Remarks on a triangulated version of Auslander-Kleiner's Green correspondence

2020/11/27 by Zimmermann, Alexander
#18E30 #20C05 #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2011.13709

Abstract

For a finite group G and an algebraically closed field k of characteristic p>0 for any indecomposable finite dimensional kG-module M with vertex D and a subgroup H of G containing N_G(D) there is a unique indecomposable kH-module N of vertex D being a direct summand of the restriction of M to H. This correspondence, called Green correspondence, was generalised by Auslander-Kleiner to the situation of pairs of adjoint functors between additive categories. In the original situation of group rings Carlson-Peng-Wheeler proved that this correspondence is actually restriction of triangle functors between triangulated quotient categories of the corresponding module categories. We review this theory and show how we got a common generalisation of the approaches of Auslander-Kleiner and Carlson-Peng-Wheeler, using Verdier localisations.

Related

• The Green correspondence for SL(2,p)
• Classification of indecomposable modules for finite group schemes of domestic representation type
• The algebraic K-theory of Green functors
• Jordan Types for Indecomposable Modules of Finite Group Schemes
• Categories of modules given by varieties of p-nilpotent operators