Rigidity degrees of indecomposable modules over representation-finite self-injective algebras

TL;DR

This paper provides explicit combinatorial formulas for calculating the rigidity degrees of all indecomposable modules over representation-finite self-injective algebras, linking these to the rigidity dimension and applying to types A and E.

Contribution

It introduces a combinatorial approach using the Euclidean algorithm to determine rigidity degrees for indecomposable modules over certain algebras, a novel method in this context.

Findings

Explicit formulas for rigidity degrees of indecomposable modules

Determination of rigidity dimensions for types A and E algebras

Development of combinatorial methods based on Euclidean algorithm

Abstract

The rigidity degree of a generator-cogenerator determines the dominant dimension of its endomorphism algebra, and is closely related to a recently introduced homological dimension -- rigidity dimension. In this paper, we give explicit formulae for the rigidity degrees of all indecomposable modules over representation-finite self-injective algebras by developing combinatorial methods from the Euclidean algorithm. As an application, the rigidity dimensions of some algebras of types $A$ and $E$ are given.