15 problems

Let $W$ be a well-generated finite complex reflection group with degrees $d_1,\dots,d_n$ and Coxeter number $h$, let $d$ be a regular degree, let $W'$ be the centralizer of a $d$-r…

Let $B$ be the associated braid group, let $R$ be its set of braid reflections, and let $B_+$ be the submonoid of $B$ generated by $R$. Define a relation $\preccurlyeq$ on $B_+$ by…

Shi-region low-element conjecture. One has $L_{\operatorname{Shi}}(W,S)=L(W,S)$. Moreover, every Shi region contains a unique low element, which is its unique element of minimal le…

Let $n\geq 3$. The monoid $\mathcal{H}_n^+$ is defined by the presentation … Let $\mathcal{B}_{n+1}$ be the braid group on $n+1$ strands, with standard generators…

Cubing conjecture. If the sliding circuit set is large, then large regions of the sliding circuit set graph should be cubical.

Internal commutativity conjecture. For every $N\geqslant 5$, there exists an integer $L_N$ such that, whenever $L\geqslant L_N$ and $x\in B_N$ is maximising among rigid pseudo-Anos…

Polynomial bound conjecture. There exists a constant $C$ such that, for every rigid braid $x$ with $N$ strands and Garside-length $L$,

Let $W$ be a finite Coxeter group with braid group $\mathbb{B}_W$, standard Garside monoid with Garside element $\Delta$, and dual positive monoid $\mathcal{B}e^+$. Write…

Let $(W,S)$ be a Coxeter system, let $\tilde S$ be its smallest Garside shadow, and let $\mathcal A_{\tilde S}(W,S)$ be the associated finite deterministic automaton. Write…

Let $(W,S)$ be a Coxeter system. For $n\in\mathbb N$, define the dominance depth of a positive root as the number of positive roots it strictly dominates, let the $n$-small roots b…

Let $B_3^+$ be the positive braid monoid on three strands, let $\Delta_3$ denote its fundamental braid, and let $\mu_3$ be the function on $B_3^+$ introduced in the paper. D., From…

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…

Let $B_n$ be the braid group on $n$ strands, let $x\in B_n$ have canonical length $r$, and let $\mathfrak{s}$ denote the cyclic sliding operation. Let $t$ be the minimal positive i…

Polynomial bound conjecture. The integer $t$ is bounded by a polynomial in $\ell$ and $n$.

Tangledness-reduction conjecture. The product $\beta\beta_+$ has smaller $\tau$-length than $\beta$.