66 problems

Kirillov–Reshetikhin crystal-base conjecture. For some $\zeta^{(i)}_s\in\mathbb{C}(q)^\times$, the module $W^{(i)}_s(\zeta^{(i)}_s)$ has a crystal base $\mathcal{B}^{i,s}$ that is…

Let $B^{2,s}$ be a crystal of type $D_n^{(1)}$, and let $B(k\varpi_2)$ denote the classical crystal of highest weight $k\varpi_2$. Let $D_{B^{2,s}}$ be an energy function on…

Let $\mathcal M(\mu,\lambda)$ be the Coulomb branch and let $\mathcal A$ be its attracting set. Consider the union, over the relevant weights $\mu$, of the sets of irreducible comp…

Anderson–Mirković conjecture. For every MV polytope $P$ and every $j\in I$, $AM_j(P)$ is an MV polytope and

Let $\mathfrak{g}$ be of type $D_{n+1}^{(1)}$, and let $\widetilde{B}^{n,s}$ be the affine crystal constructed by the stated classical decomposition and affine operators. Spin-crys…

HKOTT's perfectness conjecture. The crystal $B^{r,s}$ is perfect if and only if $\frac{s}{c_r}$ is an integer. If $B^{r,s}$ is perfect, its level is $\frac{s}{c_r}$.

Kashiwara's conjecture. Every good finite-dimensional affine crystal, that is, every good crystal arising from a $U_q(\widehat{sl_n})$-module, is a tensor product of Kirillov–Reshe…

Let $L_\theta(\lambda)$ be the crystal lattice, let $c(L_\theta(\lambda))$ denote its bar-conjugate lattice, and let $V_\theta(\lambda)^{\mathrm{up}}_{\mathbf A}$ be the dual latti…

Let $V_\theta(\lambda)$ be the symmetric-crystal representation for a dominant integral weight $\lambda$ fixed by $\theta$, and let $L_\theta(\lambda)$ be its crystal lattice with…

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

Let $B_\natural$ be the limiting crystal obtained from the crystals $B_l$, and let $g_l:B_l\to B_\natural$ be its set-theoretic embedding. For fixed $u\in B_\natural$ and…

Admissibility decomposition conjecture. For any $\xi\in(P_{cl}^+)_l$, there exists a disjoint union decomposition, not necessarily unique,

Level-restricted X=M conjecture. One has

Inhomogeneous X=M conjecture. There exists $b_0\in B_0$ with $\varphi(b_0)=k\Lambda_0$ such that

X=M conjecture. One has

Conjectural Kirillov–Reshetikhin module family. For every $(r,s)$, there exists an irreducible finite-dimensional $U'_q(X_n^{(1)})$-module $W^{(r)}_s$ such that:

Let $I=\{p^n\mathbin{;}n\in\mathbb Z_{\mathrm{odd}}\}$, with neither $1$ nor $-1$ in $I$, and let $\theta(a)=a^{-1}$. Let $K_{D,I}$ be the Grothendieck group of finite-dimensional…

Assume that no $i\in I$ satisfies $\theta(i)=i$, and use the lattice $L_\theta$, its bar-conjugate $L_\theta^-$, and the integral module $(V_\theta)_\mathbf A$ defined from the bar…

Let $I$ be the parameter set with involution $\theta$, and assume that no $i\in I$ satisfies $\theta(i)=i$. Let $V_\theta$ be the type-D module constructed in the source, with oper…

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…

Type D affine crystal conjecture. If $B^{r,s}$ exists, then

Let $D$ be a $\pm$ diagram of shape $\Lambda/\lambda$ whose ambient $r\times s$ rectangle has complement of $\Lambda$ tiled by vertical dominos. Let $m_{i}^{\circ}$, $m_{i}^{+}$,…

Branching component graph product conjecture. The branching component graph satisfies

Let $I=\{p_1^n;n\in\mathbb Z_{\mathrm{odd}}\}$, assume that none of $\pm1$ and $\pm p_0$ lies in $I$, and let $\theta(a)=a^{-1}$. Let $\mathfrak g_I$ be the associated Lie algebra…

Promotion conjecture. Theorem 4.11 remains true for any tensor product