Kaiserslautern - Fachbereich Mathematik
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In the representation theory of finite groups, the so-called local-global conjectures assert a relation between the representation theory of a finite group and one of its local subgroups. The McKay-Navarro conjecture claims that the action of a set of Galois automorphisms on certain ordinary characters of the local and global group is equivariant. Navarro, Späth, and Vallejo reduced the conjecture to a problem about simple groups in 2019 and stated an inductive condition that has to be verified for all finite simple groups.
In this work, we give an introduction to the character theory of finite groups and state the McKay-Navarro conjecture and its inductive condition. Furthermore, we recall the definition of finite groups of Lie type and present results regarding their structure and their representation theory.
In the second part of this work, we verify the inductive McKay-Navarro condition for various families of finite groups of Lie type.
In defining characteristic, most groups have already been considered by Ruhstorfer.
We show that the inductive condition also holds for the groups with exceptional graph automorphisms, the Suzuki and Ree groups, the groups \(B_n(2)\) for \(n \geq 2\), as well as for the simple groups of Lie type with non-generic Schur multiplier in their defining characteristic.
This completes the verification of the inductive McKay-Navarro condition in defining characteristic. We further consider the Suzuki and Ree groups and verify the inductive condition for all primes. On the way, we show that there exists a Galois-equivariant Jordan decomposition for their irreducible characters.
Moreover, we consider some families of groups of Lie type that do not admit a generic choice of a local subgroup.
We show that the inductive condition is satisfied for the prime \(\ell=3\) and the groups \(\text{PSL}_3(q)\) with \(q \equiv 4, 7 \mod 9\), \(\text{PSU}_3(q)\) with \(q \equiv 2, 5 \mod 9\), and \(G_2 (q)\) with \(q \equiv 2, 4, 5, 7 \mod 9\).
Further, we verify the inductive condition for the prime \(\ell=2\) and \(G_2(3^f)\) for \(f \geq 1\), \(^3 D_4(q)\), and \(^2E_6(q)\) where \(q\) is an odd prime power.