15 problems

Let $\mathbb{G}=(W,\Lambda)$ be a $\mathbb{Z}_\ell$-reflection group with $(|W|,\ell)=1$, and let $T_a=\Lambda/\ell^a\Lambda$. For $b\ge a$, regard each…

Let $\mathbb{G}=(W\phi,\Lambda)$ be a simply connected $\mathbb{Z}_\ell$-spets and let $q\in\mathbb{Z}_\ell^\times$. Let $S$ be the Sylow $\ell$-subgroup of $(\mathbb{G},q)$, let…

Let $\mathbb{G}=(W\phi,\Lambda)$ be a simply connected $\mathbb{Z}_\ell$-spets and let $q\in\mathbb{Z}_\ell^\times$. Let $S$ be the Sylow $\ell$-subgroup of $(\mathbb{G},q)$, let…

Let $\mathbb{G}=(W\phi,\Lambda)$ be a simply connected $\mathbb{Z}_\ell$-spets and let $q\in\mathbb{Z}_\ell^\times$. Let $S$ be the Sylow $\ell$-subgroup of $(\mathbb{G},q)$, let…

Let $\mathbb{G}=(W,\Lambda)$ be a simply connected $\mathbb{Z}_\ell$-spets with $\ell>2$ and $(|W|,\ell)=1$. Let $T$ be the Sylow $\ell$-subgroup of $\mathbb{G}(q)$, let…

Let $\mathbb{G}=(W,\Lambda)$ be a simply connected $\mathbb{Z}_\ell$-spets with $\ell>2$ and $(|W|,\ell)=1$. Let $T=\Lambda/\ell^a\Lambda$ be the abelian Sylow $\ell$-subgroup of…

Let $\mathbb{G}=(W,\Lambda)$ be a simply connected $\mathbb{Z}_\ell$-spets such that $\ell>2$ is very good for $\mathbb{G}$ and coprime with $|W|$. Let $q\in1+\ell\mathbb{Z}_\ell$,…

Generalised Trinh–Xue conjecture. The intersection of the $\Phi$- and $\Phi'$-Harish-Chandra series

Let $\mathbb{G}=(W\varphi^{-1},L)$ be as in the principal-block dimension conjecture, and suppose that $W$ is an $\ell'$-group, $\varphi$ is trivial, and $q\equiv1\pmod\ell$. Let…

Principal-block dimension conjecture.

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 $B_0$…

Local-global character-counting conjecture. The number of characters in $\operatorname{Irr}(\operatorname{Sol}(q))$ of defect $d$ should be determined purely by the fusion system,…

Analogue of Robinson's ordinary weight conjecture. The number of characters of defect $d$ in $\operatorname{Irr}(\boldsymbol{X}(q))$ should be countable by local data from the fusi…

Geck–Malle integrality and positivity conjecture. The Geck–Malle version of the algorithm also satisfies these integrality and positivity properties.

Spets conjecture. These constructions actually describe the representation theory of some as-yet undescribed algebraic object called a spets.