22 problems

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…

Cartan monotonicity conjecture. If

For a Lie superalgebra with Cartan matrix generated by Chevalley generators, let non-Serre-type relations mean defining relations beyond the Serre-type relations. Non-Serre encodin…

Lie superalgebras with Cartan matrix have Chevalley generators and defining relations of Serre and non-Serre type. The non-Serre relations may be complicated to express in terms of…

Let $(V,c)$ be a braided vector space with $V=\mathbb C x_1\oplus\cdots\oplus\mathbb C x_{\theta}$ and … Assume that $q_{ii}=-1$ for every $i$ and $q_{ij}\in\{\pm1\}$ for all $i,j$…

Let $B$ be a block of the twisted group algebra $S(n)$ in the usual ungraded sense. According to the classification, associate to $B$ its $p$-bar weight $d$ and its type…

Let $\boldsymbol{\text{?}}$ denote the assertion about the $(q,\widetilde{q})$-deformed Cartan matrix associated with a $(01)$-sequence $\epsilon$ introduced immediately before thi…

Cartan nonsingularity conjecture for controlled stabilizers. If $C(\mathbb C M)$ is nonsingular, then $C(kM)$ is nonsingular for every field $k$ of characteristic $p$.

Characteristic-zero Cartan nonsingularity conjecture. If $C(\mathbb C M)$ is nonsingular, then $C(KM)$ is nonsingular for every field $K$.

Let $B$ be a block of a finite group $G$ over an algebraically closed field of characteristic $p>0$, let $C\in\mathbb{Z}^{l\times l}$ be its Cartan matrix, and let $D$ be a defect…

Let $B$ be a block of a finite group $G$ over an algebraically closed field of characteristic $p>0$, let $C\in\mathbb{Z}^{l\times l}$ be its Cartan matrix, and let $D$ be a defect…

Brauer's weak Donovan conjecture. There is a bound on the Cartan invariants of blocks of finite groups with defect group $D$ that depends only on $D$.

Brauer's weak Donovan conjecture. There is a bound on the Cartan invariants of blocks of finite groups with defect group $D$ that depends only on $D$.

Conjecture 1.

Let $p$ be a prime. For a partition $\boldsymbol{\mu}=(\mu^{(1)},\dots,\mu^{(p-1)})\in M_{p-1}(d)$, let $w_H(\mu^{(p-1)})$ be the corresponding weight, and let the block of $p$-wei…

Let $I$ be the maximal graded ideal with trivial zero component defining a twisted generalized Weyl algebra, and consider the family of locally finite TGWAs constructed from a symm…

Let $G$ be an acyclic graph, or more generally an acyclic orientation of a graph associated to a Cartan matrix, and let its frise be the collection of sequences of positive rationa…

Let $B$ be an $ell$-block of $S_n$ of weight $w$, and let $C_{ell}(B)$ be its Cartan matrix. For the graded invariant factors $\vartheta_{ell}(lambda)$ and the multiplicities…

Let $ell\geq1$ and $d\geq0$, and let $X_{ell,d}$ be the matrix arising in the computation of Cartan invariants. Write the prime decomposition as $ell=\prod_i p_i^{r_i}$ and define…

Hill's conjecture. The formula for the invariant factors of $X_{p^r,d}$ established when $r\leq p$ should hold for arbitrary $r$; equivalently, the invariant factors should be the…

Külshammer–Olsson–Robinson conjecture. The Cartan matrix $C_{ell}(n)$ is unimodularly equivalent to a diagonal matrix with entries $r_{ell}(mu)$ where $mu$ runs through the set of…

Let $A$ be an $n\times n$ Cartan matrix, and let $\mathfrak{g}(A)$ be a finite-dimensional simple Lie algebra or Lie superalgebra of the form associated with $A$. A dimension-growt…