The negative $\sigma$-moment generating function

TL;DR

This paper explores the transformation of the $\sigma$-moment generating function of a pre-$\lambda$ random variable under negation, revealing a natural involution that preserves algebraic structure and applying it to compute invariants and traces in representation theory and number theory.

Contribution

It introduces a natural involution on symmetric power series that describes the $\sigma$-moment generating function of $-X$ in terms of $X$, with applications to invariant dimensions and Frobenius traces.

Findings

Derived a formula relating $\sigma$-moment generating functions of $X$ and $-X$

Computed dimensions of invariants in tensor products of classical groups

Generated functions for stable Frobenius traces in number theory

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.