Algebraic random walks in the setting of symmetric functions

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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.

Section snippets

Introduction and motivation

The subject of algebraic random walks (ARWs) is a development of recent interest in the subject of ‘quantum probability′ [1, 2]. The latter has origins in the foundations of quantum mechanics, but more recently has been influenced by trends towards noncommutative probability. ARWs use a very general algebraic framework [3, 4, 5], in formulating a variety of stochastic processes such as Markov and Lévy processes [6], via a Hopf algebraic formulation [7]. Early studies used commutative function

The ARW formalism

We work in the algebraic framework as developed especially via the study of quantum probability [3], although our current application to standard symmetric functions will of course be in a commutative context. Generically, random variables will be appropriate linear functionals on some operator *-algebra H, and probability measures are provided by states, that is, suitable normalised linear functionals. We are concerned here to develop the consequences of additional co-algebraic structure

Lemma

For a scalar product 〈· | ·〉H: HH → ℂ which is compatible with the multiplication in H (see below), then the linear functional ρ: H → ℂ defined by ρ(·) = 〈ρ | ·〉H is positive, 〈ρ | f2〉 ≥ 0 for arbitrary f ∈ H with ρ(f) ∈ ℝ, provided ρ is group-like, Δ(ρ) = ρ ⊗ ρ, or is a positive (convex) combination of group-like elements.

The property is trivially verified using the compatibility property, ρ|f2H=Δ(ρ)|ffH=ρρ|ffH=ρ|fH20.

Positivity (and normalisation) are clearly maintained by

The Hopf algebra of symmetric functions

We introduce and briefly review the salient features of symmetric functions needed for our purposes.

ARWs and symmetric functions

We wish to apply the ARW construction to the ring of symmetric functions itself, using its Hopf and co-algebraic structures, and to explore the nature of the resulting random walks. While it can be expected that the results will be analogous to those for the standard random walks on the line, the nature of the Hopf- and co-algebras involved is much richer, and provides fertile ground for new types of walks, diffusions and master equations.

In this spirit of ‘experimental’ investigation, we take

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