267 problems

For every finite Coxeter group $W$ of rank $r$, the $h$-polynomial of the order complex of its noncrossing-partition lattice admits a nonnegative real-rooted symmetric decompositio…

Let $(W,S)$ be an irreducible finite Coxeter system with Coxeter number $h$, let $\vec{c}$ be a Coxeter word in $S$, let $p>0$ be an integer coprime to $h$, and let $J\subseteq S$.…

Let $(\mathbb W,S)$ be a Coxeter system with a geometric $R$-realisation over a fusion ring $R$, and let $(\check{\mathbb W},\check{S})$ be the corresponding unfolded Coxeter syste…

Let $R$ be a fusion ring and let $(r_{s,t})_{s,t\in S}$ be a matrix satisfying the geometric conditions for a Coxeter system $(\mathbb W,S)$, as in the fusion-realisation condition…

Let $(W,S)$ be a finite Coxeter system equipped with a canonical filtration … This filtration gives a reflection-coloring map $\alpha:T\to\{1,\ldots,r\}$ by assigning to each refle…

Let $(W_1,S_1)$ and $(W_2,S_2)$ be Coxeter systems, and let $\sigma1,\omega1 easymp W1$ and $\sigma2,\omega2 easymp W2$. For each pair, let $G[sigmai,omegai]$ denote the directed Bruha…

Let $w\in\widetilde{S}^D_n$ be an affine signed permutation, and let \(w)$and$operatorname{dis}(w)…

Let $w$ be an affine signed permutation in $\widetilde{S}^B_n$. An integer $j\in[n-1]$ is good if $w$ restricts to a bijection from $\pm[j]$ to itself, and it is very good if this…

Let $W$ be a Coxeter group of type $B_n$ or $D_{2n+1}$. Let $V$ be its reflection representation, and let $U$ be the non-faithful reflection representation of dimension $n-1$ descr…

Lusztig's conjecture on the function $a$. For every $w\in W$, one has

Let $G$ be a split semisimple algebraic group with Weyl group $W$, and let $L^{u,v}=N\overline{u}N\cap B_-vB_-$ be the reduced double Bruhat cell for $u,v\in W$. For a reduced word…

PL sphere-or-ball conjecture. If $[u,v]$ is full, then $(u,v)$ is a PL sphere of dimension $\rho(v)-\rho(u)-2$. Otherwise, $(u,v)$ is a PL ball of the same dimension.

Let $W$ be an infinite Coxeter group, let $Y_W$ be the associated hyperplane complement in $\mathbb{C}\otimes V$ restricted to the Tits cone, and let $A$ be the associated Artin gr…

Integral-ratio cell conjecture. Assume that $b=ra$ for some $r\geqslant 1$. Then the Kazhdan–Lusztig left, right, and two-sided cells of $W_n$ are the smallest subsets of $W_n$ tha…

Conjecture A+. Let $r\geqslant 0$ and assume that $ra<b<(r+1)a$. Then the Kazhdan–Lusztig left, right, and two-sided cells for these parameters coincide with those obtained for the…

Conjecture A. Assume that $\Gamma=\mathbb{Z}$, $a=2$ and $b=2r+1$. Then, for $w,w'\in W_n$, they lie in the same Kazhdan–Lusztig left cell if and only if $Q^r(w)=Q^r(w')$, in the s…

Mühlherr's conjecture. There exists a reflection-preserving automorphism $\alpha$ of $W$ such that $\alpha(S)$ is sharp-angled with respect to $S'$.

Brady–McCammond–Mühlherr–Neumann conjecture. If

Let ${\mathcal{V}}$ be an $n$-dimensional, non-degenerate quadratic space, let $[n-1]=\{1,\ldots,n-1\}$, let ${\mathcal{C}}_n$ be the subgroup of chessboard elements in the symmetr…

Let $A$) be an Artin group, with corresponding Coxeter group $W$. The standard parabolic subgroups of $A$ of finite type are the parabolic subgroups associated with finite standard…

The cohomology tensor-product conjecture. There is an equality

Maximal FAn subgroup conjecture. A subgroup $H\subset W$ is maximal $\operatorname{FA}_n$ if and only if

CAT(0) conjecture. The universal cover of the $v$-splitting of $W$ determined by $S'$ is $\operatorname{CAT}(0)$.

Let $R_u$ and $R_v$ be regions of the Coxeter arrangement, and let the transformed cluster fan be the fan whose maximal cones are under consideration. Write $u\equiv v$ when $u$ an…

Let $W$ be a Coxeter group with weight function $L$, let $W_I\subseteq W$ be a parabolic subgroup, and let $Y_I$ denote the corresponding set of distinguished coset representatives…