Finding the standard representation of a group

[Gulli]

Homework Statement

In order to construct a character table I need the character of the standard representation. The only problem is I don't know how to find the standard representation of a given group.

Homework Equations

The groupprops wiki tells me: "Take a representation of degree n obtained by the usual action of the symmetric group on the basis set of a vector space. Now, look at the n − 1-dimensional subspace of vectors whose sum of coordinates in the basis is zero. The representation restricts to an irreducible representation of degree n − 1 on this subspace. This is the standard representation." Which confuses me. What exactly does it mean to "look at the n − 1-dimensional subspace of vectors whose sum of coordinates in the basis is zero". What am I suppsoed to do here and does it mean that if I chose to represent say D3 on 96x96 matrices I get a standard representation of dimension 95 and if I choose to start out with 3x3 matrices I get a standard rep. of dimension 2?

Wikipedia and Wolfram mathworld don't even have an entry on the standard representation.

The Attempt at a Solution

I have no friggin idea... I haven't found a clear definition, nor an example to demonstrate the process.

 

[Klockan3]

Are you sure that you aren't talking about regular representations?
http://en.wikipedia.org/wiki/Regular_representation

 

[Gulli]

Are you sure that you aren't talking about regular representations?
http://en.wikipedia.org/wiki/Regular_representation

I'm not sure, my reader (in Dutch) says the standard representation of Cn and Dn (with n greater than or equal to 2) is formed by complex 2x2 matrices. Does that correspond to the regular representation?

 

[Klockan3]

I'm not sure, my reader (in Dutch) says the standard representation of Cn and Dn (with n greater than or equal to 2) is formed by complex 2x2 matrices. Does that correspond to the regular representation?

No, I don't really see any pattern in that, can you give your books definition of it?

 

[Gulli]

No, I don't really see any pattern in that, can you give your books definition of it?

If G < On(R) a finite subgroup, then because On(R) < GLn(C) we get a natural representation (what's that btw?) ρ : G → GLn(C) of G on Cn. this is the so-called
standard representation of G on Cn. The cyclic group Cn and the dihedral group Dn have
a standard representation on Cs2 (complex space with dim 2), and the symmetrygroup Td of the tetrahedron has a standard representation on Cs3 (complex space with dim 3). That's from my reader.

This is from an English book I found online:

"The Standard Representation. A matrix Lie group G is by denition a subset of some GL(n;R) or GL(n;C). The inclusion map of G into GL(n) (i.e., (A) = A) is a representation of G, called the standard representation of G. Thus for example the standard representation of SO(3) is the one in which SO(3) acts in the usual way on R3. If G is a subgroup of GL(n;R) or GL(n;C), then its Lie algebra g will be a subalgebra of gl(n;R) or gl(n;C). The inclusion of g into gl(n;R) or gl(n;C) is a representation of g, called the standard representation."

 

[Gulli]

A maths professor told me the standard representation is only defined on a few groups and there is no method of deriving them: they were basically postulated as the smallest representation that gets the job done for those groups.

 

[aalaniz]

For a finite group of order n, there is a standard procedure for finding the regular representation.

Begin with the group table with elements labeled: g0=identity, g1,...,gn and construct the matrix for each element using "bra-ket" scalar product notation as follows

The matrix for g0 is

<g0|g0|g0> <g0|g0|g1> ... <g0|g0|gn>
<g1|g0|g0> <g1|g0|g1> ... <g1|g0|gn>
.
.
.
<gn|g0|g0> <gn|g0|g1> ... <gn|g0|gn>

This gives you the diagonal matrix of 1s.

The matrix element for g1 is

<g0|g1|g0> <g0|g1|g1> ... <g0|g1|gn>
<g1|g1|g0> <g1|g1|g1> ... <g1|g1|gn>
.
.
.
<gn|g1|g0> <gn|g1|g1> ... <gn|g1|gn>

and so on for g2,...,gn.

Example--Group C3 (triangle group)


<e|e|e> <e|e|c> <e|e|c>
<c|e|e> <c|e|c> <c|e|c^2>
<c^2|e|e> <c^2|e|c> <c^2 |e|c^2>

So for e we have the 3X3 matrix of 1s along the diagonal. For c we have:


<e|c|e> <e|c|c> <e|c|c^2>
<c|c|e> <c|c|c> <c|c|c^2>
<c^2|c|e> <c^2|c|c> <c^2|c|c^2>

which is


<e|c> <e|c^2> <e|e>
<c|c> <c|c^2> <c|e>
<c^2|c> <c^2|c^2> <c^2|e>

which is


0 0 1
1 0 0
0 1 0

There is so much crap out there! I just posted step-by-step notes filled with examples under my blog aalaniz in files BS1 through BS4 that build algebra, local and global topology and differential equations (ordinary, partial, linear or nonlinear) in a unified context the way mathematicians and physicists new a century ago. Now we teach all of these subjects unnaturally divorced from each other.

Warning: the notes were typed up airplanes. There are typos. I will be updating BS4 soon to add more detail on characters.

Alex