32 problems

Explicit adjoint formulas conjecture. The adjoints satisfy

Generalized centralizer theorem. For arbitrary $I\subseteq S$, if $\gamma\in\Pi^I$ and $s_\gamma\in W^{\perp I}{}_{\mathrm{fin}}$, then $w\cdot\gamma=\gamma$ for every $w\in Y_I$.…

Consider an Artin–Tits group of spherical type equipped with its classical and dual Garside presentations. A standard parabolic subgroup for the dual presentation is a subgroup gen…

Let $G$ be the group in the paper, let $P$ be a standard proper parabolic with Levi subgroup $M$, and let $W$ be an irreducible local system on $X$ of rank $r=\dim V^\gamma$. For…

A Dyer system $(D,X)$ has standard parabolic subgroups $D_Y$ generated by subsets $Y\subseteq X$. For subsets $Y,Y'\subseteq X$, let $\operatorname{Conj}(D_Y,D_{Y'})$ be the set of…

Let $A$ be an Artin group, and let parabolic subgroups mean the parabolic subgroups of $A$. The parabolic intersection conjecture. The intersection of any two parabolic subgroups i…

Let $A_S$ be an Artin group, and let RP denote the ribbon property: for all $X,Y\subseteq S$ and $g\in A_S$, one has $gA_Xg^{-1}\subseteq A_Y$ if and only if $X_{as}\subset Y$ and…

Let $A_S$ be an Artin group. It has the parabolic intersection property if the intersection of any two parabolic subgroups of $A_S$ is a parabolic subgroup of $A_S$. Parabolic inte…

Explicit parabolic basis conjecture. The resulting elements

Let $A$ be an Artin group such that every irreducible $A_Z$ with $|Z|>2$ is not of spherical type. For $X\subseteq S$, write … If $A_X$ has no cyclic irreducible component…

Let $B$ be an Artin group with standard generating set ${\mathbf S}$, let ${\mathbf b}\in B$ have minimal length in its conjugacy class, and let $\supp({\mathbf b})$ denote its sup…

Let $B$ be an Artin group, and let ${\mathbf b}\in B$. A minimal parabolic subgroup containing ${\mathbf b}$ is a parabolic subgroup containing ${\mathbf b}$ that is minimal under…

Let $B$ be an Artin group with parabolic subgroups, meaning subgroups associated with subsets of its standard generating set. Parabolic intersection conjecture. The intersection of…

Parabolic-intersection conjecture. Any intersection of parabolic subgroups of an Artin group is again parabolic.

Let $A_\Gamma$ be an Artin group, and let a parabolic subgroup mean a conjugate of a subgroup generated by a subgraph of the presentation graph. Intersection Conjecture. The inters…

Let $A_\Gamma$ be an Artin group. A standard parabolic subgroup is a subgroup generated by the standard generators associated to a subgraph of the presentation graph, and a subgrou…

Let $A_I$ be an Artin–Tits group, let $X\subseteq I$, and let $CA_I$ denote the subset of elements specified in the source. A parabolic subgroup is a conjugate of a standard parabo…

Let $A_I$ be an Artin–Tits group. For $X\subseteq I$, the standard parabolic subgroup $A_X$ is the subgroup generated by the standard generators indexed by $X$; a parabolic subgrou…

Intersection conjecture. The intersection of any two parabolic subgroups of $G_\Gamma$ is a parabolic subgroup.

Let $A_\Gamma$ be an Artin group. A parabolic subgroup is a subgroup conjugate to a standard parabolic subgroup $A_X$, where $X\subset V(\Gamma)$ and $A_X$ is generated by $X$. The…

Let $A$ be an Artin group. The paper considers three properties of $A$: existence of parabolic closures for all elements, standardisability, and the ribbon property. In particular,…

Let $V$ be the natural representation of $\operatorname{Sp}_{2m}(\mathbb{F}_q)$, let $G_{\lambda}$ be the parabolic subgroup associated to a partition…

Let $G$ be the reductive group, $G_{\ge0}$ its totally nonnegative part, $G_{\mathrm{reg},\ge0}$ the regular elements in $G_{\ge0}$, and $\mathcal{P}_{\ge0}$ the totally nonnegativ…

Let $A_S$ be an irreducible non-cyclic FC type Artin group, and let ${\mathcal{CP}_{A_S}}$ denote its complex of parabolic subgroups. Hyperbolicity and infinite-diameter conjecture…

Let $A_S$ be an irreducible FC type Artin group, and let ${\mathcal{CP}_{A_S}}$ denote its complex of parabolic subgroups. Infinite-diameter conjecture. The complex…