The inductive McKay--Navarro condition for the Suzuki and Ree groups

TL;DR

This paper verifies the inductive McKay--Navarro condition for specific finite groups, including Suzuki, Ree, and some classical groups, establishing new Galois-equivariant Jordan decompositions for their characters.

Contribution

It provides the first verification of the inductive McKay--Navarro condition for Suzuki and Ree groups and constructs Galois-equivariant Jordan decompositions for their irreducible characters.

Findings

Verified the inductive McKay--Navarro condition for Suzuki and Ree groups.

Established Galois-equivariant Jordan decomposition for their characters.

Extended results to certain classical groups with specific parameters.

Abstract

We verify the inductive McKay--Navarro condition for the Suzuki and Ree groups for all primes as well as for $p=3$ and the groups $PSL_3(q)$ with $q \equiv 4,7$ mod $9$, $PSU_3(q) $ with $q \equiv 2,5$ mod $9$, and $G_2(q)$ with $q=2,4,5,7$ mod $9$. On the way, we show that there exists a Galois-equivariant Jordan decomposition for the irreducible characters of the Suzuki and Ree groups.