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Galois action and cyclic defect groups for $\operatorname{Sp}_6(2^a)$

Andrew Peña, Frank Pryor and A. A. Schaeffer Fry

Vol. 17 (2024), No. 2, 273–292

Groups are mathematical objects used to describe the structure of symmetries, with one of the most canonical examples being the set of invertible matrices of a given size over a fixed base field. For a given group, a matrix representation leverages this by providing a way to represent each of its elements as an invertible matrix. The information about the (complex) representations of a finite group can be condensed by instead considering the trace of the matrices, yielding a function known as a character. One of the overarching themes in character theory is to determine what properties about a finite group or its subgroups can be obtained by studying its characters. We study a conjecture that proposes a correlation between the makeup of a group’s irreducible characters and the properties of certain subgroups known as defect groups. In particular, we prove the conjecture for the finite symplectic groups Sp 6(2a).

character table, irreducible characters, Galois automorphisms, McKay–Navarro conjecture, Galois–McKay conjecture, local-global conjectures, symplectic group
Mathematical Subject Classification
Primary: 20C15, 20C33
Secondary: 20C20
Received: 17 August 2022
Revised: 18 January 2023
Accepted: 8 February 2023
Published: 20 May 2024

Communicated by Scott T. Chapman
Andrew Peña
Department of Mathematics and Statistics
Metropolitan State University of Denver
Denver, CO
United States
Frank Pryor
Department of Mathematical Sciences
George Mason University
Fairfax, VA
United States
A. A. Schaeffer Fry
Department of Mathematics
University of Denver
Denver, CO
United States