94 problems

Let $k_0,\ldots,k_n$ be non-negative integers, let $E_1$ be the set of parts, and let $\mathcal{Q}^{k_0,\ldots,k_n}_{\mathds{1}}$ be the partitions into parts in $E_1$ whose freque…

Let $k_0,\ldots,k_n$ be non-negative integers. Let $\mathbb{Z}_{\mathcal{S}}^+$ be the coloured-part set, let $\Omega$ be the set of fictitious parts, and let…

Let $B$ be a perfect crystal, let $\{w^{(k)}\}$ be a sequence of Weyl group elements, and let $\lambda=l\Lambda_0$ be the level-$l$ dominant integral weight for which the sequence…

Let ${\cal L}(\Lambda_n)$ be the level $1$ integrable module of $A^{(2)}_{2n}$, and let $Y(B_n)$ be the Yangian of type $B_n$. Suppose … where $BS$ is the set of border strips with…

Let ${\cal L}(\Lambda_k)$ be the level $1$ integrable module of $\widehat{sl}_n$, and let the character decomposition in equation (26a) be the decomposition obtained from the irred…

Let $n$ and $l$ be positive integers, and let $X^{(l)}{ u}(\rho)$, $X^{(l)'}{ u}(\rho)$, $F^{(l)}{ u}(q)$, and $F^{(l)'}{ u}(q)$ denote the restricted one-dimensional sums and fermio…

Let $B^{r,s}$ be a crystal of type $D_n^{(1)}$. Let $B(\Lambda)$ denote the classical $D_n$-crystal of highest weight $\Lambda$, and let the complement of a partition $\Lambda$ in…

Let ${\mathfrak g}={\mathfrak gl}_r$, let $\xi$ be a partition, let $W^2(\xi)$ be the two-dimensional Weyl module, and let $V_\xi$ be the…

The paper considers the affine Lie algebra $\hat{\mathfrak g}$, its central element $\mathbf k$, and truncated vertex operator algebras $\hat{\mathbf F}_{k+h^\vee}$ and…

Principal-subspace basis conjecture. The basis

Let $R$ be a dominant sequence of rectangles, let $\lambda$ be a partition, and let $\diamondsuit$ denote one of the four stable families. Write $R^t$ and $\lambda^t$ for the trans…

Let $R$ be a dominant sequence of rectangles and let $\lambda$ be a partition. For each of the four stable families indexed by…

Let $\fg$ be the Lie algebra under consideration, with index set $I$, and let $\fF^\lambda_{\fg,\fb}$ be the corresponding fiber space with open stratum…

Let $\fg$ be a Kac–Moody algebra, $\fb$ a Borel subgroup, $\bC$ the curve, and $\mu$ a dominant coweight. Let $\on{Maps}^\mu(\bC,\CG_{\fg,\fb})$ be the corresponding space of maps,…

Let $P^{(p,p')}$ be the parameter set used for the affine weights, and let $X_{\eta,\Lambda,\Lambda'}(q)$ be the corresponding configuration sum. Vanishing conjecture. For…

Let $P^{(p,p')}$ be the parameter set used for the affine weights, let $P_{+}^{r}$ denote the dominant weights at level $r$, and let $X_{\eta,\Lambda,\Lambda'}(q)$ be the configura…

Let $\ell\in\mathbb{Z}_{+}$, $k\in Q\cap P_{+}$, and $\eta\in\mathcal{P}$ satisfy $\eta_1\leq n-1$ and $|\eta|\equiv\ell\pmod n$. Let $\Lambda\in P_{+}^{N}$ satisfy…

Intersection-cohomology stalk conjecture. Its stalk at $(\phi,\underline{c},\underline{s})$ is isomorphic to

Let $\Sigma$ be the underlying compact Riemann surface of genus $g$, let $\mathfrak{g}$ be the finite-dimensional Lie algebra used to construct the affine Krichever–Novikov algebra…

Let $\mathfrak{g}$ be an (almost) simple finite-dimensional Lie superalgebra of positive defect, namely one of $\mathfrak{sl}(m,n)$ with $m,n\geq 1$, $\mathfrak{osp}(m,2n)$ with…

Let $\widetilde{\mathfrak{g}}$ be a twisted loop algebra with $\epsilon(\mathfrak{h})=0$. Let $\widehat{\rho}$ be the Weyl vector determined by the simple roots of…

Let $I'=I\setminus\{p\}$ index the irreducible components $Y_k$ of the exceptional variety $\mathfrak{L}=\mathfrak{L}_m$, and let $\widetilde E_k(m)$ be the characteristic function…

Affine theta-function expansion conjecture. The $A_{N-1}$ theta function in the blow-up formula satisfies

Rogers–Ramanujan type character identity. For $n\geq 1$ and $0\leq b\leq 2n$, the identity

Let $\mathbb{N}$ denote the nonnegative integers, and let $\mathcal{C}_0$ be the set of core abaci of affine type $C_3^{(1)}$ with charge $0$. The preceding proposition identifies…