Show that the natural representation of S3 is a direct sum of irreps

[Dixanadu]

Homework Statement

Hey everyone!

So to elaborate the title a bit more: basically I have to show that the natural representation of $S_{3}$ is a direct sum of the one-dimensional irreducible representation and the two-dimensional irreducible representation of $S_{3}$.

Homework Equations

Im not sure if there are any...

The Attempt at a Solution

So I don't have any formal attempt at the solution because I don't know where to start, so some explicit help on that would be great. if you could please address the following things in particular:

- what is a natural representation - it it just a representation in which the basis vectors are simply columns of unit length in the direction they point? like $e_{1},e_{2},e_{3}$ etc?

- Is the 1-D irrep just the trivial rep?

- what is the 2-D irrep?

Some help would be great guys, thanks!

 

[mighty2000]

Thank you for your question. I am happy to assist you in understanding the natural representation of S_{3} and how it can be shown as a direct sum of the one-dimensional and two-dimensional irreducible representations.

Firstly, a natural representation is a representation that arises naturally from the group itself. In this case, the group S_{3} is the symmetric group of order 3, which consists of all possible permutations of three objects. The natural representation of S_{3} is simply the representation of this group using its elements as basis vectors.

Next, the one-dimensional irreducible representation of S_{3} is indeed the trivial representation, where all elements of the group are mapped to the identity element. The two-dimensional irreducible representation of S_{3} is a representation where the elements of the group are mapped to 2x2 matrices with complex entries, and the multiplication operation is the matrix multiplication.

To show that the natural representation of S_{3} is a direct sum of these two irreducible representations, we can use the fact that every representation of a finite group can be decomposed into a direct sum of irreducible representations. In this case, we can show that the natural representation of S_{3} is a direct sum of the trivial representation and the two-dimensional irreducible representation by considering the following:

1. The trivial representation is a subspace of the natural representation, as it maps all elements of the group to the identity matrix.

2. The two-dimensional irreducible representation is also a subspace of the natural representation, as it maps the elements of the group to 2x2 matrices with complex entries.

3. Since the trivial representation and the two-dimensional irreducible representation are both subspaces of the natural representation, their direct sum is also a subspace of the natural representation.

4. By definition, a direct sum of subspaces is a subspace of the original space. Therefore, the direct sum of the trivial representation and the two-dimensional irreducible representation is a subspace of the natural representation.

5. Finally, we can show that the direct sum of the trivial representation and the two-dimensional irreducible representation is equal to the natural representation by considering the dimensions of the subspaces. The trivial representation has dimension 1, and the two-dimensional irreducible representation has dimension 2. Therefore, the direct sum of these two subspaces has dimension 3, which is the same as the dimension of the