12 problems

Finite-dimensional quotient conjecture. The module $\mathbb{M}(r+\kappa,r,[\iota;n],\zeta)$ has a finite-dimensional irreducible quotient $L$ if and only if this condition holds. I…

Let a good Verma module mean a Verma module with a weight space decomposition, and let an exceptional Verma module be a module of the form…

Mixed tensor product conjecture. There is an isomorphism of $U(\mathfrak{gl}(m+1|n))$-modules

Let $(W,S)$ be a Weyl group with Bruhat order. For $w\in W$, let $\mathbf{JM}(w)$ be the distinguished set of Bruhat-maximal join-irreducible elements whose join is $w$, and let…

Let $\chi$ be semisimple and lie in a Cartan subalgebra $\mathfrak{h}$, and let $T$ be the corresponding torus. The $T$-action gives graded module categories…

Dual-morphism conjecture. The linear map $\psi:M(W^*)\rightarrow M(V^*)$ associated to

Let $M(\boldsymbol a)$ be the Verma module, let ${}^{\mathbf R}M(\boldsymbol a)$ be its relevant $\mathbf R_T$-form, and let $\widehat W[\boldsymbol\lambda]$ be the operators index…

Let ${}_\mathbf A M(\boldsymbol a)$ be the $\mathbf A_T$-form of the Verma module generated by $|\boldsymbol a\rangle$, and let ${}_\mathbf A F(\boldsymbol a)$ be the $\mathbf A_T$…

Let $p\in\mathbb{N}$, and let $c,h,h_W,r$ be the parameters of the module $L^{\prime}(c,h,h_W)$. Assume … Reducibility criterion. The module $L^{\prime}(c,h,h_W)$ is reducible if a…

Let $V(c,\frac{1-r}{2},0)$ be the module in question, let $J^{\prime}(c,\frac{1-r}{2},0)$ denote the specified submodule, and let $u$ be a subsingular vector in…

Let $E(5,10)$ be the exceptional Lie superalgebra, and consider its degenerate (minimal) Verma modules. The source constructs morphisms of degrees $2$, $3$, and $5$ in addition to…

Let $L=E(5,10)$, let $\mathfrak g_0\simeq \mathfrak{sl}_5(\mathbb C)$ be its degree-zero component, and let $F(n_1,n_2,n_3,n_4)$ be the finite-dimensional irreducible…