38 problems

M=N conjecture. The computer experiments suggest

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 $N\geq 1,n\geq 2$ be integers, $\lambda$ a partition with $\operatorname{height}(\lambda)\leq n$, and $\boldsymbol L$ an $n\times N$ matrix with nonnegative integer entries. Le…

Let $n,l,r$ be as in the source, let ${\cal H}(l\Lambda_r,\mu)$ be the set of paths defined there, let $E(p)$ be their energy, and let $\operatorname{af}(\operatorname{wt}p)$ denot…

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…

The $X=M$ conjecture. For all affine Kac–Moody algebras $\mathfrak{g}$,

Let $B_\mu=B_{\mu_1}\otimes\cdots\otimes B_{\mu_m}$ be an affine crystal, and let $X$ denote the associated one-dimensional sum, obtained by grading highest-weight vertices in the…

Let $\overline{K}^{\diamondsuit}_{L,\lambda}(t)$ and $\overline{M}^{\diamondsuit}_{L,\lambda}(t)$ be the $K$- and fermionic formulas associated with an allowed diagram shape…

Let $W^L=\bigotimes (W_s^{(r)})^{\otimes L_s^{(r)}}$ be a finite tensor product of Kirillov–Reshetikhin modules, with $L=(L_s^{(r)})$ a collection of nonnegative integers. Let…

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 $\mathfrak{g}$ be a nonexceptional affine Lie algebra of type $X_N^{(r)}$, with canonical simple Lie subalgebra $\overline{\mathfrak{g}}$ of rank $n$. Let $lambda$ be a partiti…

Let $B$ be a tensor product of finite-dimensional crystals and let $X(B,\lambda)$ denote the one-dimensional configuration sum, while $M(B,\lambda)$ denotes the corresponding fermi…

Let $\eta\in\mathcal{P}$ satisfy $\eta_1\leq n-1$. For $s,u\in\{0,\dots,N\}$ and $r,v\in\{0,\dots,n-1\}$, set $\Lambda=(N-s)\Lambda_r+s\Lambda_{r+1}$ and…

Let $\mathfrak{g}$ be a nonexceptional affine Kac–Moody algebra other than $A^{(2)\dagger}_{2n}$. Fix $\lambda\in\overline{P}^+$ and a matrix $\mu=(L_i^{(a)})$ of nonnegative integ…

Let $B_R$ be a crystal of type $A_{2n}^{(2)\dagger}$ and let $4\Lambda4$ be a dominant integral weight. With $X(B_R,\Lambda;q^{-1})$ the one-dimensional configuration sum, let…

Let $4\mathfrak{g}4$ be a nontwisted affine Lie algebra, and consider the finite-dimensional $U'_q(\mathfrak{g})$-representations naturally associated with multiples of fundamental…

Let $4\mathfrak{g}4$ be one of the affine types considered in the paper, let $J$ index the classical simple roots, and let $4\overline{P}4$ be the classical weight lattice with dom…

Let $\hat{\mathfrak{g}}=X_n^{(1)}$ be as in the classically restricted setting, let $B=\bigotimes_{a=1}^n\bigotimes_{i\ge1}(B^{a,i})^{\otimes L_i^{(a)}}$, and let…

Let $\hat{\mathfrak{g}}=X_n^{(1)}$, with $X=A,B,C,D$ or $E$ for $n=6,7,8$, $F$ for $n=4$, or $G$ for $n=2$. Let … where $L_i^{(a)}\in\mathbb{N}$ and only finitely many are nonzero.…

Let $U'_q(\hat{\mathfrak{g}})$ be the quantum affine algebra and let $U_q(\mathfrak{g})$ be its finite-type subalgebra. Consider the finite-dimensional representations naturally as…

Let $B^{r,s}$ denote a Kirillov--Reshetikhin crystal in an exceptional affine type. The fermionic formula $M(B,\lambda;q)$ is the generating function obtained from rigged configura…