Orbital Decomposition of the Wronskian and Orbital Vandermonde Polynomials

Abstract

This paper introduces an orbital decomposition of the Wronskian determinant induced by the right action of the cyclic subgroup CnC_nCn on the symmetric group SnS_nSn. For functions f1,…,fnf_1,\ldots,f_nf1,…,fn of class Cn−1C^{n-1}Cn−1 on an interval, the n!n!n! terms in the Leibniz expansion of the Wronskian split canonically into (n−1)!(n-1)!(n−1)! orbits of cardinality nnn, yielding an exact decomposition into orbital sums. For exponential families fj(x)=eλjxf_j(x)=e^{\lambda_jx}fj(x)=eλjx, each orbital sum factors as a common exponential factor multiplied by an orbital Vandermonde polynomial. The sum of all orbital polynomials recovers the classical Vandermonde determinant. The family of orbital Vandermonde polynomials is proved to be linearly independent for every nnn, and hence spans a space of dimension (n−1)!(n-1)!(n−1)!. The paper further establishes a dihedral reciprocity law, complete low-dimensional factorizations, and a representation-theoretic description of the orbital space as an induced SnS_nSn-module whose irreducible multiplicities are governed by the Kraśkiewicz–Weyman rule. The intersection of the orbital space with the ordinary immanant space is determined according to the parity of nnn. An orbital condition number is introduced to quantify cancellation among orbital contributions and to distinguish orbitally coherent linear independence from delicate independence sustained by strong cancellations. At the nnn-th roots of unity, the Vandermonde matrix becomes the unnormalized discrete Fourier matrix, and every orbital polynomial collapses to nnn times a root of unity. This yields an orbital refinement of the classical evaluation of the Fourier determinant and a closed formula for its orbital condition number. The construction is algebraic-combinatorial and acts directly on the Leibniz expansion of the ordinary Wronskian determinant. It is distinct from the geometric theory of Wronski maps, Grassmannians, and generalized Wronskians.