16 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 $n$ 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…

Let ${\mathfrak g}$ be a simple Lie algebra with Cartan matrix $C$, let ${\mathbf R}=\{a_p\omega_{i_p}:1\leq p\leq N\}$ be a collection of dominant integral weights, and let…

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 $R$ 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 $\eta\in\mathcal{P}$ 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…

X=M conjecture. One has

Let $\mathfrak g$ be a Kac–Moody algebra, let $\beta$ be a positive-root-lattice element, and let $X^{\lambda_1,\lambda_2}_\beta$ be the bilinear fermionic quantity defined from th…

Let $C$ be the matrix defining the fermionic sums, let $m$ be a multiplicity vector, let $r\leq s$, and let $W_{C,m}$ denote the quadratic weight appearing in the fermionic sum. Fo…

Let $\mathfrak{g}$ be a simple Lie algebra. For dominant integral $\lambda$ and multiplicity data $\mathbf n$, let $M_{\lambda,\mathbf n}^{(k)}$ and $N_{\lambda,\mathbf n}^{(k)}$ b…