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Archive for the ‘algebra’ Category

I have written many papers in the last 20 years. Recently, I was taking a trip down memory lane and was reading some of my older stuff. I happened on a pretty identity on one of my papers. Something I had forgotten. That particular identity probably has someone elses name attached to it. I don’t believe that I was the first person who discovered this algebraic identity, but I wouldn’t know where to start looking for the correct precedence.  If the identity has a name attached to it, I wouldn’t be surprised if the culprit for finding it first is pushing daisies.

Mathematics keeps on getting rediscovered after all.

 

The identity is as follows.

Consider a set of of s numbers (they can be real, rational, algebraic, or even more messy commutative algebra objects so long as for the most part their multiplicative inverses are well defined). Let us call this set

S= \{ \alpha_1, \dots , \alpha_s\}

And consider the set of permutations of the first s integers, where \sigma is one such permutation

\sigma\in Perm\{1, \dots, s\}

We are then instructed to take the sum

{\sum_{\sigma\in Perm\{1, \dots, s\}} } {\frac 1{(\alpha_{\sigma(1)}+\alpha_{\sigma(2)}+\dots +\alpha_{\sigma(s)}) }}{\frac 1{(\alpha_{\sigma(2)}+\dots+\alpha_{\sigma(s)})}}\dots {\frac 1{\alpha_{\sigma(s)}}}

The stipulation is that the sum has no infinities (the numbers are generic).

This sum is equal to

{\frac 1{\alpha_1 \cdot \alpha_2 \dots \alpha_s}}

As everyone can see, it’s an  obvious identity so the proof is left to the reader 😉

For some reason, looking at it I can imagine it appearing miraculously in the twistor formulation of  Yang Mills scattering amplitudes… well, this is just a random speculation.

 

 

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