472 problems

Kazhdan–Lusztig positivity conjecture. The coefficients of the Kazhdan–Lusztig polynomials are non-negative, namely $h_{y,x}\in v\mathbb{Z}_{\geq0}[v]$, and the structure constants…

Explicit adjoint formulas conjecture. The adjoints satisfy

Let $W$ and $W'$ be Coxeter groups, with $u,v\in W$ and $u',v'\in W'$. Write $[u,v]$ and $[u',v']$ for the corresponding Bruhat intervals, and let $P_{u,v}(q)$ and…

Reineke's conjecture. For every standard Coxeter element $c$, there exists $\gamma\in V^*$ such that $Z_{\rho,\gamma}$ is totally stable for $\mathcal{Q}^c$, meaning that every ind…

Quasi-isometry classification conjecture. The groups $W_\Gamma$ and $W_{\Gamma'}$ are quasi-isometric if and only if they are isomorphic.

For a Coxeter system $(W,S)$, the extended weak order is the partial order introduced by Dyer that generalizes the weak order to possibly infinite Coxeter groups. Dyer's lattice co…

The WDVV equations are the associativity equations for a Frobenius manifold prepotential, and a polynomial WDVV solution means a polynomial prepotential satisfying these equations.…

Mühlherr's twist conjecture. There exists a possibly trivial element $w\in W$ such that $S'$ is obtained from the conjugate $S^w$ by applying finitely many elementary twists.

Lusztig's boundedness conjecture. The number $N_L(W)$ is a bound for any positively weighted Coxeter group $(W,S,L)$ of finite rank. This is a weighted generalization of Xi's bound…

Let $W_n^{\mathsf{BC}}$ be the type $\mathrm{B}$ Coxeter group equipped with a weight function determined by a parameter $s$, and let its cells be the Kazhdan–Lusztig cells. The do…

Let $[u,v]$ be a Bruhat interval in a Coxeter group, and let $P_{u,v}(q)$ denote its Kazhdan–Lusztig polynomial. Combinatorial Invariance Conjecture. The polynomial $P_{u,v}(q)$ de…

Kellerhals–Perren's conjecture. These growth rates should be Perron numbers.

Monotonicity conjecture. If $u,v,w\in W$ and $u\leq v\leq w$, then

Let $W$ be a finite Coxeter group, let $C$ be a conjugacy class of involutions in $W$, and let $V_C$ be the real vector space with basis $\{a_w\mid w\in C\}$ equipped with the $W$-…

Finite Coxeter Cambrian mutation conjecture. The Cambrian lattice $\operatorname{Camb}(Q)$ is mutable.

Let $K$ be a field and let $\lambda\in K\setminus\{0,1,-1\}$. Let $(W,S)$ be an arbitrary Coxeter system. Write $\mathcal{A}_{\pm 1}=\mathbb{Z}[u,u^{-1},(u+1)^{-1},(u-1)^{-1}]$, le…

Parabolic combinatorial invariance conjecture. Under these hypotheses, $P_{u_1,v_1}^{H_1,x}(q)=P_{u_2,v_2}^{H_2,x}(q)$, equivalently $R_{u_1,v_1}^{H_1,x}(q)=R_{u_2,v_2}^{H_2,x}(q)$…

Calogero–Moser cell coincidence conjecture. If $W$ is a Coxeter group, the Calogero–Moser notions coincide with the corresponding Hecke-algebra notions defined by Kazhdan and Luszt…

Weighted Singer Conjecture. The weighted $L^2$-homology vanishes above the middle dimension:

Let $W$ be a finite crystallographic Coxeter group with Coxeter number $h$, let $J(\Phi^+(W))$ be the order ideals of its positive root poset, and let $d_1,d_2,\ldots,d_n$ be the d…

Let $(W,S)$ be a Coxeter system, let $I\subseteq S$ be a subset of simple reflections, and let $W_I$ be the parabolic subgroup generated by $I$. Let $\mathfrak{h}$ be a realization…

Dual EL-shellability conjecture. The poset $\widehat{\operatorname{Cox}(\mu)^K}$ is dual EL-shellable.

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…

Geodetic-number conjecture. If ${\boldsymbol{\alpha}}$ is a reduced expression, then the geodetic number of $\operatorname{\mathcal{B}}({\boldsymbol{\alpha}})$ is two. The source s…

Isometric-dimension conjecture. If $(W,S)$ is type $\Lambda$ and ${\boldsymbol{\alpha}}$ is a reduced expression, then