Representation dimensions linked by Frobenius bimodules with applications to group algebras

TL;DR

This paper explores how Frobenius bimodules relate the representation dimensions of different algebras, providing bounds and formulas, especially for group algebras and subgroup relations.

Contribution

It establishes new connections between representation dimensions via Frobenius bimodules and derives bounds and equalities for group algebra cases.

Findings

Representation dimensions of group algebras of G and H are equal if [G:H] is invertible in the field.

Upper bounds for representation dimensions are obtained for symmetric separably equivalent algebras.

Formulas relating the representation dimensions of connected algebras are established.

Abstract

We establish relations between representation dimensions of two algebras connected by a Frobenius bimodule or extension. Consequently, upper bounds and equality formulas for representation dimensions of group algebras, symmetric separably equivalent algebras and crossed products are obtained. Particularly, for any subgroup H of a finite group G, if [G:H] is invertible in a field, then the representation dimensions of the group algebras of G and H over the field are the same.