20 problems

Let $W$ be a finite Coxeter group, let $w\in W$, and let $\Omega_W(w;t)$ denote the order polynomial associated with $W$ and $w$. Assume that $t_W$ is a linear function depending o…

Let $W$ be a finite Coxeter group of rank $n$, with $R$ its representation on the set of roots. Let $G'$ be the virtual $W$-module obtained by evaluating at $t=1$ the graded quotie…

Let $W$ be a finite Coxeter group with simple generators $S$. For $I,J\subset S$, let $X_{IJ}$ be the set of distinguished representatives used in the double-coset decomposition, a…

Let $W$ be a finite Coxeter group, and let $\Sigma W$ be the Solomon descent algebra of $W$. The Solomon homomorphism $[...]$ maps elements of $\Sigma W$ to class functions on $W$.…

Euler-number specialization conjecture.

Rotational $\Lambda\Sigma=\Delta$ conjecture.

Let $D_n$ be the finite Coxeter group of type $D_n$. A minuscule middle order is a middle order arising from the minuscule construction described in the paper. Type D minuscule cla…

Let $B_n$ be the type $B$ finite Coxeter group, with $n\geq3$, and let $J$ be a maximal parabolic subgroup. A middle order is a distributive lattice that refines the weak order and…

Let $W$ be any finite Coxeter group. Let $\operatorname{Irr}(W)$ be its irreducible characters, and call a family $F$ of $\operatorname{Irr}(W)$ good if it contains a character occ…

Let $W$ be a non-crystallographic finite Coxeter group, let $c$ be a two-sided cell in $W$, and let $\mathcal{U}_W^c=\mathcal{Z}(\mathcal{H}_W^c)$ be the Drinfeld center of the asy…

Let $W$ be a finite Coxeter group. Write $\mathrm{Weak}(W)$ for its right weak order, $\mathrm{Tamari}(W)$ for a Cambrian lattice associated to a Coxeter element obtained from a li…

Unimodality conjecture. (i) If $X$ is a conjugacy class of involutions in $W(A_n)$ or $W(B_n)$, then the even/odd length profile of $X$ is unimodal. (ii) If $X$ is the set of invol…

Unimodality and log-concavity conjecture. The even and odd involution length profiles of $W(A_n)$ are log-concave, while the even and odd involution length profiles of $W(B_n)$ and…

Let $W$ be a finite Coxeter group with weight function $L$, and let $E,E'\in\text{Irr}(W)$. The preorder $\preceq_L$ is defined using standard representation-theoretic operations,…

Let $W$ be a finite Coxeter group with weight function $L$. Lusztig's families partition the irreducible representations $\text{Irr}(W)$, and the Kazhdan–Lusztig preorder…

Unique-sign conjecture. There is a unique $\lambda_0\in\Lambda$ such that $E^{\lambda_0}\sim_L {\mathfrak{c}}$ and $c_{w,\lambda_0}$ has the same sign for all $w\in {\mathfrak{c}}$…

Multi-cluster polytopality conjecture. The multi-cluster complex $\Delta_c^k(W)$ is the boundary complex of a simplicial polytope.

Facet-maximality conjecture. The number of facets of $\Delta(Q,w_\circ)$ is at most the number of facets of $\Delta_c^k(W)$. Moreover, if the two numbers are equal, then $Q$ has th…

SIN-property characterization conjecture. The subword complex $\Delta(Q,\pi)$ is isomorphic to a multi-cluster complex if and only if $Q$ has the SIN-property and

Minimal non-face conjecture. Every minimal non-face of $\Delta_c^k(W)$ has cardinality $k+1$.