Averages of ratios of characteristic polynomials in circular beta-ensembles and super-Jack polynomials

We study the averages of ratios of characteristic polynomials over circular $\beta$-ensembles, where $\beta$ is a positive real number. Using Jack polynomial theory, we obtain three expressions for ratio averages. Two of them are given as sums of super-Jack polynomials and another one is given by a hyperdeterminant. As applications, we give dual relations for ratio averages between $\beta$ and $4/\beta$.