Nonrealizability of certain representations in fusion systems

2022/09/30 by Bob Oliver, Oliver, Bob
Mathematics · #Advanced Topics in Algebra #Advanced Differential Equations and Dynamical Systems #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2209.15375

Abstract

For a finite abelian p-group A and a subgroup Γ\leAut(A), we say that the pair (Γ,A) is fusion realizable if there is a saturated fusion system F over a finite p-group S≥ A such that C_S(A)=A, \textrmAut_F(A)=Γ as subgroups of Aut(A), and A is not normal in F. In this paper, we develop tools to show that certain representations are not fusion realizable in this sense. For example, we show, for p=2 or 3 and Γ one of the Mathieu groups, that the only \mathbbF_pΓ-modules that are fusion realizable (up to extensions by trivial modules) are the Todd modules and in some cases their duals.

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