Self-Dual Symmetric Polynomials and Conformal Partitions

A conformal partition function ${\cal P}_n^m(s)$, which arose in the theory of Diophantine equations supplemented with additional restrictions, is concerned with {\it self-dual symmetric polynomials} -- reciprocal ${\sf R}^{\{m\}}_ {S_n}$ and skew-reciprocal ${\sf S}^{\{m\}}_{S_n}$ algebraic polynomials based on the polynomial invariants of the symmetric group $S_n$. These polynomials form an infinite commutative semigroup. Real solutions $\lambda_n(x_i)$ of corresponding algebraic Eqns have many important properties: homogeneity of 1-st order, duality upon the action of the conformal group ${\sf W}$, inverting both function $\lambda_n$ and the variables $x_i$, compatibility with trivial solution, {\it 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 ${\cal P}_n^m(s)$. The unimodality indices for the reciprocal and skew-reciprocal equations were found. The existence of algebraic functions $\lambda_n(x_i)$ invariant upon the action of both the finite group $G\subset S_n$ and conformal group ${\sf W}$ is discussed.