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

*Posted by Sarah in Uncategorized.*

Tags: representation theory

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Tags: representation theory

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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

We want to find a natural basis of eigenvectors, and explain how to compute the DFT fast. Related is the following theorem:

Fix an additive character. There is a unique up to isomorphism irreducible representation such that

Let’s show first:

In particular,

To show this, let us answer

.

How to compute ?

Define

Properties of intertwining numbers :

1.

2.

3.

1 is clear; 3 is proven by Schur’s Lemma.

To prove 2, observe

The result is, if

then

The application to the first theorem is that if

and given a representation then

Proof:

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:

This is an isomorphism:

Assume Then . But .

The left hand side are the functions constant on conjugacy classes of G, while the right hand side are

.

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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