Reductions to simple fusion systems

We prove that if $\mathcal{E}\trianglelefteq\mathcal{F}$ are saturated fusion systems over $p$-groups $T\trianglelefteq S$, such that $C_S(\mathcal{E})\le T$, and either $Aut_{\mathcal{F}}(T)/Aut_{\mathcal{E}}(T)$ or $Out(\mathcal{E})$ is $p$-solvable, then $\mathcal{F}$ can be "reduced" to $\mathcal{E}$ by alternately taking normal subsystems of $p$-power index or of index prime to $p$. In particular, this is the case whenever $\mathcal{E}$ is simple and "tamely realized" by a known simple group. This answers a question posed by Michael Aschbacher, and is useful when analyzing involution centralizers in simple fusion systems, in connection with his program for reproving parts of the classification of finite simple groups by classifying certain 2-fusion systems.Comment: 13 page

POVMs and Naimark's theorem without sums

We provide a definition of POVM in terms of abstract tensor structure only. It is justified in two distinct manners. i. At this abstract level we are still able to prove Naimark's theorem, hence establishing a bijective correspondence between abstract POVMs and abstract projective measurements on an extended system, and this proof is moreover purely graphical. ii. Our definition coincides with the usual one for the particular case of the Hilbert space tensor product. We also point to a very useful normal form result for the classical object structure introduced in quant-ph/0608035

Fusion systems realizing certain Todd modules

We study a certain family of simple fusion systems over finite $3$-groups, ones that involve Todd modules of the Mathieu groups $2M_{12}$, $M_{11}$, and $A_6=O^2(M_{10})$ over $\mathbb{F}_3$, and show that they are all isomorphic to the $3$-fusion systems of almost simple groups. As one consequence, we give new $3$-local characterizations of Conway's sporadic simple groups

A Krull-Remak-Schmidt theorem for fusion systems

We prove that the factorization of a saturated fusion system over a discrete $p$-toral group as a product of indecomposable subsystems is unique up to normal automorphisms of the fusion system and permutations of the factors. In particular, if the fusion system has trivial center, or if its focal subgroup is the entire Sylow group, then this factorization is unique (up to the ordering of the factors). This result was motivated by questions about automorphism groups of products of fusion systems

Nonrealizability of certain representations in fusion systems

For a finite abelian $p$-group $A$ and a subgroup $\Gamma\le\text{Aut}(A)$, we say that the pair $(\Gamma,A)$ is fusion realizable if there is a saturated fusion system $\mathcal{F}$ over a finite $p$-group $S\ge A$ such that $C_S(A)=A$, $\textrm{Aut}_{\mathcal{F}}(A)=\Gamma$ as subgroups of $\text{Aut}(A)$, and $A$ is not normal in $\mathcal{F}$. 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 $\Gamma$ one of the Mathieu groups, that the only $\mathbb{F}_p\Gamma$-modules that are fusion realizable (up to extensions by trivial modules) are the Todd modules and in some cases their duals