17 problems

Hanlon–Wales semisimplicity conjecture. The algebra $\mathfrak{B}_n(\delta)$ is always semisimple if $\delta \in \mathbb{C}$ is not an integer.

Let $\mathcal B_f^{(x)}$ be a Brauer algebra with parameter $x$, and consider the generic indecomposable $\mathcal B_f^{(x)}$-modules and the standard filtration of the algebra. Th…

Brauer specialization conjecture. There is a subalgebra $B^{(0)}$ of $B$ defined over $E$, containing a spanning set of $B$, such that after reduction modulo $(l-1)$ one obtains a…

BMW deformation conjecture. A similar deformation exists for the spherical simply laced types, with the pictures forming the monomial basis of the Brauer algebra indexed by a combi…

Tangle algebra conjecture. There is a tangle algebra $\widetilde{\mathcal P}$ with a basis labelled by set partitions of $\{1,2,\ldots,2r\}$ that specializes to the partition algeb…

Hu's generalized surjectivity conjecture. The natural map $\psi'_{f,K}$ is surjective. The paper states that this conjecture implies the preceding algebraically closed-field conjec…

Hu's surjectivity conjecture. The map $\psi_{f,K}$ is surjective. This conjecture asks for the symplectic double-centralizer property on the quotient by the $f$-th Brauer ideal; it…

Brauer full-commutant conjecture. The full commutant of the action of ${\boldsymbol U}_{\star}'$ on

Brauer-projector decomposition conjecture. The tensor power

Brauer-centralizer conjecture. The action of ${\boldsymbol U}_{\star}$ on

Let $W$ be a 2-reflection group, and let $\mathfrak{Q}(W)$ be the algebra generated by $t_H$ for $H\in\mathcal{A}$ and by $W$, subject to the relations described above. Finite-dime…

Lehrer–Zhang's kernel threshold conjecture. The kernel is nonzero if and only if $r\geq r_c$:

Let $R$ be a field, let $n$ be a positive integer, and let $W({\rm I}_2^n)$ be the corresponding dihedral group. The characteristic of $R$ is assumed not to divide $n$. Cellularity…

Let $m<n$, let $K$ be a field with $\operatorname{char} K\neq 2$, and let $\mathfrak{B}_n(m)$ be the Brauer algebra with representation map $\varphi$. Define $E_i$ as in the preced…

Let $f$ be an integer with $1\leq f\leq [n/2]$, let $K$ be an infinite field, let $V$ be the natural symplectic $K$-vector space, and let $\mathfrak{B}_n^{(f)}$ denote the relevant…

Semisimplicity conjecture. (i) If $n=2k+1$, then $B_n(\hat{z})$ is semisimple only if $\kappa\mathfrak{S}_n$ is semisimple and $p_{2k-1}(\hat{z})\ne 0$. (ii) If $n=2k$, then…

Non-degeneracy conjecture. (i) If $n=2k+1$, then the bilinear form on the $B_n(\hat{z})$-module $S^{(1)}$ determined by the referenced construction is non-degenerate if and only if…