159 problems

Let $G$ be a finite group, $p$ a prime, and $B$ a $p$-block of $G$. A $p$-weight of $B$ is a $p$-weight belonging to the block $B$, and the irreducible $p$-Brauer characters of $B$…

Let $G$ be a finite group, let $\b5$ be a field of characteristic $p>0$, let $V$ be a $\b5 G$-module, and let $P$ be a Sylow-$p$-subgroup of $G$. For each subgroup $H\leq G$, let…

Let $p$ be a prime, let $k$ be an algebraic closure of $\mathbb{F}_p$, let $\Gamma={\rm Gal}(k/\mathbb{F}_p)$, let $G$ be a finite group, and let $\mathcal{O}_G$ be its $p$-orbit c…

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

Let $G$ be a reductive algebraic group over a field $k$ of characteristic $p$, let $G_rT$ be the corresponding Frobenius kernel together with a maximal torus, and let $X_r$ denote…

Benson's conjecture. All indecomposable summands of $V\otimes V^*$ have dimension divisible by $2$, except for the single summand isomorphic to the trivial representation $\Bbbk$;…

Weak Donovan conjecture. There exists a constant $c(D)\in\mathbb N$ such that, if $G$ is a finite group and $B$ is a block of $kG$ with defect group isomorphic to $D$, then the ent…

Donovan's conjecture. Amongst all finite groups $G$ and blocks $B$ of $kG$ with defect group isomorphic to $D$, there are only finitely many Morita equivalence classes.

Let $p$ be a prime, let $k$ be an algebraic closure of $\mathbb{F}_p$, let $\Gamma$ be the Galois group of $k/\mathbb{F}_p$, and let $G$ be a finite group. Let…

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…

Projective-module Loewy-length conjecture. If $\chi$ has standard Levi form with $I=\Pi$ and $\lambda$ is $p$-regular, then

Bounded Morita Frobenius number conjecture. Amongst all finite groups $G$ and blocks $B$ of $kG$ with defect group isomorphic to $D$, the Morita Frobenius number…

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 $G$ be a group, let $\Bbbk$ be a field of characteristic $l$, and let $(C,\mathcal{E})\in\mathscr{I}(G)$ be a $0$-cuspidal pair. A prime $l$ is rather good for $G$ if it is goo…

Periodicity conjecture. Under this isomorphism,

Strict inequality conjecture. If $G$ is not $p$-central, then

Trivial-source conjecture. Indecomposable, self-dual $k\Sigma_d$-modules with Specht filtrations are trivial source modules. This is a proposed weaker replacement for the earlier c…

Let $R$ be a reduced, irreducible, finite root system, $X$ its weight lattice, and $k$ a field whose characteristic is bigger than the Coxeter number of $R$. Let $\mathfrak g$ be t…

Let $G$ be a finite group, let $D$ be an abelian Sylow $\ell$-subgroup, let $b$ and $c$ be the principal block idempotents of $kG$ and $kH$ with $H=N_G(D)$, and form the groups and…

Let ${\mathcal{O}}Gb$ be a block of a finite group $G$ with defect group $D\neq 1$. For a chain ${\mathcal{S}}$ of subgroups … let $l({\mathcal{S}})=n\geq 1$, let the sum run over…

Geck–Geck–Hiss conjecture. Assume that $cell$ is not “too small” and that the center of $G$ is connected. For every $\varphi\throw in \operatorname{UBr}_{cell}(G^F)$, there is a uni…

Let $k$ be the underlying field, let $\N\in\operatorname{mod} k\Sigma_d$ be indecomposable and self-dual, and suppose that $N$ has a Specht, and hence also a dual Specht, filtratio…

Height-three induction conjecture.

Restriction branching conjecture. Except for $A$, the only $\lambda^A$-normal node of residue $\alpha$ is $B$, and