159 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 $G$ 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

Induction branching conjecture. There are exactly two such conormal nodes, and

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…

Divided-differential-operator dimension conjecture. One has

Let $\mathbb{F}$ be a field of characteristic $p>2$, let $c\in\mathbb{F}$, and let $L_{\operatorname{Vir}}(c,0)_{\mathbb{F}}$ denote the simple quotient Virasoro vertex operator al…

Let $H^{(2^m;k)}$ denote the twisted Foulkes module for the symmetric group $S_{2m+k}$. Let $\tau$ be the $p$-content and $\theta$ the odd sequence of a partition of $2m+k$, with…

Let $p$ be a prime, let $k$ be an algebraic closure of $\mathbb{F}_p$, let $\Gamma={\rm Gal}(k/\mathbb{F}_p)$, and let $\mathcal{C}$ be a finite category. If $\mathcal{C}$ is a fin…

Let $p$ be a prime, let $k$ be an algebraic closure of $\mathbb{F}_p$, and let $\mathcal{C}$ be a finite category. Let $\operatorname{Aut}(\mathcal{C})$ denote its automorphism gro…

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 $p$ be a prime, let $k$ be an algebraic closure of $\mathbb{F}_p$, let $\Gamma={\rm Gal}(k/\mathbb{F}_p)$, and let $\mathcal{C}$ be a finite EI-category, meaning that every end…

Let $p$ be a prime, let $k$ be an algebraic closure of $\mathbb{F}_p$, let $\Gamma={\rm Gal}(k/\mathbb{F}_p)$, and let $\mathcal{C}$ be a finite category with $p$-orbit category…

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…

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 $D$-…