The negative $\sigma$-moment generating function

Abstract

For $X$ a pre-$\lambda$ random variable, we show the $\sigma$-moment generating function of $-X$ can be obtained from the $\sigma$-moment generating function of $X$ by applying the composition of the standard and degree flip involutions on symmetric power series. This isometric involution is natural as it preserves the pre-$\lambda$ ring structure on symmetric power series with pre-$\lambda$ coefficients, thus this formula provides a simple description of the $\sigma$-moment generating function of $-X$ whenever the $\sigma$-moment generating function of $X$ has a simple description using the pre-$\lambda$ structure. As an application we compute, in a natural range, the dimensions of orthogonal and symplectic group invariants in tensor products of exterior powers of their standard representations on $\mathbb{C}^n$. We also compute a generating function for stable traces of Frobenius related to the moment conjecture for prime-order function field Dirichlet characters.