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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 |
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Vol. 17 (2024), No. 2, 273–292
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Abstract |
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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 . |
Keywords
character table, irreducible characters, Galois
automorphisms, McKay–Navarro conjecture, Galois–McKay
conjecture, local-global conjectures, symplectic group
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Mathematical Subject Classification
Primary: 20C15, 20C33
Secondary: 20C20
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Milestones
Received: 17 August 2022
Revised: 18 January 2023
Accepted: 8 February 2023
Published: 20 May 2024
Communicated by Scott T. Chapman
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Authors |
| Andrew Peña | |
| Department of Mathematics and
Statistics Metropolitan State University of Denver Denver, CO United States |
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| Frank Pryor | |
| Department of Mathematical
Sciences George Mason University Fairfax, VA United States |
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| A. A. Schaeffer Fry | |
| Department of Mathematics University of Denver Denver, CO United States |
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| © 2024 MSP (Mathematical Sciences Publishers). |
