12 problems

Kirillov–Reshetikhin conjecture. For every $(a,m)\in H$,

Let $\mathcal{Q}^{\nu}(y)$ and $\mathcal{K}^{\nu}_{D,G}(y)$ be the normalized KR-character product and associated power series. Type-$B_n$ and type-$D_n$ KR conjectures. For…

Let $X_N^{(r)}=A_{2n}^{(2)}$, so that $\mathfrak{g}_0=B_n$, and let $\mathcal{Q}^{\nu}(y)$ and $\mathcal{K}^{\nu}_{D,G}(y)$ be the normalized KR-character product and associated po…

Let $X_N^{(r)}\neq A_{2n}^{(2)}$, let $\Delta_+^{\mathfrak{g}_0}$ be the positive roots of $\mathfrak{g}_0$, and let $\mathcal{Q}^{\nu}(y)$ and $\mathcal{K}^{\nu}_{D,G}(y)$ be the…

Let $\mathfrak{g}_0$ be the finite-dimensional Lie algebra associated with $U_q(X_N^{(r)})$, let $H$ be the index set of pairs $(a,m)$ for the KR $Q$-system, and let…

Let $\mathfrak{g}$ be a complex simple Lie algebra of rank $n$. Set $y=(y_a)_{a=1}^n$ with $y_a=e^{-\alpha_a}$ for the simple roots $\alpha_a$ of $\mathfrak{g}$. For the KR module…

Let $X_n$ be a finite-dimensional simple Lie algebra of classical or exceptional type, let $W_m^{(a)}$ denote the Kirillov–Reshetikhin module associated with node $a$ and positive…

Fix $a\in I_0$, set $c=\min\{c_b\mid A_{ba}\neq 0\}$, and let $\ell\geq\lceil m/c\rceil$. Write $L(k)=\left\lfloor\frac{mA_{ba}-k}{A_{ab}}\right\rfloor$. Q-system Demazure decompos…

Let $A_1,\dots,A_{2r}$ be coordinates on the phase space of the $B_r$ Q-system, and let $B$ be the signed adjacency matrix of its quiver. Define the Poisson bracket by … where … The…

The KNS conjecture. The collection of real numbers $Q^{(i)}_k$ is a solution of the $\ell$-restricted $Q$-system. Moreover, for every $i\in I$:

Let $I$ be the node set, let $t_a$ and $h^{\vee}$ be the parameters associated with the Dynkin diagram, and let $W^{(a)}_m$ be the elements of the WZW fusion ring defined from the…

Fix a level $\ell$ and let $Q^{(a)}_m=\dim_q\mathrm{res}\,W^{(a)}_m$ be the root-of-unity quantum dimensions of the restricted Kirillov–Reshetikhin modules, with $a\in I$ and…