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Representation Theory: Basics and Heisenberg Representation (3) May 25, 2010

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We return to our regularly scheduled programming — more notes from the seminar.

The next example is the Heisenberg group. Recall that this is the group of 3-by-3 upper triangular matrices with ones on the diagonal. What are the irreducible representations of this group H?

Let (V, \omega) = (\mathbb{F}_p \times \mathbb{F}_p; \omega(v, v') = det(v v'). V is denoted a symplectic vector space, and \omega is its form. The Heisenberg group H is
H = V \times \mathbb{F}_p
endowed with the group law
(v, z) * (v', z') = (v + v', z + z' + 1/2 \omega (v, v'))

One-dimensional representation:
H \to V \to \mathbb{C}^\times
\rho_\theta(h) = \theta(Pr(h))
where \theta is the function from V to \mathbb{C}^\times. So this is just the projection.

Stone-von Neumann Theorem:

There exists a unique up to isomorphism irreducible representation $latex(\pi, H, \mathcal{H})$ such that
\pi |_z = \psi(z) Id|_{\mathcal{H}}

Lagrangian models.

Space:
\mathcal{H}_L = \mathbb{C}(L \ H, \psi) = \{ f: H \to \mathbb{C}, f(z l h) = \psi(z) f(h)
$dim \mathcal{H}_L = p$
We can think of L as a line through the space V.

Action
\pi_L = H \to \mathcal{H}_L
[\pi_L(h') f](h) = f(h h')

Claim (\pi_L, H, \mathcal{H}) irreducible.
We obtain (p-1) \times p-dimensional irreducible representations
P^2 * 1^2 + (p-1)*p^2 = p^3 = |H|

Intertwiners:
F_{M, L} \in Hom_H(\mathcal{H}_L, \mathcal{H}_M) \simeq \mathbb{C}
F_{M, L}[f] (h) = \sum_{m \in M} f(mh)

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Representation Theory: Basics and Heisenberg Representation (2) May 21, 2010

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More notes from Shamgar Gurevich’s lectures bringing representation theory to the applied-math populace.

We start with an example. Let G = S_4 = Aut\{a, b, c, d\}. Denote the irreducible representations of G by G^v. There are 5 of these, one for each conjugacy class (in the case of the symmetric group, a conjugacy class is a partition.)

Constructions: S_4 goes into the complex numbers over the tetrahedron, which is V_0 \oplus V_{const}, where V_0 is the three-dimensional space of the front face of the tetrahedron. The representation is
\rho_3: S_4 \to GL(V_0).
Why is \rho_3 irreducible?

General tool: the intertwining number, \langle \rho, \pi \rangle.

Proposition: \rho is irreducible iff \langle \rho, \rho \rangle =1.
Proof: \rho = \oplus m_i \rho_i.
1 =  = \sum m_i^2
Since these are integers, the equation is true iff there is only one subrepresentation.

In our case,
\rho = \rho_3 + \rho_1
\langle \rho, \rho \rangle = \langle \rho_3, \rho_3 \rangle  + 2\langle \rho_3, \rho_1 \rangle  +\langle \rho_1, \rho_1 \rangle .
We have X, Y as G-sets: $ latex (\pi_X, G, \mathbb{C}(X))$.

Proposition:
\langle X, Y \rangle is the number of G orbits in X \times Y.
In our case, \langle \rho, \rho \rangle= 2 implies \langle \rho_3, \rho_3 \rangle= 1.

Proof of proposition:
K: Hom(\mathbb{C}(X), \mathbb{C}(Y)) \to \mathbb{X \times Y}
both groups are isomorphic to G. This implies
Hom_G (\mathbb{C}(X), \mathbb{C}(Y)) \to \mathbb{C}(X \times Y)^G.

Another 3-dimensional representation:
\pi_3=  sgn \otimes \rho_3: S_4 \to GL(V_0).
Clearly this is also irreducible.
How do we know it’s not equivalent to \rho_3?

Well, take three function f_1, f_2, f_3 defined on the triangle (all entries have to add to 1.) f_1 is 0 on both vertices a, b, f_2 is 1 on both, and f_3 is 1 on a and -1 on b.
These form a basis. The trace of \rho_3 in this basis is 1 + 1 + -1 = 1, and the trace of \pi_3 in this basis is sign(a, b) = -1. So they are not identical.

(To be continued — I’m posting from Montreal, so I don’t have long blocks of computer time.)

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

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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.

Representation Theory for cryo-EM April 14, 2010

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I’m going to a seminar on representation theory taught by Shamgar Gurevich, in particular as it applies to the cryo-electron microscopy problem.

Cryo-EM is actually what I’ve been spending this whole year on, for my senior thesis. The short story is this. One method of determining protein structure is to encase the proteins in a thin film of ice, stick them in an electron microscope, and take pictures. (Really they aren’t photographs but measurements of electrical potential; but this gives a picture of the density of a cross section of the molecule.) The trouble is, these are two-dimensional images, and we want a three-dimensional model.

It would be straightforward to do this if we knew the direction the images were taken at; we could reconstruct the 3-d structure using the Fourier Projection Slice Theorem, the same way tomography works. But we don’t know the angles. Another hurdle is that experimentally, the images are extremely noisy, with an SNR of about 1/60. So, we need mathematics to come to the rescue. It turns out to be very pretty math, involving graph theory, the eigenvalues of large sparse matrices, and some random matrix theory.
For more thorough explanations, check out some of my advisor’s recent papers.

Anyhow, all that is by way of introduction. Cryo-EM is also intimately related to representation theory, since it deals with the symmetries of groups. For example, if the molecule happens to have internal symmetries, and many such molecules do, then the simplest version of the reconstruction algorithm falls apart. We need representation theory to explain generalizations of the problem.

So I’m learning. At the moment we’re only working with finite groups; today we proved Schur’s Lemma, which gives an important relationship — a little like orthogonality, to my mind — between irreducible finite-dimensional representations of a group. If the two representations are not isomorphic then there is no linear map between them that commutes with the action of the group; if they are isomorphic, then the linear map is a scalar operator.

For us, the motivation is that this allows us to diagonalize matrices. If an operator T from V to V is diagonalizable, and V has a group representation, and T commutes with the representation, then T preserves all the irreducible subrepresentations. This, and Schur’s lemma, allows us to conclude that T restricted to a subrepresentation is some (complex) eigenvalue multiple of the identity on that subrepresentation.

We like diagonalizing big matrices for cryo-EM because a key method is to make enormous matrices based on the correlations between images, and determine the viewing directions of the images from the eigenvectors of those matrices. These matrices are sparse, which makes computation easier, but they have to be very large to achieve accuracy. So methods of diagonalization — and methods of identifying irreducible subrepresentations, a related problem — are invaluable.