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

*Posted by Sarah in Uncategorized.*

Tags: representation theory

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Tags: representation theory

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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 . Denote the irreducible representations of G by . There are 5 of these, one for each conjugacy class (in the case of the symmetric group, a conjugacy class is a partition.)

Constructions: goes into the complex numbers over the tetrahedron, which is , where is the three-dimensional space of the front face of the tetrahedron. The representation is

Why is irreducible?

General tool: the intertwining number, .

Proposition: is irreducible iff .

Proof: .

Since these are integers, the equation is true iff there is only one subrepresentation.

In our case,

.

We have X, Y as G-sets: $ latex (\pi_X, G, \mathbb{C}(X))$.

Proposition:

is the number of G orbits in .

In our case, implies .

Proof of proposition:

both groups are isomorphic to G. This implies

Another 3-dimensional representation:

Clearly this is also irreducible.

How do we know it’s not equivalent to ?

Well, take three function defined on the triangle (all entries have to add to 1.) is 0 on both vertices a, b, is 1 on both, and is 1 on a and -1 on b.

These form a basis. The trace of in this basis is 1 + 1 + -1 = 1, and the trace of 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.)

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