Cyclotomic identity

In mathematics, the cyclotomic identity states that

${1 \over 1-kz}=\prod _{j=1}^{\infty }\left({1 \over 1-z^{j}}\right)^{M_{j}(k)}$

where M is Moreau's necklace-counting function,

$M_{n}(k)={1 \over n}\sum _{d\,|\,n}\mu \!\left({n \over d}\right)k^{d},$

and μ is the classic Möbius function of number theory.

The name comes from the denominator, 1 − z ^j, which is the product of cyclotomic polynomials.

The left hand side of the cyclotomic identity is the generating function for the free associative algebra on k generators, and the right hand side is the generating function for the universal enveloping algebra of the free Lie algebra on k generators. The cyclotomic identity witnesses the fact that these two algebras are isomorphic.

Another interpretation is as the Hasse–Weil zeta function of the affine line over the finite field $\mathbb {F} _{q}$ with $k=q$ elements. The exponent $M_{n}(k)$ counts the number of number of maximal ideals of degree n in the coordinate ring $\mathbb {F} _{q}[x]$, i.e. the number of monic irreducible polynomials of degree n.

There is also a symmetric generalization of the cyclotomic identity found by Strehl:

$\prod _{j=1}^{\infty }\left({1 \over 1-kz^{j}}\right)^{M_{j}(\ell )}=\prod _{j=1}^{\infty }\left({1 \over 1-\ell z^{j}}\right)^{M_{j}(k)}$

References

• Metropolis, N.; Rota, Gian-Carlo (1984), "The cyclotomic identity", in Greene, Curtis (ed.), Combinatorics and algebra (Boulder, Colo., 1983). Proceedings of the AMS-IMS-SIAM joint summer research conference held at the University of Colorado, Boulder, Colo., June 5–11, 1983., Contemp. Math., vol. 34, Providence, R.I.: American Mathematical Society, pp. 19–27, ISBN 978-0-8218-5029-9, MR 0777692