63 problems

For $n\geq 0$, let $S_n$ be the symmetric group, let $\mathrm{Inv}_n$ be its set of involutions, and let $\mathbf{K}_n(w)=-$ mean that Kostant's problem for the simple highest weig…

Let $\mathfrak{g}$ be an arbitrary quasi-reductive Lie superalgebra and let $\mathfrak p$ be a parabolic subalgebra of $\mathfrak{g}$. Write $\mathcal{PI}^{\mathfrak p}$ for the fu…

Let $R$ be the deformation ring, let $\overline R$ be the corresponding deformation ring after replacing $e$ by $e+1$, and let $O^{\nu,\Delta}_{\mu,R}$ and…

Let $\lambda$ be an integral parameter and let $\zeta$ be a generic cocharacter. Define the long twist and long shuffle by … These are endofunctors of the relevant bounded derived…

Let $\mathcal{D}$ be an integral quantization of a conical symplectic resolution, let $\mathcal{I}$ index the simple objects of its category $\mathcal{O}$, and define two indices t…

Let $A$ be the algebra governing the principal block of category $\mathcal{O}$, let $\Delta$ be the direct sum of the relevant Verma modules, and let $\mathtt{E}$ be the graded ext…

Let $\mathfrak{g}$ be a semi-simple complex finite-dimensional Lie algebra, let $W$ be its Weyl group with longest element $w_0$, and let $\mathfrak{t}(w)$ and $\mathfrak{i}(w)$ de…

Let $A_{\kappa}$ be a finite-dimensional algebra with $A_{\kappa}\text{-mod}\cong\mathcal{O}^{\kappa}$, and let $e\in A_{\kappa}$ be the maximal idempotent such that the left…

Let $\kappa=(k_1,\dots,k_m)$ be a decomposition of $n$, let $\mathcal{O}^{\kappa}$ be the corresponding parabolic highest weight category for $\mathfrak{sl}_n$, and let…

Let $\mathcal{O}^{n,n}$ be the parabolic highest weight category for $\mathfrak{sl}_{2n}$, and let $A_{n,n}$ be its finite-dimensional algebra model. Let $e\in A_{n,n}$ be the idem…

Let $(i,a)\in I\times_2\mathbb Z$ and $w\in W$, and let $Q_{w\varpi_i,a}$ be the corresponding formal solution of the extended $QQ$-system. Let $\chi_{\ell}$ denote the $\ell$-char…

Let $(i,a)\in I\times_2\mathbb Z$, let $w\in W$, and let $Q_{w\varpi_i^{\vee},a}$ be the corresponding $Q$-variable. Let $\chi_{\ell}$ denote the $\ell$-character and let…

Let $\mathcal{O}_{sh}^{\mathbb Z}$ be the integral finite-length subcategory of shifted category $\mathcal{O}_{sh}$, and let $\mathscr A$ be the cluster algebra associated with the…

Let $V_1,V_2,V_3$ be objects of the shifted category $\mathcal{O}_{sh}$. Associativity conjecture. There exists an isomorphism … Thus, $k$-fold tensor products in…

Fix $J\subseteq I$, and let $V_J$ be an object of $\mathcal{O}_{\nu}^{\lambda}(\mathbf R,\mathfrak g_J)$ for some $\lambda$, $\nu$, and $\mathbf R$. An inflation of $V_J$ is a cons…

Let $\mathfrak{r}$ be the group equipped with the involution $d$. The group-automorphism conjecture. The involution $d$ of $\mathfrak{r}$ is a group automorphism. This is proposed…

Assume that ${\mathfrak{g}}$ is of finite type. Let $W$ be the Weyl group, let $i\in I$, let $a\in\mathbb{C}^*$, and let $\boldsymbol{\psi}_{w(\omega_i^\vee),a}$ be the correspondi…

Assume that ${\mathfrak{g}}$ is of finite type. Let $I$ index the simple roots, let $W$ be the Weyl group, and let $\mathcal{Q}_{w(\omega_i^\vee),a}$ and…

Assume that ${\mathfrak{g}}$ is simply laced. Let $\mathcal{A}$ be the infinite-rank cluster algebra embedded in $K_0(\mathcal{O}^{\mathrm{sh}})$ by … and let $I^{-1}\circ i$ be th…

Let $\mathrm{Inv}_n$ be the set of involutions in $\mathrm S_n$, let $L_z^{(n)}$ be the simple highest weight module indexed by $z$, let $\theta_x^{(n)}$ be the projective functor…

Let $\theta_x^{(n)}$ be the projective functor indexed by $x\in\rm S_n$, and let $L_z^{(n)}$ be the simple highest weight module indexed by $z\in\rm S_n$ in the principal block of…

Let $\scr{X}_n$ be the parameter space with strata $W_I$, and let $W_I^{\natural}\subset W_I$ consist of points $(\bt,\bs)$ whose coordinates $t_i$ are not integers. A point of…

Let $\Phi^{-,\mathrm{re}}_\mu$ be the relevant set of negative real roots, let $\langle\alpha,\mu\rangle$ denote the root–coweight pairing, and let $L$ be the unique irreducible ob…

Let $G$ be a semisimple group with Langlands dual group $G^\vee$, let $\mathfrak{u}_q$ be the small quantum group of $G^\vee$, and let…

Let $\mathfrak{g}$ be the Lie-theoretic datum of the paper, let $\mathcal O_{\mathrm{sh}}$ be the shifted category O, and let $R=\bigoplus_{\nu\in Q_+}R_\nu$ be the associated KLR…