbb{C}^\times$ an additive character. There is a unique up to isomorphism irreducible representation $(\pi, H, \mathcal{H})$ such that
$\pi |_{\mathbb{Z}(H)} = \psi(\cdot) Id_{\mathcal{H}}.$

Let’s show first:
$R \to V = \mathbb{C}(G) \simeq \oplus_{\rho \in Irr(G)} dim \rho V_{\rho}.$

In particular,

$|Irr(G)| < \infty$
$|G| = \sum_{\rho \in Irr(G)} dim(\rho)^2$
To show this, let us answer
$G \to_\pi V = \oplus_{\rho \in Irr(G)} m_\rho V_{\rho}$.
How to compute $m_\rho$?

Define
$\langle G, \pi \rangle = dim Hom_G(G, \pi)$

Properties of intertwining numbers $\langle G, \pi \rangle$:

1. $G \simeq \hat{G}, \pi \simeq \hat{\pi}; \implies \langle G, \pi \rangle = \langle \hat{G}, \hat{\pi}; \rangle$

2. $\langle G_1 \oplus G_2, \pi \rangle = \langle G_1, \pi \rangle + \langle G_2, \pi \rangle$

3. $G, \pi \in Irr(G) \langle G, \pi \rangle = \delta_{G, \pi}$

1 is clear; 3 is proven by Schur’s Lemma.
To prove 2, observe
$Hom(V_1 \oplus V_2, W) = Hom(V_1, W) \oplus Hom(V_2, W)$
$Hom_G(V, W) = Hom(V, W)^G$
$(V \oplus W)^G = V^G \oplus W^G$

The result is, if
$\pi \simeq \oplus m_\rho * \rho$
then
$m_\rho = \langle \rho, \pi \rangle$

The application to the first theorem is that if
$R: G \to \mathbb{C}(G)$ and given a representation $(\rho, G, V)$ then
$\langle R, \rho \rangle = dim V.$

Proof: $Hom(\mathbb{C}(G), V) = V$

We are now ready to prove the next theorem: the number of irreducible representations of G equals the number of conjugacy classes of G. The idea is an isomorphism between the “geometric side” and the “spectral side.”
Example:

$\mathbb{C}(G) \to \oplus_{\rho \in Irr(G)} Hom(V_\rho, V_\rho)$
This is an isomorphism:
$R(f) = \sum_{\rho \in Irr(G)} dim \rho * \rho(f)$
$= \sum_{g \in G} f(g) \rho(g)$
Assume $\rho(f) = 0 \forall \rho.$ Then $R(f) = 0$. But $R(f) \delta_e = f$.
The left hand side are the functions constant on conjugacy classes of G, while the right hand side are
$Hom_G (V_\rho, V_\rho)$.
In particular, the number of conjugacy classes is the number of irreducible representations.
And G is abelian if every irreducible representation has dimension 1.