Exploring Representations: Understanding Fundamental and Unitary Representations

[vertices]

These are probably a bit stupid, so I hope you don't mind me asking them...

1)what is a fundamental representation?

2)what is a unitary representation? (Is it just the identity matrix?)

3)What is meant by the 'orthogonal complement' in the following context? "If $$W\subset{V}$$ is an invariant subspace for a unitary representation, $$\pi$$ on V, then the orthogonal complement of W inside V is also an invariante subspace for $$\pi$$"

 

[Fredrik]

1. I don't know.

2. A representation of a group G is a group homomorphism U into GL(V) where V is a vector space, and GL(V) is the group of invertible linear operators on V. A unitary representation is a representation such that U(g) is unitary for every g in G.

3. The set of all vectors in V that are orthogonal to all the vectors in W.

 

[vertices]

1. I don't know.

2. A representation of a group G is a group homomorphism U into GL(V) where V is a vector space, and GL(V) is the group of invertible linear operators on V. A unitary representation is a representation such that U(g) is unitary for every g in G.

3. The set of all vectors in V that are orthogonal to all the vectors in W.

Thanks for this explanation.

Can I ask a further question please:

The following is defined to be a trivial representation:

$$G \rightarrow GL(V)$$

$$g\leftharpoondown Id_v$$

what is the point of this representation exactly? It seems to me that we lose all information about the group if all the elements just map to the identity..

Also, an unrelated question - what does this statement mean exactly: "... elements of G act in a natural way"?

 

[Fredrik]

The following is defined to be a trivial representation:

$$G \rightarrow GL(V)$$

$$g\leftharpoondown Id_v$$

what is the point of this representation exactly? It seems to me that we lose all information about the group if all the elements just map to the identity..

I don't know that "leftharpoondown" means, but it sounds like you meant "mapsto":

$$g\mapsto\mbox{Id}_V$$

They're calling this function a "trivial representation" because it satisfies the definition of a representation, but is completely useless (for the reason you stated). It's like calling a set with a single point a trivial vector space.

Also, an unrelated question - what does this statement mean exactly: "... elements of G act in a natural way"?

It means that elements on G act on [whatever they say it's acting on] in the first way you would think of if someone asks you to think of a way that elements of G can act on [whatever they say it's acting on]. For example, the natural left action of GL(V) on V is $(A,v)\mapsto Av$, and the natural right action of GL(V) on the set of linear functions from V into $\mathbb R^n$ is $(f,A)\mapsto f\circ A$.

 

[vertices]

Fantastic - I'm finally beginning to understand the basics of this strange subject. As ever, thanks for your help Fredrik:)

(BTW: as regards the last post, yes I did mean 'maps to'... the symbol for "maps to" gives the code for left harpoon for some reason!)