Tokens: An Algebraic Construction Common in Combinatorics, Analysis, and Physics

TL;DR

This paper introduces tokens, an algebraic construction that generalizes intertwining operators, enabling the expression of semigroup structures across different sets with applications spanning combinatorics, analysis, and physics.

Contribution

It presents the concept of tokens as a flexible algebraic tool to relate semigroups on different sets, expanding the framework of intertwining operators with broad interdisciplinary applications.

Findings

Tokens generalize intertwining operators for semigroups.

Tokens facilitate expressing semigroup convolutions across sets.

Applications include quantum physics, wavelets, and combinatorics.

Abstract

We give a brief account of a construction called tokens here, which is significant in algebra, analysis, combinatorics, and physics. Tokens allow to express a semigroup on one set via a semigroup convolution on another set. Therefore tokens are similar to intertwining operators but are more flexible. Keywords: semigroups, hypergroups, tokens, poset, multiplicative functions, polynomial sequence of binomial type, integral kernel, wavelets, refinement equation, special functions, quantum propagator, path integral, quantum computing.