Refinements of the orthogonality relations for blocks

TL;DR

This paper refines orthogonality relations for blocks in finite groups by expressing decomposition numbers via cyclotomic bases, leading to new bounds on irreducible characters and applications to specific block types.

Contribution

It introduces refined orthogonality relations using cyclotomic bases and applies these to derive bounds on irreducible characters, extending previous results and addressing a recent conjecture.

Findings

Derived upper bounds for the number of height 0 irreducible characters.

Generalized results to blocks with abelian defect groups of rank 2.

Addressed a recent conjecture by Navarro.

Abstract

For a block B of a finite group G there are well-known orthogonality relations for the generalized decomposition numbers. We refine these relations by expressing the generalized decomposition numbers with respect to an integral basis of a certain cyclotomic field. After that, we use the refinements in order to give upper bounds for the number of irreducible characters (of height 0) in B. In this way we generalize results from [H\'ethelyi-K\"ulshammer-Sambale, 2014]. These ideas are applied to blocks with abelian defect groups of rank 2. Finally, we address a recent conjecture by Navarro.