21 problems

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

Type-A twelve-period homomorphism conjecture. Define

Interval-family homomorphism conjecture. The assignments

B2 classification conjecture. Homomorphisms from any of those three stated families exhaust, up to diagram automorphisms, all fully supported standard homomorphisms from…

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…

For the infinite series of types $A_l$, $B_l$, and $D_l$, let $G_P^{\mathrm{dual}+}$ be the corresponding dual Artin monoid and let $N_{G_P^{\mathrm{dual}+}}(t)$ be its skew-growth…

Let $P$ be a finite-type Coxeter graph with rank $l=\operatorname{rank}(P)$, let $G_P^{\mathrm{dual}+}$ be its dual Artin monoid, and let $N_{G_P^{\mathrm{dual}+}}(t)$ be its skew-…

Conjecture 1. The polynomial $\widehat{N}_{G_P^{\mathrm{dual}+}}(t)$ is irreducible over $\mathbb Z$, up to the trivial factor $1-2t$ for types $A_l$ with even $l$ and type $D_4$.

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…

Weak left-Garside conjecture. The monoid $A_n$ is a weak kind of left-Garside monoid.

An Artin monoid is a monoid associated with an Artin system; the source considers cancellative monoids equipped with a discrete degree map and the class of pairs $(M,\deg)$ satisfy…

F-conjugacy extension conjecture. The results on conjugacy of periodic elements extend to the case of $F$-conjugacy.

Let $\Gamma_M$ be an Artin monoid of affine type, and let $h_{\Gamma_M,G}$ denote the period associated with its growth function, equivalently the number governing the poles on the…