Universal deformation rings and tame blocks

TL;DR

This paper classifies certain modules over tame blocks of finite groups and explicitly determines their universal deformation rings, advancing understanding of modular representation theory in positive characteristic.

Contribution

It identifies all modules with endomorphism ring k in tame blocks and computes their universal deformation rings, a novel explicit classification and computation.

Findings

All modules with endomorphism ring k in tame blocks are classified.

Universal deformation rings for these modules are explicitly determined.

Provides new insights into deformation theory in modular representation contexts.

Abstract

Let k be an algebraically closed field of positive characteristic, and let W be the ring of infinite Witt vectors over k. Suppose G is a finite group and B is a block of kG of infinite tame representation type. We find all finitely generated kG-modules V that belong to B and whose endomorphism ring is isomorphic to k and determine the universal deformation ring R(G,V) for each of these modules.