The partial derivative of ratios of Schur polynomials and applications to symplectic quotients

Abstract

We show that a ratio of Schur polynomials $s_{\lambda}/s_{\rho}$ associated to partitions $\lambda$ and $\rho$ such that $\lambda\subsetneq\rho$ has a negative partial derivative at any point where all variables are positive. This is accomplished by establishing an injective map between sets of pairs of skew semistandard Young tableaux that preserves the product of the corresponding monomials. We use this result and the description of the first Laurent coefficient of the Hilbert series of the graded algebra of regular functions on a linear symplectic quotient by the circle to demonstrate that many such symplectic quotients are not graded regularly diffeomorphic. In addition, we give an upper bound for this Laurent coefficient in terms of the largest two weights of the circle representation and demonstrate that all but finitely many circle symplectic quotients of each dimension are not graded regularly diffeomorphic to linear symplectic quotients by $\operatorname{SU}_2$.