21 problems

Let $(W,S)$ be a finite Coxeter system. For a subset $J\subseteq S$, write $W_J$ for the corresponding standard parabolic subgroup. For pairs in finite Coxeter systems, let…

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 $w\in Y_I$.…

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…

Explicit adjoint formulas conjecture. The adjoints satisfy

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…

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 $A_\Gamma$ 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. For $X\subseteq I$, the standard parabolic subgroup $A_X$ is the subgroup generated by the standard generators indexed by $X$; a parabolic subgrou…

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 $P_{u,w}$ be the Kazhdan--Lusztig polynomials for symmetric groups, and let $H\simeq S_m\times\dots\times S_m$ be the parabolic subgroup of $S_{mn}$ of type $(m,\dots,m)$. For…

Let $P_\alpha$ be the parabolic subgroup of $Gl_n(\mathbb{F}_q)$ associated with a composition $\alpha$ of $n$, and let … Let $a_n$, $b_{r,k}$, $c_{r,s,k}$, and $d_r$ be the famili…

Global-nilpotent gluing conjecture. This functor is fully faithful.

Let $G$ be an algebraic group, let $P$ be a parabolic subgroup with unipotent radical $N$, and let $C$ be a conjugacy class in $P/N$. Let $D$ be the inflation of $C$ to $P$, and le…

Let $G$ be a split reductive group over a number field $F$, let $B/F$ be a fixed Borel subgroup, and let $P=UM$ be a standard parabolic subgroup with Levi factor $M$. If…

Let $W$ be a finite Coxeter group, let $W_L$ be a proper parabolic subgroup of rank $r$, and let $\mathcal C_L$ represent the cuspidal conjugacy classes of $W_L$. For each…

Let $W$ be a finite Coxeter group generated by $S$, let $L\subseteq S$, and let $W_L$ be the corresponding parabolic subgroup. Let $\mathcal{C}_L$ be the set of cuspidal conjugacy…

Let $q$ be a prime power and let $\mathbf l=(l_1,\ldots,l_s)$ be a tuple of nonnegative integers. For $m=l_1+\cdots+l_s$, let $P^{\mathbf l}(q)$ be the parabolic subgroup of…