Algebraic random walks in the setting of symmetric functions

TL;DR

This paper explores algebraic random walks within the ring of symmetric functions, analyzing how different algebraic structures induce various transformations like translations and scalings in the context of symmetric functions.

Contribution

It introduces a framework for algebraic random walks based on co- and Hopf-algebraic structures in symmetric functions, highlighting new types of transformations induced by these walks.

Findings

Outer, inner, and plethystic ARWs produce translations, dilations, and inflations.

Standard coordinates ensure positivity and proper normalization of measures.

Walk steps correspond to specific geometric transformations of height or occupancy coordinates.

Abstract

Using the standard formulation of algebraic random walks (ARWs) via coalgebras, we consider ARWs for co-and Hopf-algebraic structures in the ring of symmetric functions. These derive from different types of products by dualisation, giving the dual pairs of outer multiplication and outer coproduct, inner multiplication and inner coproduct, and symmetric function plethysm and plethystic coproduct. Adopting standard coordinates for a class of measures (and corresponding distribution functions) to guarantee positivity and correct normalisation, we show the effect of appropriate walker steps of the outer, inner and plethystic ARWs. If the coordinates are interpreted as heights or occupancies of walker(s) at different locations, these walks introduce translations, dilations (scalings) and inflations of the height coordinates, respectively.