Formal Group-Theoretic Generalizations of the Necklace Algebra, Including aq-Deformation

Abstract

N. Metropolis and G.-C. Rota (Adv. Math.50, 1983, 95–125) studied thenecklace polynomials, and were lead to define thenecklace algebraas a combinatorial model for the classical ring ofWitt vectors(which corresponds to the multiplicative formal group lawX+Y−XY). In this paper, we define and study a generalized necklace algebra, which is associated with an arbitrary formal group lawFover a torsion free ringA. The map from the ring of Witt vectors associated withFto 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 theVerschiebungandFrobeniusoperators, as well as of thep-typification idempotentare described and interpreted combinatorially. Aq-analogue and other generalizations of thecyclotomic identityare also presented. In general, the necklace algebra can only be defined over the rationalizationA⊗Q. Nevertheless, we show that for the family of formal group laws over the integersFq(X,Y)=X+Y−qXY,q∈Z, we can define the corresponding necklace algebras over Z. 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 lawsFq. Theq-necklace polynomials, which turn out to benumerical polynomialsin two variables, can be interpreted combinatorially in terms of so-calledq-words, and they satisfy an identity generalizing a classical one.