Classifying blocks with abelian defect groups of rank 3 for the prime\n 2

2017/09/13 by Charles W. Eaton, Eaton, Charles, Michael Livesey +2
Computer Science · Mathematics · #20C20 (Primary) 16D90 (Secondary) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1709.04331

openalex publication_date 2017/09/13 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

Abstract

In this paper we classify all blocks with defect group C_2ⁿ\×\nC₂\× C₂ up to Morita equivalence. Together with a recent paper of Wu,\nZhang and Zhou, this completes the classification of Morita equivalence classes\nof 2-blocks with abelian defect groups of rank at most 3. The\nclassification holds for blocks over a suitable discrete valuation ring as well\nas for those over an algebraically closed field. The case considered in this\npaper is significant because it involves comparison of Morita equivalence\nclasses between a group and a normal subgroup of index 2, so requires novel\nreduction techniques which we hope will be of wider interest. We note that this\nalso completes the classification of blocks with abelian defect groups of order\ndividing 16 up to Morita equivalence. A consequence is that Broue's abelian\ndefect group conjecture holds for all blocks mentioned above.\n

Related

• Morita equivalence classes of 2-blocks of defect three
• Morita equivalence classes of blocks with elementary abelian defect\n groups of order 16
• 2-blocks with abelian defect groups
• Broué's Conjecture for 2-blocks with elementary abelian defect groups of order 32
• Some examples of Picard groups of blocks