12 problems

Morton's conjecture. These central elements generate the entire center $Z(B_{r,s})$.

Jung and Kim's conjecture. For every $\delta \in \mathbb{C}$, the center of the walled Brauer algebra $B_{r,s}(\delta)$ is generated by the supersymmetric polynomials in the Jucys–…

Let $\mathsf{qw}\kern-1.8pt\mathscr{B}_{m,n}$ be the quantum walled Brauer algebra with $\theta=\frac{\delta}{\gamma}$, let $\mathscr{C}\mathrm{r}_{m,n}$ denote the set of cross bi…

Let $\mathsf{qw}\kern-1.8pt\mathscr{B}_{m,n}$ have special parameter $\theta=-(-\frac{\delta}{\gamma})^{M-N}$, let $\Lambda_{m,n}$ be the set of bipartitions, and let an $(M,N)$-cr…

Let $\frac{\theta}{\text?}$ be the parameter ratio $\frac{\theta}{\text?}$—more precisely, let $\frac{\theta}{\text?}$ denote the generic parameter $\frac{\theta}{\text?}$ of the q…

Decomposition-number conjecture. The decomposition numbers of cyclotomic walled Brauer algebras over $\mathbb C$ can be determined by those in Theorem 7.19 together with some resul…

Sartori–Stroppel's conjecture. For every $\delta\in\mathbb C$, the center of $B_{r,s}(\delta)$ is generated by the supersymmetric polynomials in $L_1,\ldots,L_{r+s}$.

Brundan–Stroppel's conjecture. The symmetric polynomials in $x_1^R,\ldots,x_{r+s}^R$ generate the center of $B_{r,s}(\delta)$.

Generation conjecture. The center of the walled Brauer category ${\mathrm{Br}}_{r,t}(\delta)$ is the subalgebra generated by these elements.

The symmetric-polynomial conjecture. Every element in the center of the walled Brauer category can be expressed as a symmetric polynomial in the Jucys–Murphy elements.

Let $R\in\mathscr{R}_{r,s}$, and let $\Theta_R^{(k)}$ be the homomorphisms constructed using choices of isomorphisms in the definition of $\Theta_R^{(k)}$. Stabilisation conjecture…

Let $\delta\in\mathbb{Z}$, let $\underline{\operatorname{Re}}\!\operatorname{p}(GL_\delta)$ denote Deligne's tensor category, and let $K(\delta)$ be the algebra whose category of l…