14 problems

James's conjecture. James's conjecture holds for all weight $5$ blocks of $\hhh$.

Geck's conjecture. The $\ell$-decomposition matrix of $G$ is unitriangular.

Let $G={\mathbf{G}}^F$ for ${\mathbf{G}}$ connected reductive defined over ${\mathbb{F}}_q$. Let $\ell$ be a prime such that Sylow $\ell$-subgroups of $G$ are abelian, and let…

Let $k$ be a field of characteristic $\ell$, let $q\in k$ have multiplicative order $e$, and let $\Lambda(n,r)$ label the standard and simple modules of the specialised $q$-Schur a…

Let $m$ be a positive integer, let $p$ be an odd prime, and let ${\rm DP}(m)$ denote the relevant set of strict partitions of $m$. Let ${\rm DPR}_p(m)$ denote the indexing set for…

Genericity conjecture for unipotent decomposition matrices. For $\ell$ sufficiently large, this square part depends only on $(W,\phi)$ and $d$; equivalently, it is independent of…

Isolated-block unitriangularity conjecture. The sum $B_t$ has an $\operatorname{Aut}(\mathbf{G}^F)_{B_t}$-stable unitriangular basic set.

Geck–Hiss unitriangularity conjecture. The restriction of the $ll$-decomposition matrix to the set of unipotent characters should have unitriangular shape.

Let $G$ be a finite reductive group with Weyl group $W$, defined over a field of characteristic $p$, and let $k$ be a field of characteristic $ell$ with $ell eq p$. Let $d$ denote…

Let $G$ be a finite group of Lie type, let $w$ be an element of its Weyl group, and let $Q_w$ be the virtual character afforded by the Alvis–Curtis dual of the intersection cohomol…

Let $G$ be the finite group of Lie type under consideration, with Weyl group ${\mathbf{W}}$, Frobenius-induced automorphism $\gamma$, and unipotent ordinary representations labelle…

Let $G$ be a finite reductive group, let $\ell$ be a good prime for $G$, and consider the results established in the surrounding discussion for the principal $\ell$-block, includin…

Let $W$ be a complex reflection group. Let $\theta:q\mapsto\xi$ be a specialization of ${\mathcal H}_q(W)$, and let…

Let $G=GL_n$, let ${\mathfrak P}_n$ be the set of partitions of $n$, and let $d_{\lambda,\mu}$ and $d^{S(n)}_{\lambda,\mu}$ denote respectively the decomposition matrices for perve…