26 problems

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 $H$ be the KR index set, let $\mathcal{Q}^{\nu}(y)=\prod_{(a,m)}(\mathcal{Q}_m^{(a)}(y))^{\nu_m^{(a)}}$, and let $\mathcal{K}^{\nu}_{D,G}(y)$ and $\mathcal{R}^{\nu}_{D,G}(y)$ b…

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…

Extended QQ-system categorification conjecture. The module $V_w$ is of highest $\ell$-weight and fits into a non-split short exact sequence

Let $G$ be a semisimple algebraic group with Lie algebra $\mathfrak{g}$, and let the $Q$-system be the functional relations satisfied by the relevant generalized-minor functions. F…

Let an on-shell overlap be the overlap between a two-site state and a Bethe eigenstate, and let the $Q$-system be the system of Baxter $Q$-functions associated with the integrable…

Let a weight subspace be the relevant subspace of the Hilbert space, and let the Bethe algebra be the algebra of commuting charges. Consider the algebra of Q-functions of analytic…

Let an extended Q-system be a collection of Q-functions satisfying the extended Q-system relations and the analytic structure. For a fixed corresponding weight subspace of the Hilb…

Let $F_0(u),\ldots,F_m(u)$ be the functions in the determinant representation of the $A_m^{(1)}$ Q-system, and set $t\equiv e^u$. The Bethe roots are called admissible when they sa…

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…

A Q-system is associated with a Young diagram whose distinguished Q-functions have the form … where the $q_{a,s}$ are the corresponding reduced functions. A solution is called phys…

Let solutions of the Q-system equations be polynomial solutions associated with a spin chain, and let irreducible multiplets denote the irreducible representation-theoretic multipl…

$AH+2$ Q-system conjecture. The construction of $AH+1$ can be iterated once more: there is a Q-system for

Infinite log-concavity conjecture. This sequence is infinitely log-concave.

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…

KNS quantum-dimension conjecture. For every $a\in I$, the following properties hold: $z^{(a)}_m>0$ for $0\leq m\leq k$; $z^{(a)}_m=z^{(a)}_{k-m}$ for $1\leq m\leq k-1$;…

Iterated Q-system conjecture. The plus-one procedure should be iterable once more to produce a Q-system for the object

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…

Let $M^Y_{\lambda,\mathbf n}$ denote the multiplicity of the irreducible $\mathfrak g$-module $V(\lambda)$ in a generic tensor product of Kirillov–Reshetikhin modules. Let…