
TL;DR
This paper characterizes groups with blocks having only linear characters or Brauer characters, linking these properties to group structure and providing a formula relating block algebra dimension to irreducible Brauer characters.
Contribution
It establishes new criteria for blocks with linear characters and derives a dimension formula for block algebras in such groups.
Findings
Groups with only linear ordinary characters are p-nilpotent with abelian Sylow p-subgroups.
Groups with only linear Brauer characters satisfy specific subgroup inclusion conditions.
Dimension of block algebra relates to sum of squares of irreducible Brauer character degrees.
Abstract
This paper will prove that: 1. has a block only having linear ordinary characters if and only if is a -nilpotent group with an abelian Sylow -subgroup; 2. has a block only having linear Brauer characters if and only if , where is the principal block of and is the -module affording the Brauer character ; 3. if satisfies the conditions above, then for any block algebra of , we have where is the defect group of .
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsFinite Group Theory Research · Algebraic structures and combinatorial models · Rings, Modules, and Algebras
