As the title suggests I am working on some general relativity and combinatorics seems to be my ever-returning Achilles heel. I have a four dimensional tensor, denoted by g_abcd with a,b,c,d ranging between 0 and 3, which is fully antisymmetric, i.e.: it is zero if any of the two (or more) indices are equal. Intuitively, I know that this tensor has only one independent component, but I would like to prove it using combinatorics.(adsbygoogle = window.adsbygoogle || []).push({});

My idea is as follows: the tensor has 4^4 = 256 components --> calculate all zero components N --> (256 - N)/2^6 should be one, as 4 unequal indices can be arranged in 6 ways, so that every arrangements cuts the number of independent components in half.

I believe this is correct, but correct me if I am wrong. The tricky part is calculating the number N. My idea: N = #(2 indices equal) + #(3 indices equal) + #(4 equal).

Obviously: #(4 equal) = 4 and #(3 equal) = 4*4*3 = 48. Then:

#(2 indices equal) = #(2 equal, other two equal) + #(2 equal, other not equal) = 36 + 12*4*3*2*0.5 = 180.

This would yield N = 180 + 4 + 36 = 232, which is obviously not correct.

Can anyone help me? Thanks in advance!

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# How many independent components has a four-dimensional fully antisymmetric tensor?

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**