31 problems

Coxeter polynomial conjecture. If

Let $(V,c)$ and the matrix $(A_{ij})$ be as above: $V=\mathbb C x_1\oplus\cdots\oplus\mathbb C x_{\theta}$, $c(x_i\otimes x_j)=q_{ij}x_j\otimes x_i$, $q_{ii}=-1$,…

Minimal-odd-root Dynkin diagram conjecture. There always exists exactly one Dynkin diagram with the minimal number of odd roots satisfying the definition of a hyperbolic Kac–Moody…

ADET effective-central-charge conjecture. The effective central charge is

ADET torsion conjecture. When $U=A(X,Y)V$, the element $(U,V)$ is a torsion element of $\hat B(\C)$.

Let $\Delta$ be the root system under consideration, let $I$ index the complete flag $\mathcal{S}_\bullet^1$, and for each $i$ with $1\leq i\leq |I|$ define … The roots…

Let $G=ADE$ and let $G'$ be a folding of $G$. Write $\mathrm{CFT}(A_1,G)$ and $\mathrm{CFT}(A_1,G')$ for the two-dimensional conformal field theories associated with these Dynkin d…

Let $X$ and $Y$ be Dynkin diagrams of type $ABCDEFGT$, with Cartan matrices $C(X)$ and $C(Y)$. Let $D(X)$ be the diagonal matrix whose $j$-th diagonal entry is $\alpha_j^2/\beta^2$…

Let $\mathcal{C}(D_k)$ be the Cartan matrix of the Dynkin diagram of type $D_k$, and let $f_{A,b,C}(q)$ denote the corresponding Nahm sum. Zagier-duality conjecture. For every…

Vertical-sum folding conjecture. The vertical sum operation is a kind of the dual version of the folding map of Dynkin diagrams.

Mizuno's conjecture. The following identity holds:

Let $X$ be a simply-laced connected Dynkin diagram of finite type with rank $r$, Coxeter number $h=h(X)$, and vertex set $I=1,\dots,r1$. Let $\ell\geq 2$ and let…

Use the displayed labeling of the Coxeter graph of type $D_{n+1}$, let $K_1,K_2\subset[0,n]$, and let $w_\circ^K$ denote the longest element associated with $K$. Conjectural…

Let $A_S$ be an irreducible spherical Artin group. Let $S'\subset S$ be such that $A_{S'}$ has Dynkin diagram isomorphic to type $D_n$ for $n\geq 3$, without requiring the isomorph…

Let a positive integral frieze be a frieze of Dynkin type $E_8$ whose entries are positive integers. The $E_8$ frieze count conjecture. The number of positive integral friezes of t…

Let a positive integral frieze be a frieze of Dynkin type $E_7$ whose entries are positive integers. The $E_7$ frieze count conjecture. The number of positive integral friezes of t…

Corank bound conjecture. Then

Let $\{\widetilde\Gamma,\Gamma\}$ be a homogeneous pair of Carter diagrams. Associate signed enhanced Dynkin diagrams $\Delta(\widetilde\Gamma)$ and $\Delta(\Gamma)$ to them. Homog…

Carter diagrams with cycles and extra nodes in the Dynkin–Minchenko completion procedure are the objects under consideration. Correspondence conjecture. There is a correspondence b…

Let $v$ be a vertex in a classified $T$-system outside the tensor-product case with finite factor of type $A_m$, $D_{2m+1}$, or $E_6$, and let $J_v(z)$ be its recurrence polynomial…

Assume the $T$-system has tensor-product form $\Lambda'\otimes\widehat{\Lambda}$, where $\Lambda'$ is of type $A_m$, $D_{2m+1}$, or $E_6$, and let $\eta$ be the canonical involutio…

Unitary-frieze bound conjecture. The value of a frieze at any node of a Dynkin diagram is less than the maximal value of that node over the set of unitary friezes.

Frieze-count conjecture. The numbers of friezes of types $E_6$, $E_7$, $E_8$ and $F_4$ are $868$, $4400$, $26592$ and $112$, respectively.

Let $\mathfrak{g}$ be a semisimple Lie algebra with Weyl group $W$, let $\overline{\mathfrak{g}}$ be the ambient semisimple Lie algebra with dominant weights $\overline{P}_{+}$ and…

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 $1\leq m\leq k-1$;…