Chebyshev polynomials on symmetric matrices

TL;DR

This paper evaluates Chebyshev polynomials on symmetric matrices related to Dynkin diagrams and applies these results to compute minimal projective resolutions of modules in certain symmetric algebras.

Contribution

It introduces a method to evaluate Chebyshev polynomials on specific symmetric matrices and applies it to representation theory of symmetric algebras.

Findings

Explicit calculations of Chebyshev polynomials on adjacency matrices of Dynkin diagrams

Derivation of minimal projective resolutions for modules in finite and tame symmetric algebras

Connection between polynomial evaluations and algebraic module resolutions

Abstract

In this paper we evaluate Chebyshev polynomials of the second-kind on a class of symmetric integer matrices, namely on adjacency matrices of simply laced Dynkin and extended Dynkin diagrams. As an application of these results we explicitly calculate minimal projective resolutions of simple modules of symmetric algebras with radical cube zero that are of finite and tame representation type.