59 problems

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 $(W,S)$ 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…

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 $\mathsf{M}$ be a loopless matroid of rank $k$ on a ground set of size $n$. Let $\mathsf{U}_{k,n}$ be the uniform matroid of rank $k$ on $n$ elements, and write…

Let $n\geq 2$, and let $e\leq \sigma_1\leq \sigma_2\leq \Omega_n$. For the Kazhdan–Lusztig $R$-polynomial $\widetilde{R}_{\sigma_1,\sigma_2}(q)$, write $F_j(q)$ for the $q$-Fibonac…

Let $\boldsymbol{\boldsymbol{\text?}}$ be a tensor product parameter with associated free abelian group $S^{\boldsymbol{\boldsymbol{\text?}}}(V_{\mathbb Z})$, and let…

Let $R$ be a reduced root system with equal parameters $t_s=t_\ell=:t$. For a weight $\lambda$, let $C'_{\lambda}$ be the canonical basis element and let $P_{v_\mu,v_\lambda}(t)$ b…

First-$n$-coefficients conjecture. If $(\Phi_+,w)$ is not a rationally smooth pair, then comparing the first $n$ coefficients and the last $n$ coefficients of $P_w(t)$ suffices to…

Let $W$ be a Coxeter group of type $A$, let $u,v\in W$, and let $M$ be a special matching of the Bruhat interval $[u,v]$. For $x,y\in [u,v]$ with $x\le y$, write $R_{x,y}(q)$ for t…

Let $W$ be a Coxeter group of type $A$, and fix a Bruhat interval $[u,v]$. Consider the equivalence relation on the amazing hypercube decompositions of $[u,v]$ generated by … where…

Double-shortcut triviality conjecture. The above equivalence relation is trivial, meaning that it has only one equivalence class. This is a conjecture of Ellenberg and McNamara and…

Fix $n\ge 0$, and let $T_n$ be the thagomizer matroid with the action of the hyperoctahedral group $\mathcal{B}_n$. Write $P_{T_n}^{\mathcal{B}_n}(t)$ and…

Graph Combinatorial Invariance conjecture. The $\widetilde{R}$-polynomial $\widetilde{R}_{u,v}(q)$ depends only on the isomorphism type of the graph $B([u,v])$.

Flip Combinatorial Invariance conjecture. Flip Combinatorial Invariance holds for flipclasses of Weyl type.

Combinatorial Invariance Conjecture. If two Bruhat intervals, potentially from different Coxeter groups, are isomorphic as posets, then

Let $M$ be a matroid, and let $P_M(t)$ and $Q_M(t)$ denote its Kazhdan–Lusztig and inverse Kazhdan–Lusztig polynomials, respectively. Degree equality conjecture. For any matroid…

Let $[u,v]\subset S_n$ be a Bruhat interval, let $I$ be a hypercube decomposition of $[u,v]$, and let $\ell(u,v)$ denote the length of the interval. The polynomials $N_{u,v,I}$ and…

For each $n$, let $\mathrm{uni}_{n}$ be the proportion of distinct first-row Kazhdan–Lusztig polynomials for $\mathrm{S}_{n}$ that are unimodal, and let $\mathrm{uni}_{n}^{mult}$ b…

Product-of-projective-spaces conjecture. The Kazhdan–Lusztig polynomial indexed by $w_{l}^{k}$ is the Hilbert–Poincaré polynomial of $(\mathbb{C}\mathbb{P}^{l})^{\times(k-1)}$:

For $w\in\mathrm{S}_{n}$, let $\mathrm{P}_{w}$ be the first-row Kazhdan–Lusztig polynomial, and let $\mathrm{coeff}_{n}$ denote the maximal coefficient among these polynomials; let…

For each $n$, let $\mathrm{num}_{n}$ be the number of distinct polynomials among $\{\mathrm{P}_{w}\mid w\in\mathrm{S}_{n}\}$, and let $\mathrm{num}_{n}^{\%}$ be the ratio of this n…

Let $\mathrm{den}_{n}$ denote the density quantity defined in the paper for Kazhdan–Lusztig polynomials indexed by permutations in $\mathrm{S}_{n}$. Asymptotic density conjecture.…

Hypercube-decomposition conjecture. For any hypercube decomposition $[u,z]$ of $[u,v]$,

Parabolic Combinatorial Invariance Conjecture. Then

Microlocal multiplicity conjecture. For every loopless matroid $M$,