27 problems

Let $W$ be a Coxeter group, let $\widetilde{S}$ be its smallest Garside shadow, and consider the automaton constructed from $\widetilde{S}$ that recognises the language of reduced…

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 $W$ be an irreducible complex reflection group that need not be well-generated. The proposed general construction associates a dual braid monoid to a regular degree and maps it…

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…

Let $B_n^+$ be the positive braid monoid, let $n$ be fixed, and let $a\in B_n^+$ have word length $l$. Write $C^+(a)=C(a)\cap B_n^+$ for the positive conjugacy class of $a$. Thurst…

Parkinson–Yau's conjecture. The set $\Gamma$ is the smallest Garside shadow in $(W,S)$.

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…

Ouroboros conjecture. The presence of ouroboroi is essentially the only reason sliding circuit sets can become large: for sufficiently long braids, if a braid has a large sliding c…

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 $(W,S)$ be a Coxeter system with positive roots $4\Phi^+4$. For $n\in\mathbb N$, let $n$-small roots be the positive roots of dominance depth at most $n$, and let $n$-low eleme…

For each $n$, let the $n$-low elements be the elements defined using the notion of $n$-small roots in the associated Coxeter group, and let the associated Artin--Tits monoid be the…

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…

Let $B_n$ be the $n$-strand braid group. Its classical and dual minimal Garside systems are $(B_n,\Delta_n)$ and $(B_n,\delta_n)$, respectively, where … and … Here the $\sigma_i$ a…

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.

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 bound $T_{n,\ell}$ is a polynomial in $n$ and $\ell$.

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

A dual Garside structure for an Artin–Tits group is a Garside structure analogous to the dual structures known for spherical-type Artin–Tits groups. Crisp–McCammond conjecture. No…