66 problems

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

Twisted Kirillov–Reshetikhin crystal conjecture. The perfect $U_q^{\prime}(\widehat{\mathfrak{g}})$-crystal $\widehat{\mathcal{B}}^{i,s}$ is isomorphic to the conjectural crystal b…

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…

Type-C identities.

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 $\Lambda_k^A$ and $\Lambda_{n-k}^A$ be the $k$-th and $(n-k)$-th fundamental weights of $U_q(\mathfrak{sl}_n)$, and set … Let $K_{\Lambda_{2k},0}(q)$ be the type $C_n$ Kostka–F…

Let $\beta$ be a dominant weight for the root system $C_n$, let $B(\beta)$ be the corresponding crystal, and let … be its subset of tableaux of weight $\mu$. Let…

Let $(V,\widehat{V})$ and $(V',\widehat{V}')$ be simple aligned virtual crystals, where $V$ and $V'$ are $X$-crystals and the hatted crystals are $Y$-crystals. Let … be the type…

Let ${\mathcal M}$ be the set of monomials in variables $Y_i(n)$, with $i\in I$ and $n\in\mathbb{Z}$, equipped with the crystal structure defined by the functions and operators in…

Let $4\mathfrak{g}4$ be of type $D_{n+1}^{(2)}$, $A_{2n}^{(2)}$, $A_{2n}^{(2)\dagger}$, or $C_n^{(1)}$, let $1\leq r\leq n$ and $s\geq1$, and let $V^{r,s}$ be the virtual crystal w…