Monday, September 19, 2005

A Certain 24th Root

This paper by Heninger, Rains and Sloane (that's Sloane as in the famous Encyclopedia) is interesting. It's about the conditions required for a formal power series with integer coefficients, to be the nth power of another such series. In particular it focuses on theta functions for lattices.

The theta function of a lattice is simply the sum of xu.u for all u in the lattice. Clearly it is a formal power series with integer coefficients if the lattice is such that u.u is always an integer. These are often pretty amazing functions, for example for certain types of lattice they are modular forms. It turns out that the theta function for the Leech lattice is a power of 24. More amazingly: this is a new result!

It's clear that the theta function is the generating function for the number of lattice elements of a given length. So if it's a 24th power you'd expect this to have a combinatorial explanation too. Unfortunately the proof isn't combinatorial. So even though the 24th root appears to be counting something, nobody knows what it is that is being counted!

Anyway, Sloane and friends had fun trawling through their sequence database trying to find other sequences that were powers. Another interesting one they found was the exponential generating function for the number of partitions of {1,...,n} into an odd number of ordered subsets. It turns out this is a perfect square. This is begging for a combinatorial explanation but it hasn't yet been found. I'll have to see if I can reproduce the square root of this series with my (unfinished) Haskell code for formal power series.

And of course this paper is interesting to me because of the prominent role of the number 24 :-)



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