How Many Parameters Define O(N) and SO(N) Groups?

[Norman]

A little intro: This is for a whirlwind intro to Group Theory as part of another class (QFT) in which we are not proving anything, simply introducing definitions and theorems. We are not using a textbook, simply some notes the professor has written up for this intro and the notes are very incomplete. They also do not give any proof or explanation. So here is the problem:

Show that the number of group parameters for $$O(N)$$ and $$SO(N)$$ are $$\frac{N(N-1)}{2}$$

Does this make sense as the solution:

We have $$n^2$$ parameters. For $$O(N)$$ we have the condition that $$O^T O = 1$$. This condition places $$\frac{n(n+1)}{2}$$ constraints on the parameters. This leaves $$n^2 - \frac{n(n+1)}{2} = \frac{n(n-1)}{2}$$

Since $$O(N)$$ has $$detO(N) = \pm 1$$ and $$SO(N)$$ has $$detSO(N)=+1$$, simply picking the sign of the determinent doesn't force any additional constraint on the parameters.

I can only show this statement: "This condition places $$\frac{n(n+1)}{2}$$ constraints on the parameters." for specific examples, like N=2 and 3. I can see how it is extended to higher orders, but I cannot prove it in the general case for N. Any ideas? Also, any good books on Group Theory and Lie Algebras for Physicists?
Thanks,
Ryan

 

[George Jones]

The matrix equation O^T O = 1 gives n^2 scalar equations - one for each element of the identity matrix. By symmetry, the equations for the elements below the main diagonal are the same as the equations above the main diagonal, so the number of constraint equations is the the number of elements on the main diagonal, n, plus the number of elements above the main diagonal, say x. x equals the total number of elements in the matrix minus the number of elements on the diagonal, all divided by 2, i.e., x = (n^2 - n)/2.

Cosequently, the number of contraint equations is

n + x = n(n + 1)/2.

Another way to calculate this is using an arithmetic series. The number of constraint equation for the top row of the identity matrix is n. The number of constraint equations for the second row of the identity matrix is n - 1, the number of elements in this row starting with the main diagonal and moving right, etc. So the total number of constraint equations is

n + (n -1) + (n - 2) + ... + 2 + 1 = n(n + 1)/2.

Regards,
George

 

[quicknote]

Yes, your solution makes sense. You have correctly identified the constraints imposed by the condition O^T O = 1 for O(N) and SO(N) groups, and you have correctly calculated the number of remaining parameters for each group.

To prove this in the general case for N, you can use the fact that the matrix O(N) can be written as a product of N(N-1)/2 independent rotations and reflections, each of which contributes one parameter. This can be shown using the fact that the determinant of O(N) is equal to the product of the determinants of these individual rotations and reflections.

As for books on Group Theory and Lie Algebras for physicists, some popular choices are "Group Theory in a Nutshell for Physicists" by A. Zee and "Lie Algebras in Particle Physics" by H. Georgi. However, there are many other good resources available, so it's best to explore and find what works best for you.

 

[quicknote]

Yes, your solution makes sense. The idea is that for O(N) and SO(N), we have n^2 parameters but the condition O^T O = 1 places constraints on these parameters. This reduces the number of independent parameters to n^2 - \frac{n(n+1)}{2} = \frac{n(n-1)}{2}. This is because the condition O^T O = 1 means that we only need to specify the upper or lower triangular elements of the matrix, and the diagonal elements are determined by the other elements. This results in n(n+1)/2 constraints.

As for proving it in the general case for N, you can use mathematical induction. Start with the base case N = 2 and show that it holds. Then assume it holds for N = k and show that it also holds for N = k+1. This will prove it for all N.

As for books on Group Theory and Lie Algebras for physicists, some good options are "Lie Algebras in Particle Physics" by Howard Georgi and "Group Theory in Physics" by Wu-Ki Tung. These books cover the basics of group theory and its applications in physics.