p-Blocks Relative to a Character of a Normal Subgroup II

Abstract

Let p be a prime number, let G be a finite group, let N be a normal subgroup of G, and let \(\theta \) be a G-invariant irreducible character of N. In Rizo (J Algebra 514:254–272, 2018), we introduced a canonical partition of the set \(\mathrm{Irr}(G|\theta )\) of irreducible constituents of the induced character \(\theta ^G\), relative to the prime p. We call the elements of this partition the \(\theta \)-blocks. In this paper, we construct a canonical basis of the complex space of class functions defined on \(\{ x \in G \, |\, x_p \in N\}\), which supersedes previous non-canonical constructions. This allows us to define \(\theta \)-decomposition numbers in a natural way. We also prove that the elements of the partition of \({\text {Irr}}(G|\theta )\) established by these \(\theta \)-decomposition numbers are the \(\theta \)-blocks.

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Acknowledgements

Most of this paper is part of my Ph.D. thesis under the direction of Gabriel Navarro. I would like to take this opportunity to warmly thank him for thoroughly reading this manuscript. I would also like to thank Radha Kessar and Charles Eaton for many useful comments on this work and, in general, on my thesis.

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Correspondence to Noelia Rizo.

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The author acknowledges support by Ministerio de Ciencia e Innovación PID2019-103854GB-I00 and FEDER funds.

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Rizo, N. p-Blocks Relative to a Character of a Normal Subgroup II. Mediterr. J. Math. 17, 171 (2020). https://doi.org/10.1007/s00009-020-01599-z

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Keywords

  • Brauer p-blocks
  • Brauer’s first main theorem
  • Block orthogonality
  • Central extensions

Mathematics Subject Classification

  • Primary 20D
  • Secondary 20C15