9 problems

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 $i,j$…

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

Cartan monotonicity conjecture. If

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…

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 $D$.

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 $ell\geq1$ 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…