13 problems

Type-D family conjecture. The following assignments define standard homomorphisms:

Type-A twelve-period homomorphism conjecture. Define

Interval-family homomorphism conjecture. The assignments

A2 classification conjecture. Up to diagram automorphisms, either $M=A_{3m-1}$ for some $m\ge1$ and $\Phi=\Phi_{m,J}$ with $J\subset[1,m-1]$ as in Theorem, or $M=E_6$ and…

Dihedral injectivity criterion. Both induced maps $\overline\Phi$ and $\overline\Phi_\star$ are injective if and only if either both $\Phi$ and $\Phi$ are self-orthogonal, or $K$ s…

Generalized Tits injectivity conjecture. All Tits homomorphisms $\mathbf T_{\mathbf d}$ are injective.

Optimal injectivity conjecture. Any optimal and injective on generators homomorphism in $\mathscr{Ast}$ is injective.

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…

For $i\in[2,n-2]$, let $d=1+\delta_{2i,n}+\delta_{2i,n-1}+\delta_{2i,n+1}$. $12/d$ family conjecture. The assignments … define a standard homomorphism … This is one of the conjectu…

Let $1\le b\le a<n-1$ and $a>1$. Let $w_\circ^{[r,s]}$ denote the longest element associated with the interval $[r,s]$. Three-family conjecture for $I_2(N)$ homomorphisms. The assi…

Let $M\in\mathbf{Cox}(I)$ be irreducible of finite type, and let $\Phi\in\operatorname{Hom}_{\mathscr A}(A_2,M)$ be fully supported and standard with $\Phi\ne\Phi$. Classification…

Let $M(\mathbf d)$ be a Coxeter monoid and let $\mathbf T_{\mathbf d}$ be a Tits homomorphism associated with it. Injectivity conjecture. All Tits homomorphisms…

Lattice and normal-form conjecture. 1. The ordered set $(A_n,\leq)$ is a lattice. 2. Let $x\in A_n$. Then $x\leq\Delta_n$ if and only if there exist…