39 problems

Let $\lambda=l_1\omega_1+l_2\omega_2\in P^+$, and let $N^{i;\lambda}$, for $i\in\{1,2\}$, be the matrices of the cactus-group generators $\sigma^i$ with respect to the basis…

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…

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…

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…

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 $m\in\mathbb Z_{>0}$, 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 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…

Vexillary involution crystal conjecture. There is an isomorphism of $\mathfrak q^+_n$-crystals

Symplectic reduced-tableau conjecture. There is a $\mathfrak q_n$-crystal isomorphism

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

Let $\mathcal{B}$ be a crystal of type $C_3$, let $\mathcal{B}_+(\lambda)$ denote the relevant positive part, and let $\epsilon_i$ and $s_i$ be the crystal statistics and simple re…

Let $\lambda$ be a partition, and let $Skeleton(\lambda)$ be the skeleton whose vertices are standard tableaux of shape $\lambda$. For each positive integer $s$, let $H_s$ be the i…

Let $\lambda$ be a partition, and let $Skeleton(\lambda)$ be the graph obtained from the crystal of tableaux $B(\lambda)$ by replacing each quasicrystal with its associated standar…

Let $v=v_1\otimes v_2\otimes\cdots\otimes v_{\mathfrak{d}}\in(B^{r,1})^{\otimes\mathfrak{d}}$ be a soliton. For each factor $v_j$, let $k_j$ be the value of $k$ assigned by the pre…

Let $\lambda$ be a partition, let $\mu\subseteq\lambda$, and let $\mathcal{H}_{\lambda/\mu}$ be the set of recording tableaux arising from the uncrowding map on edge labeled tablea…

Type $D_m$ cactus-generator conjecture. The action of the cactus group on $B(n\omega_1)$ is generated by the actions of cactus generators corresponding to one-element and two-eleme…

Let $X$ be a smooth space in the setting of the paper, and consider the projective maps … Let $M$ be a corresponding $Z$-projective differential crystal spectrum. Logarithmic cryst…

Let $\overline{M_{0,n+2}^{comp}}$ be the compact real moduli space, and let the eigenlines of Bethe subalgebras on a tensor product of Kirillov–Reshetikhin representations form the…

Chen–Haiman's conjecture. The Catalan functions $H(\Psi;\mu)$ are Schur positive:

Let $B^{1,s}$ be the Kirillov–Reshetikhin crystal of type $E_7^{(1)}$, and let $\eta_i$ denote the fundamental weights of $A_7$. For $a,b,c,d\in\mathbb{Z}_{\geq 0}$ satisfying…

Critical-resonance conjecture. If

Soliton characterization conjecture. The state $p$ corresponds to a soliton of length $\ell$ if and only if

Permutation-equivariance conjecture. The diagram relating $\operatorname{RC}(B)$ to $\operatorname{RC}(B')$ by the identity and $B$ to $B'$ by $R$, with vertical maps given by…