Algebras with two multiplications and their cumulants

TL;DR

This paper extends the concept of cumulants from probability theory to algebraic structures with two multiplications, providing combinatorial formulas and insights into Jack characters.

Contribution

It introduces an algebraic formula involving two multiplications and cumulants, using combinatorial tools like forests, and relates to Jack characters.

Findings

Derived an algebraic cumulant formula for algebras with two multiplications

Connected cumulant formulas to combinatorial structures such as forests

Provided insights into the structure constants of Jack characters

Abstract

Cumulants are a notion that comes from the classical probability theory, they are an alternative to a notion of moments. We adapt the probabilistic concept of cumulants to the setup of a linear space equipped with two multiplication structures. We present an algebraic formula which involves those two multiplications as a sum of products of cumulants. In our approach, beside cumulants, we make use of standard combinatorial tools as forests and their colourings. We also show that the resulting statement can be understood as an analogue of Leonov--Shiraev's formula. This purely combinatorial presentation leads to some conclusions about structure constant of Jack characters.