Path decompositions of digraphs and their applications to Weyl algebra

TL;DR

This paper explores how decomposing directed graphs into paths relates to the Weyl algebra of differential operators, offering a combinatorial perspective on normal ordering and polynomial identities.

Contribution

It introduces a graph-theoretic framework for analyzing the Weyl algebra, including path decompositions and G-Stirling functions, connecting combinatorics with algebraic properties.

Findings

Path decompositions provide insights into minimal polynomial identities.

G-Stirling functions enumerate path decompositions by sources and sinks.

Graph-theoretic methods relate to classical algebraic theorems like Eulerian tours.

Abstract

We consider decompositions of digraphs into edge-disjoint paths and describe their connection with the $n$-th Weyl algebra of differential operators. This approach gives a graph-theoretic combinatorial view of the normal ordering problem and helps to study skew-symmetric polynomials on certain subspaces of Weyl algebra. For instance, path decompositions can be used to study minimal polynomial identities on Weyl algebra, similar as Eulerian tours applicable for Amitsur--Levitzki theorem. We introduce the $G$-Stirling functions which enumerate decompositions by sources (and sinks) of paths.