76 problems

Kashiwara crystal conjecture. should have the structure of a Kashiwara crystal isomorphic to $B(\lambda)$, the crystal of the integrable highest weight module of the quantized enve…

Let $m\in\mathbb Z_{>0}$, let $\mathcal C\subseteq\mathsf{RPD}_{m,n}$ be a $\sqrt{\mathfrak{gl}_n}$-crystal, and let $w\in S_n$. Write … A polynomial is Lascoux positive if it is a…

Let $\mathfrak g$ be an affine Lie algebra, let $k\in I\setminus\{0\}$, and let $W(l\varpi_k)$ be the Kirillov–Reshetikhin module parametrized by $k$ and a positive integer $l$. Wr…

Let $\mathfrak{g}$ be a simple Lie algebra, let $\lambda$ be a minuscule dominant weight, and let $w_0^J$ be the corresponding Weyl group element. Write $C_{\mathfrak{g}}$ for the…

Symplectic Demazure crystal conjecture. The crystal $\mathrm{BRF}^{\mathsf{Sp}}_n(z,\phi)$ is a direct sum of Demazure $\mathfrak q_n$-crystals.

Let $\Phi$ be the bijection between rigged configurations and tensor products of the Kirillov–Reshetikhin crystal $B^{1,1}$ in the setting of affine crystals. General KR-crystal ex…

Let … be a tensor product of KR crystals, with virtualization map $v$ to the corresponding virtual crystal, and let $\Phi$ be the rigged-configuration bijection. Virtualization com…

Let $\mathfrak{g}$ be of type $D_4^{(3)}$, and let $\widehat{\mathfrak{g}}$ be the folded type $D_4^{(1)}$. Write $\widehat{B}^{r,s}$ for KR crystals of type $D_4^{(1)}$. KR virtua…

Let $\mathfrak{g}$ be of type $D_4^{(3)}$, and let $W^{r,s}$ be a Kirillov–Reshetikhin module. KR perfect-crystal conjecture. The module $W^{r,s}$ admits a crystal basis $B^{r,s}$,…

Let $\{B_l\}_{l\ge 1}$ be a coherent family of perfect crystals, let $B_\infty$ be its limit, and let $B_l^{\operatorname{min}}=\{b\in B_l\mid\langle c,\varepsilon(b)\rangle=l\}$.…

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 $\mathfrak{g}=C^{(1)}_n$, let $B$ be the perfect crystal considered for this case, and set $\lambda=l\Lambda_i$ with $i\neq 0,n$. For the associated sequence $\{w^{(k)}\}$ and…

Let $G=GL_n$, let $I$ specify a Levi subgroup, and let $K_{\lambda,\mu}^{GL_n,I}(q)$ be the corresponding Kostant-type $q$-analogue of the generalized Littlewood–Richardson coeffic…

Let $D_q(m,\lambda)$ be a quantum Demazure module whose crystal graph is a subgraph of the crystal graph of the corresponding irreducible $U_q(\widehat{\mathfrak{g}})$-representati…

Chain-independence conjecture. The colored directed graph defined by the action of root operators on the admissible subsets corresponding to any $\lambda$-chain does not depend on…

Right-splitting morphism conjecture. The stated right-splitting morphism and its extension after tensoring with any such $B$ exist with the asserted injectivity and preservation of…

Uniqueness of the star involution. There is a unique involution $*:B\rightarrow B$ satisfying these weight and crystal-operator relations.

Danilov–Koshevoy's conjecture. The Yang–Baxter equation holds for any crystals $A$, $B$, and $C$.

Let $U_{\lambda,\mu}(q)$ be the $q$-analogue of the multiplicity of the representation $V(\lambda)$ in $\frak{H}^{C_m}(\mu)$, and let $X_{\lambda,\mu}(q)$ be the corresponding crys…

Thermodynamic-limit independence conjecture. The final results—the energy per unit cell and the ground-state density of the crystal—should not depend on the chosen thermodynamic-li…

Crystal-lowering conjecture. The operation $$ can be completed to a lowering operation $f^1$ that picks out which Kronecker tableaux should be considered lowest weight.

Coxeter-element periodicity conjecture. Let $m\in\mathbb{Z}_{>0}$ satisfy

Minuscule or cominuscule periodicity conjecture. Let $t\in I$. Then $\PD(x_t)$ is finite if and only if $t\in I_{\mathrm{MC}}$. Moreover, if $t\in I_{\mathrm{MC}}$, then the follow…

Let $n$ and $k$ be as in the construction of the crystal embeddings, let $\mathcal{B}(k\theta)$ be the crystal of highest weight $k\theta$, and let $\psi_j^n$ denote the specified…

Let $B_{\mu}^{t}=B_{\mu}^{t}(\scalebox{0.4}{\ydiagram{2}})$ be the indicated tensor product of crystals, and let the augmentation operation be the operation discussed immediately b…