280 problems

Let $p$ be a prime, let $(\mathcal{K},\mathcal{O},k)$ be a complete $p$-modular system with algebraically closed residue field $k$ of characteristic $p$, and let $G$ be a finite gr…

Let $F$ be a field of characteristic $p$, and let two Specht modules for a symmetric group correspond to partitions. The $p$-core of a partition is obtained by removing rim $p$-hoo…

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

Let $e$ be a prime, let $n$ be a positive integer, and let $\lambda$ be an $e$-regular partition of $n$. Over a field of characteristic $e$, write $D^{\lambda}$ for the irreducible…

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…

Assume that $k$ is an algebraically closed field, and let $P$ be a fixed finite $p$-group. A source algebra of a block is regarded as an interior $P$-algebra when the block has def…

Malle–Robinson conjecture. One has

Blockwise Alperin weight conjecture. For every $p$-block $B$ of every finite group,

Let $B^{\rho,d}$ be a RoCK block of a symmetric group, and let $T^{\mathsf{Br}(p)}(d,d)$ be Turner's algebra, which contains an idempotent corner isomorphic to…

Broué's conjecture. Any $ell$-block of a finite reductive group is Morita equivalent to a unipotent $ell$-block of a possibly disconnected group, and in this case the decomposition…

Let $p$ be a prime, let $G$ be a finite $p$-solvable group, and let $\varphi\in\operatorname{IBr}_p(G)$ be an irreducible Brauer character with vertex $Q$. Let $L_\varphi$ denote t…

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 $G$ and $H$ be finite groups with a common abelian Sylow $p$-subgroup $D$ and common $p$-local structure. For every $p$-subgroup $Q$ of $D$, consider the principal blocks of…

Let $p$ be a prime and let $G$ be a finite $p$-group. The Loewy length $\mathsf{L}(G)$ is the nilpotency index of the Jacobson radical of the group algebra $\mathbb{F}_p[G]$, and…

Let $\mathcal{B}^{\rho,d}$ and $\mathcal{B}^{\rho',d'}$ be spin blocks of symmetric groups, and let $\mathcal{B}^{\rho,d}_{0}$ and $\mathcal{B}^{\rho',d'}_{0}$ be the corresponding…

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 $\mathfrak{S}_n$ be the symmetric group, let $\mathbb{F}$ be an algebraically closed field of characteristic $p$, and let $\lambda\vdash n$. For $m\in\mathbb{N}$ with $m\leq n$…

Let $p$ be a prime, let $S_n$ be a symmetric group, and let $B$ be a block of $\mathbb{Z}_pS_n$ of weight $w$ with $w<p$. Martin's conjecture. All projective indecomposable $B$-mod…

Let $G$ be a finite group, let $p$ be a prime, and let $B$ be a Brauer $p$-block of $G$ with defect group $D$. Write $D'=[D,D]$ for the derived subgroup of $D$. Let $k(B)$ be the n…

Hiss's conjecture. Every finite group has a $p$-basic set.

James–Mathas conjecture. The Specht module $S^{\lambda}$ remains irreducible over $F$ if and only if $\lambda$ is a $p$-JM partition.

Broué's perfect-isometry conjecture. There should be a perfect isometry between $K_0(B)$ and $K_0(b)$.

Let $\lambda=(\lambda_1,\ldots,\lambda_r)$ be a partition. Bessenrodt–Morris–Olsson's conjecture. For $p=5$, the labels for irreducible modular spin representations are the partiti…

Let $m$ be a degree, let $n$ be a positive integer, and let $H_m$ be the type $A$ Hecke algebra specialized at an $n$th root of unity. Consider the degree $m$ part of the transitio…