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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.

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