37 problems

Let $b$ be a block of $\mathcal{O}G$ with defect group $D$, let $A=l\mathcal{O}Gl$ be a source algebra, and let $\mathcal{F}$ be the saturated fusion system of $A$ on $D$. Write…

Aschbacher–Kessar–Oliver's conjecture. If a fusion system $\mathcal{F}$ on $S$ is exotic, then $\mathcal{F}$ is block-exotic.

Oliver's Schur multiplier conjecture. $\ker(\mu_{\mathcal L})$ is isomorphic to a subgroup of the Schur multiplier of $\mathcal F$.

Existence conjecture. There are $p'$-reduced locally finite $p$-artinian groups that are not locally finite, strongly $p$-artinian groups, and there are fusion systems that are LF-…

Let $\mathcal{F}$ be a saturated fusion system on a finite $p$-group $S$ of order $p^e$, and let $\alpha$ be a compatible family. Write $s$ for the sectional rank of $S$, and let…

Serwene's conjecture. Let $\mathcal{F}$ be a saturated fusion system. For any finite group $G$ having an $\mathcal{F}$-block $b$, it follows that $\mathcal{F}$ is realizable.

Let $S$ be a finite $p$-group, let $\mathcal{F}$ be a fusion system over $S$, and let $M=(M_{*},M^{*})$ be a Mackey functor over $\mathcal{F}$ on $\mathbb{F}_{p}$. Write…

Let $G$ be a finite group, $D$ a defect group, $B$ a block algebra, and ${\cal B}$ the corresponding block. Let ${\cal L}{\cal P}={\cal L}{\cal P}(B)$ be the pointed fusion system…

Let $G$ be a finite group, $D$ a defect group, $B$ a block algebra, and ${\cal B}$ the corresponding block. Write … for a finitely generated algebra $\Lambda$ over ${\cal O}$. The…

Sharpness problem. For every $p>0$ and $q\geq 0$,

Let $F$ be an algebraically closed field of characteristic $p>0$, let $G$ be a finite group, and let $B$ be a block of $FG$ with defect group $D$ and fusion system…

Let $F$ be an algebraically closed field of characteristic $p>0$, let $G$ be a finite group, and let $B$ be a block of $FG$ with fusion system $\mathcal{F}=\mathcal{F}_D(B)$. Write…

Let $F$ be an algebraically closed field of characteristic $p>0$, let $G$ be a finite group, and let $B$ be a block of $FG$ with defect group $D$. The fusion system…

Alperin's Weight Conjecture. If $\mathcal F$ is connected, meaning every element of $S$ is $\mathcal F$-conjugate to an element of $T$, then

Non-exoticity conjecture. The generalised block fusion system $\mathcal F_{(P,e_P)}(H,G,b)$ is non-exotic.

The two-element base conjecture for saturated fusion systems. Every saturated fusion system has a base of size $2$.

Let $X$ be a simply connected $\ell$-compact group with Weyl group $(W,L)$, with $\ell>2$ very good, and let $\tau$ be an automorphism of $X$ of finite order prime to $\ell$. Let…

Let ${\mathbb{G}}$ be a simply connected ${\mathbb{Z}}_\ell$-spets, let $\ell>2$ be very good for ${\mathbb{G}}$, and let $q$ be a power of a prime different from $\ell$. Let $B_0$…

Let $p$ be a prime, and let $\mathcal{S}$ be the collection of pairs $(G,S)$ such that $G$ is a simple Lie type group defined in characteristic $p$, $G$ is not isomorphic to…

Let $p\geqslant 5$, let $S$ be a $p$-group of order $p^6$, and let $\mathcal{F}$ be a saturated fusion system on $S$ such that $O_p(\mathcal{F})=1$ and…

Polynomiality conjecture. For each $d\geqslant 0$, the invariant $\mathbf{m}(\mathcal{F}(q),\alpha,d)$ is a rational polynomial in $p^l$.

Robinson's ordinary weight conjecture. One has

Let $p$ be a prime, let $P$ be a finite $p$-group, and let $\mathcal F$ be a saturated fusion system on $P$. The fusion system $\mathcal F$ is exotic if it is not of the form…

Let $S$ be a nontrivial finite $p$-group, let $\mathcal{F}$ be a saturated fusion system on $S$, and let $\alpha$ be an $\mathcal{F}$-compatible family. For each fully normalized n…

Let $S$ be a finite $p$-group of order $p^d$, let $\mathcal{F}$ be a saturated fusion system on $S$, and let $\alpha$ be an $\mathcal{F}$-compatible family. Let…