23 problems

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 $\mathcal{O}^{n,n}$ 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…

Assume $n\geqslant m$, with $\kappa_c$ the critical level, and let $\mathfrak{z}$ denote the commutative center. Let $\mathcal{A}^m_{\mathrm{deg}}$ be the degenerate $\beta\gamma$-…

Pfaffian generator conjecture. A Pfaffian generator exists if and only if either ${\mathscr{Y}}_{N,\ell}^{+}(\sigma)$ has $N$ even and $\ell$ odd, or…

Center conjecture. The assertion that the centers of ${\mathscr{Y}}_{N,\ell}^{\pm}(\sigma)$ are polynomial algebras in distinguished generators should hold without the stated assum…

Let $\mathfrak{g}=\mathfrak{gl}(m\cdot\1+nP)$, or let $\mathfrak{g}$ be any restricted subalgebra of it, and let $X\in\mathfrak{g}$ with $X\notin\ker d$. Write $X^{[2]}$ for the re…

Let $C_k$ be the central elements of $\mathrm{U}_q(\mathfrak{gl}_{m,n})$ defined for $k\in\mathbb{Z}_+$ by the quantum trace construction in Theorem, and regard…

Chavli–Pfeiffer's integral basis conjecture. There exist an $R$-basis $\{\mathbf{b}_w\mid w\in W\}$ of $\mathscr{H}(W)$ and conjugacy-class representatives…

Center conjecture. The center of $\mathscr{R}_{\alpha,K}^{\Lambda}$ consists precisely of the symmetric elements in $\mathscr{R}_{\alpha,K}^{\Lambda}$.

McNamara's center conjecture. The center $Z(\mathscr{H}_{n,K}^\Lambda)$ is precisely the set of symmetric polynomials in $\mathcal{L}_1,\ldots,\mathcal{L}_n$.

Let $\mathfrak{g}$ be the Lie superalgebra and $e$ the nilpotent element defining the finite $W$-superalgebra $U(\mathfrak{g},e)$. Let $Z(U(\mathfrak{g}))$ and…

Polynomial class-sum basis conjecture. There exists a choice of a basis $\{b_w:w\in W\}$ of $H$ and a choice of conjugacy class representatives $\{w_C:C\in\operatorname{Cl}(W)\}$ s…

Let $f\in \operatorname{V}_{n,d}$ be a nondegenerate form, and let $Z(f)$ denote its center. O'Ryan–Shapiro dimension conjecture. It was conjectured that … The conjecture concerns…

Godelle–Paris conjecture. Every irreducible non-spherical Artin group has trivial center:

Let $q$ be a root of unity, and let ${\mathsf u}_q(\mathfrak{sl}_n)$ and ${\mathsf U}_q(\mathfrak{sl}_n)$ denote the small and big quantum groups, respectively. Their centers are d…

Let ${\rm U}_q(\mathfrak{g})$ be a quantum group, let $V(\lambda)$ be a finite-dimensional simple ${\rm U}_q(\mathfrak{g})$-module, and suppose that tensor powers of $V(\lambda)$ s…

Let $J$ be the two-sided ideal of $\mathcal A_q$ generated by $\mathcal W_0$ and $\mathcal W_1$, and let $Z(\mathcal A_q)$ denote the center. Ideal–center direct-sum conjecture. Th…

Let $\mathcal A_q$ be the current algebra, let $Z(\mathcal A_q)$ denote its center, let $\Delta_{k+1}\in Z(\mathcal A_q)$, and let $\langle\mathcal W_0,\mathcal W_1\rangle$ be the…

Let $A$ be a symmetric generalized Cartan matrix, and let $cGamma$ be its associated quiver with no loops. Let $cmathscr{R}_{cbeta}(cGamma)$ and…

For $k>0$, let $Y_3^+(k)$ be the $k$-shifted twisted Yangian for $\mathfrak{so}_3(\mathbb C)$, namely the subalgebra of $Y_3^+$ generated by … The preceding corollary gives…

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 $B_{r,s}(\delta)$.

Conjecture B. If $\Gamma$ is non-spherical and connected, then