Independent components of tensor

[cscott]

Homework Statement

For indices running from 0-3:
$$R_{iklm} = -R_{kilm} = -R_{ikml}$$

With the above conditions how do I know the number of independent component is reduced from 4^4 = 256 to 36.


No idea how to figure this out.

 

[HallsofIvy]

Homework Statement

For indices running from 0-3:
$$R_{iklm} = -R_{kilm} = -R_{ikml}$$

With the above conditions how do I know the number of independent component is reduced from 4^4 = 256 to 36.


No idea how to figure this out.

When in doubt, look at a simpler case. If you know that $$R_{ij}= -R{ji}$$, then knowing $$R_{ij}$$ for i< k immediately gives you the value of $$R_{ji}$$: Of the 16 possible values of $$R_{ij}$$, there are 4 with i= j so 16- 4= 12 where i is not equal to j and so 12/2= 6 where i< j. Knowing those 6 tells you the other 6. Further, if i= j then $$R_{ii}= -R_{ii}$$ so that must be 0: The 4 "diagonal" terms must be 0. You can choose 6 of the possible 16 terms of $$R_ij$$<sub>.

Now, for this problem you know that R_{iklm} = -R_{kilm}[/itex]. Of the 4⁴= 256 terms, how many have i< k? Knowing that many automatically gives you an equal number. How many have i= k? You know those are 0.

You also know $$R_{iklm}= -R_{ikml}$$. Do the same thing with m and l.</sub>