Download this article
 Download this article For screen
For printing
Recent Issues

Volume 17
Issue 2, 183–362
Issue 1, 1–182

Volume 16, 5 issues

Volume 15, 5 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 8 issues

Volume 11, 5 issues

Volume 10, 5 issues

Volume 9, 5 issues

Volume 8, 5 issues

Volume 7, 6 issues

Volume 6, 4 issues

Volume 5, 4 issues

Volume 4, 4 issues

Volume 3, 4 issues

Volume 2, 5 issues

Volume 1, 2 issues

The Journal
About the journal
Ethics and policies
Peer-review process
 
Submission guidelines
Submission form
Editorial board
Editors' interests
 
Subscriptions
 
ISSN (electronic): 1944-4184
ISSN (print): 1944-4176
 
Author index
To appear
 
Other MSP journals
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
Abstract

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).

Keywords
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
Milestones
Received: 17 August 2022
Revised: 18 January 2023
Accepted: 8 February 2023
Published: 20 May 2024

Communicated by Scott T. Chapman
Authors
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