Self-Dual Symmetric Polynomials and Conformal Partitions

A conformal partition function Pm n (s), which arose in the theory of Diophantine equations supplemented with additional restrictions, is concerned with self-dual symmetric polynomials – reciprocal R {m} Sn and skew-reciprocal S {m} Sn algebraic polynomials based on the polynomial invariants of the symmetric group Sn. These polynomials form an infinite commutative semigroup. Real solutions λn(xi) of corresponding algebraic Eqns have many important properties: homogeneity of 1-st order, duality upon the action of the conformal group W, inverting both function λn and the variables xi, compatibility with trivial solution, etc. Making use of the relationship between Gaussian generating function for conformal partitions and Molien generating function for usual restricted partitions we derived the analytic expressions for Pm n (s). The unimodality indices for the reciprocal and skew-reciprocal equations were found. The existence of algebraic functions λn(xi) invariant upon the action of both the finite group G ⊂ Sn and conformal group W is discussed.