Permutation representation (symmetric group)
Okay kiddo, have you ever played with toys that can be put together in different ways, like building blocks or Legos? Imagine you have a bunch of letters, let's say A, B, and C, and you want to put them in different orders. You could write out all the possible ways they could be arranged: ABC, ACB, BAC, BCA, CAB, CBA.
Now, let's pretend these letters are actually people, and we want to see how they move around. We can make a group called the symmetric group that keeps track of all the different ways we can rearrange these people. Each way is called a permutation.
We can represent each permutation with a matrix. The rows and columns in the matrix correspond to the letters or people, and the numbers in the cells tell us where each person goes. For example, the permutation ABC becomes the matrix:
1 0 0
0 1 0
0 0 1
This tells us that person A stays in the first position, person B stays in the second position, and person C stays in the third position.
Now, let's say we want to see how the people move when we apply a permutation. We can take a different permutation, like BAC, and multiply it by our original matrix:
0 1 0 1 0 0 0 1 0
1 0 0 x 0 1 0 = 1 0 0
0 0 1 0 0 1 0 0 1
The result is a new matrix that shows where each person is after the permutation is applied. In this case, person A moved to the second position, person B moved to the first position, and person C stayed in the third position.
We can keep doing this for all the possible permutations, and we get a lot of matrices that show us how the people move around. This is called a permutation representation of the symmetric group. It's like a big map that tells us how everything is connected and where everything goes when we move it around.
Now, let's pretend these letters are actually people, and we want to see how they move around. We can make a group called the symmetric group that keeps track of all the different ways we can rearrange these people. Each way is called a permutation.
We can represent each permutation with a matrix. The rows and columns in the matrix correspond to the letters or people, and the numbers in the cells tell us where each person goes. For example, the permutation ABC becomes the matrix:
1 0 0
0 1 0
0 0 1
This tells us that person A stays in the first position, person B stays in the second position, and person C stays in the third position.
Now, let's say we want to see how the people move when we apply a permutation. We can take a different permutation, like BAC, and multiply it by our original matrix:
0 1 0 1 0 0 0 1 0
1 0 0 x 0 1 0 = 1 0 0
0 0 1 0 0 1 0 0 1
The result is a new matrix that shows where each person is after the permutation is applied. In this case, person A moved to the second position, person B moved to the first position, and person C stayed in the third position.
We can keep doing this for all the possible permutations, and we get a lot of matrices that show us how the people move around. This is called a permutation representation of the symmetric group. It's like a big map that tells us how everything is connected and where everything goes when we move it around.
Related topics others have asked about: