Finite groups whose real irreducible representations have unique dimensions
Thomas Breuer, Frank Calegari, Silvio Dolfi, Gabriel Navarro, Pham Huu Tiep
TL;DR
This work classifies finite groups whose real irreducible representations have pairwise distinct degrees by introducing Hypothesis C, which ties equal-degree, real-valued characters with real realizability via the Frobenius–Schur indicator. The authors separate the problem into solvable and non-solvable cases, proving that solvable groups must be factor groups of $ (C_2\times C_2\times C_2)\rtimes (C_7\rtimes C_3)$ or $(C_p\times C_p)\rtimes SL_2(3)$ with $p\in\{3,5\}$, and that non-solvable groups are almost simple, lying in a finite list of groups including ${A}_8$, ${ m SL}_3(2)$, ${ m M}_{11}$, ${ m M}_{22}$, ${ m M}_{23}$, ${ m M}_{24}$, ${ m SU}_3(3)$, ${ m McL}$, ${ m Th}$, ${ m SL}_2(8).3$, and ${ m O}_8^+(2).3$. A highlighted exception is ${ m McL}$, which uniquely has two real irreducibles of the same degree with distinct FS-indicators. The paper develops a cohesive blend of Clifford theory, orbit-counting for regular orbits on modules, and extensive computational checks (GAP, table of marks) to rule out other configurations, culminating in a definitive classification of all such groups. This advances understanding of real representation degrees in finite groups and links deep structural properties with representation-theoretic constraints.
Abstract
We determine the finite groups whose real irreducible representations have different degrees.