19 problems

Let $\mathfrak{g}$ be a simple Lie algebra, let $V(\lambda_1),\ldots,V(\lambda_n)$ be irreducible representations, write…

Let $k\in\mathbb{N}$ and let $c_1,\dots,c_k$ be arbitrary constants. For each $i$, let $\{\vec{J}_i\}$ denote the set of vectors satisfying … For $\vec{J}_i$, write…

Let $r=d_0-d_1$ and use the polynomials $W_r,U_r$ and the Hermite-Wronskian polynomial $V_\mu$ defined in the source. Unequal-degree large-$l$ conjecture. For generic $l,m$, the Ga…

Let $W_r(x)$ and $U_r(x)$ be the recursively defined polynomials associated with the unique exponential-Gaudin solutions indexed by $r\in\mathbb Z$. Polynomial-root conjecture. For…

Let the exponential Gaudin Bethe Ansatz system be as in the source, and let $d_0=r^2$, $d_1=r(r-1)$ characterize the Virasoro highest-weight locus. Exponential Gaudin nonexistence…

Let $d_0=d_1$ and let $\operatorname{Part}(d_0)$ denote the number of partitions of $d_0$. Large-$l$ Gaudin counting conjecture. For generic $l,m$, the Gaudin Bethe Ansatz system h…

Let $L_0$ be the displayed BLZ-type oper, with distinct points $t_j\notin\{0,1\}$, satisfying the condition $b_0+b_1+\sum_iq_i=0$ and having monodromy-free singularities at $x=1$ a…

Let $L_1$ be the displayed scalar oper with distinct nonzero points $s_i$, integer $r$, and parameters $k,l,n_1$ as in the source. Gaudin no-monodromy conjecture. For generic $k,l$…

Completeness conjecture. The operator $\mathscr H(x)$ has simple spectrum on $(L_{\bm\Lambda}(\bm b))^{\mathrm{sing}}$. There is a bijection between the monic divisors $y$ of…

Let $\mathfrak{g}$ be a Lie algebra, let $\mu\in\mathfrak{g}^*\cong\mathfrak{g}$, and choose $\mu\in\mathfrak b_- = \mathfrak n_-\oplus\mathfrak h$ after conjugation. Let…

Let $\mathfrak{g}$ be a Lie algebra with Cartan subalgebra $\mathfrak{h}$ and Cartan subgroup $H$, let $M$ be the finite-dimensional representation under consideration, and let…

Let $\mathfrak{g}$ be a Lie algebra, let $M$ be the finite-dimensional representation under consideration, let $M^{\mathrm{sing}}$ be its singular subspace, and let…

Let $\boldsymbol\lambda$ be a sequence of arbitrary $\mathfrak{gl}_{1|1}$ weights, and let $L(\boldsymbol\lambda)^{{\boldsymbol s}ing}$ be its singular subspace. Let…

Consider a tensor product $L(\boldsymbol\lambda)$ of typical polynomial $\mathfrak{gl}_{1|1}$ modules, with singular subspace $L(\boldsymbol\lambda)^{{\boldsymbol s}ing}$, and let…

Let $\boldsymbol\lambda=(\lambda^{(1)},\dots,\lambda^{(n)})$ be a typical sequence of polynomial $\mathfrak{gl}_{M|N}$ weights, let $\boldsymbol z=(z_1,\dots,z_n)$ be a sequence of…

Let $\fg=\mathfrak{gl}_N$, let the $V_{\l_i}$ be symmetric powers of the standard representation, and let $PJ_n$ act on the associated Gelfand–Tsetlin polytope through Bethe-vector…

Let $\overline{M_{0,n+1}}(\mathbb{R})$ carry Speyer's covering and the covering constructed in the source, denoted the source's covering from Corollary. Speyer covering conjecture.…

Let $M_{\l,\mu,\nu}$ be the set of monodromy-free ${}^L G$-opers on $\mathbb{C}\mathbb{P}^1$ with regular singularities of residues $\l,\mu,\nu$ at $0,1,\infty$, respectively. Defi…

Let $\fg$ be the Lie algebra in the Gaudin model, and let the theorem referred to in the source as Theorem assert the cyclic monodromy property for the associated Bethe vectors. Cy…