FORMAL GROUP-THEORETIC GENERALIZATIONS OF THE NECKLACE ALGEBRA, INCLUDING A q-DEFORMATION

N. Metropolis and G.-C. Rota Adv. Math, 50, 1983, 95{125] studied the necklace polynomials, and were lead to deene the necklace algebra as a combinatorial model for the classical ring of Witt vectors (which corresponds to the multiplicative formal group law X + Y ? XY). In this paper, we deene and study a generalized necklace algebra, which is associated with an arbitrary formal group law F over a torsion free ring A. The map from the ring of Witt vectors associated with F to the necklace algebra is constructed in terms of certain generalizations of the necklace polynomials. We present a combinatorial interpretation for these polynomials in terms of words on a given alphabet. The actions of the Verschiebung and Frobenius operators, as well as of the p-typiication idempotent are described and interpreted combinatorially. A q-analogue and other generalizations of the cyclotomic identity are also presented. In general, the necklace algebra can only be deened over the rationalization A Q. Nevertheless, we show that for the family of formal group laws over the integers F q (X; Y) = X+Y ?qXY , q 2 Z, we can deene the corresponding necklace algebras over Z. We classify these algebras , and deene isomorphic ring structures on the groups of Witt vectors and the groups of curves associated with the formal group laws F q. The q-necklace polynomi-als, which turn out to be numerical polynomials in two variables, can be interpreted combinatorially in terms of so-called q-words, and they satisfy an identity generalizing a classical one. In 8], Metropolis and Rota studied the properties of the so-called necklace polynomials, which are deened for every positive integer n by M(x; n) := 1 n X djn n d x d in Qx] ; as usual, denotes the classical MM obius function. For every positive integer m, M(m; n) represents the number of primitive necklaces (i.e. asymmetric under rotation) with n colored beads, where the colors are chosen from a set of size m. Hence, M(x; n) are numerical polynomials (i.e. they take integer values for integer x). Metropolis and Rota were lead to deene for every torsion free commutative ring A with identity the necklace algebra Nr(A) (over A). This algebra is the set A 1 of innnite sequences of elements of A with componentwise addition, and multiplication deened by