Abstract:

N. Metropolis and G.-C. Rota (Adv. Math. 50, 1983, 95–125) studied the necklace polynomials, and were lead to define 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 define 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-typification 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 defined over the rationalization A . Nevertheless, we show that for the family of formal group laws over the integers F_q(X, Y) = X + Y - qXY, q , we can define the corresponding necklace algebras over . We classify these algebras, and define 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 polynomials, 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. Copyright 1998 Academic Press.

Language: English

Document Type: Research article