Duality for regular systems of weights

Abstract

Regular systems of weights are certain combinatorial and arithmetic objects related to a generalization of Coxeter elements [S6,7,8 and 11], and introduced in motivation to understand the flat structure for primitive forms for isolated hypersurface singularities [S3] (cf. [Man],[S11]). In the present article, the theory is applied to explain the self-duality of ADE (=simply laced Dynkin diagrams) and the strange duality of Arnold. Beyond the original applications, the study gives further class of dual weight systems, which, for instance, has close connection with Conway group and seems interesting to be studied yet further. On the other hand, the duality of weight systems has an interpretation in terms of certain products of Dedekind eta functions. We give a conjecture on the non-negativity of the Fourier coefficients of the eta-products. The conjecture is solved affirmatively for the cases corresponding to elliptic root systems [S45]. But the meaning is not yet clear. Recently, one finds an equivalence between the duality in the present article and certain string duality in mathematical physics [T]. 0. Introduction. The present article gives a general frame work on the duality of regular systems of weights. For a sake of self-containedness, all proofs are given or sketched except for some basic facts. For simplicity, we shall call a regular system of weights a weight system unless otherwise is stated. A weight system W := (a, b, c;h) is a system of 4 positive integers with some arithmetic constraint (see (1.0)). To W , we attach a cyclotomic polynomial φW , called the characteristic polynomial of the weight system (see (2.1)). The duality we study in the present article appears as a duality between the cyclotomic polynomials. Let us explain this by examples. Let h be a positive integer and let φ(λ) and φ∗(λ) be cyclotomic polynomials whose roots are h-th roots of unity. The polynomials can be decomposed in the form: