On tensor products of path algebras of type A

TL;DR

This paper provides a formula for the Coxeter polynomial of tensor products of path algebras of type A quivers, enabling the recovery of quiver weights and linking algebraic properties to singularity theory applications.

Contribution

It introduces a formula connecting Coxeter polynomials to tensor product weights of path algebras of type A, with implications for singularity theory and vector bundle categories.

Findings

Coxeter polynomial formula for tensor products of type A path algebras

Weights n_i can be recovered from the Coxeter polynomial

Applications to endomorphism algebras in singularity theory

Abstract

We derive a formula for the Coxeter polynomial of the s-fold tensor product F[A_{n_1-1}] x ... x F[A_{n_s-1}] of path algebras of linearly oriented quivers of Dynkin type A_{n_i-1}, in terms of the weights n_1, ..., n_s > 1, and show that conversely the weights can be recovered from the Coxeter polynomial of the tensor product. Our results have applications in singularity theory, in particular these algebras occur as endomorphism algebras of tilting objects in certain stable categories of vector bundles.