102 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…

Let $\mathcal{O}$ be a complete local noetherian commutative ring with algebraically closed residue field of prime characteristic $p$. Let $G$ be a finite group with $p\mid |G|$, l…

Let $H$ be a finite group, let $B$ be an $ell$-block of $H$, and let $D$ be a defect group of $B$. The sectional $ell$-rank $s(D)$ is the maximum of the ranks of elementary abelian…

Let $G$ be a finite group and $B$ a block of $kG$ over a field $k$ of characteristic $p$, with defect group $D$. Benson's conjecture. The Loewy length of $B$ should satisfy … This…

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

Let $G$ be a finite group, let $p$ be a prime, and let $B$ be a $p$-block of $G$ with defect group $D$. Write $\operatorname{ht}(B)$ for the set of heights of the irreducible chara…

Let $G$ be a finite group, let $p$ be a prime, and let $B$ be a $p$-block of $G$ with defect group $D$. Write $\operatorname{ht}(B)$ for the set of heights of the irreducible chara…

Brauer height-set conjectures. The derived length $\operatorname{dl}(D)$ is bounded in terms of $e$; $b$ is bounded in terms of $e$; and $\operatorname{dl}(D)$ is bounded in terms…

Let $G$ be a finite group, let $p$ divide $|G|$, and let $\bar{b}$ be a block idempotent of $kG$ with non-trivial defect group. A source algebra of $kG\bar{b}$ is the interior $D$-…

Let $e$ be the quantum characteristic, let $\mathcal{S}_n$ be the level 2 cyclotomic $q$-Schur algebra, and let $B$ be a block of $\mathcal{S}_n$. Schurian-finiteness conjecture. F…

Let $e$ be the quantum characteristic, let $\hhh$ be an integral type B Hecke algebra, and let $B$ be a block of $\hhh$ with defect at least $2$. Schurian-finiteness conjecture. If…

Fix a prime $p$, a sufficiently large $p$-modular system $(\mathcal{K},\mathcal{O},F)$ with finite residue field, and a finite abelian group…

Let $B^{\eta,\rho,d}_{R_p}$ and $B^{\eta,\rho',d'}_{R_p}$ be arbitrary spin blocks of symmetric groups, or let $(B^{\eta,\rho,d}_{R_p})_{0}$ and $(B^{\eta,\rho',d'}_{R_p})_{0}$ be…

Let $\mathcal{O}$ be an arbitrary complete discrete valuation ring, let $G$ be a finite group, and let $b$ be a block idempotent of $G$ over $\mathcal{O}$ with an abelian defect gr…

Let $p$ be a prime and let $B$ be a $p$-block of a finite group. An irreducible character in $B$ is almost $p$-rational if, writing $c$ for its conductor, $p^2$ does not divide $c$…

Functorial Alperin weight conjecture. In $K_0(\mathcal{F}\mathcal{B}\ell_{\mathbb{F},k})$,

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 $q\in1+\ell\mathbb{Z}_\ell$,…

Let $G$ be a finite group, let $p$ be a prime, let $B$ be a $p$-block of $G$, and let $b$ be its Brauer correspondent. Write $\operatorname{Irr}_0(B)$ for the set of height-zero ir…

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

Trinh–Xue's conjecture. There exists a choice of complete HC-parametrisation $\Psi=\{\Psi_{{\mathbf L},\lambda}\}$ such that, for every such pair $({\mathbf L},\lambda)$ and…

Let $G\unlhd A$ be finite groups, let $B$ be a block of $G$ with defect group $D$, and let $b$ be its Brauer correspondent in a subgroup $M$ as below. Write $\operatorname{IBr}(B)$…

Let $p$ be a prime and $G$ a finite group. Let $B$ be a $p$-block of $G$ with defect group $D$, let $b$ be its Brauer correspondent in $\mathbf{N}_G(D)$, and let $\mathcal{H}_B$ be…

Let $p$ be a prime, $G$ a finite group, and let $B$ be a $p$-block of $G$ with defect group $D$. Let $u$ be the $p$-adic valuation, let $c(\rho)$ denote the conductor of the values…

Let $p$ be a prime, let $G$ be a finite group, and let $B$ be a $p$-block of $G$ with defect group $D$. For a finite $p$-group $D$, define … with value $\infty$ when $D$ is abelian…

Normal Sylow subgroup conjecture. $G$ has a normal Sylow $q$-subgroup if and only if, for any $p\neq q$, $q$ does not divide the degree of any $p$-height zero irreducible character…