93 problems

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…

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…

Let $\mathsf{M}$ be a matroid, and write $Q_{\mathsf{M}}(t)=\sum_i a_i t^i$. A sequence is log-concave with no internal zeros if $a_i^2\geq a_{i-1}a_{i+1}$ whenever defined, and it…

Gedeon's coefficient bound conjecture. The coefficients of the equivariant Kazhdan–Lusztig polynomial of $\mathrm{M}$ are bounded above by those of the equivariant Kazhdan–Lusztig…

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

Xie–Zhang's conjecture. For every matroid $\mathsf{M}$, the polynomial $\mathscr{B}(Q_{\mathsf{M}}(t))$ is real-rooted. Real-rootedness would imply strong coefficient inequalities…

Let $\mathsf{M}$ be a matroid, and let $Q_{\mathsf{M}}(t)$ denote its inverse Kazhdan–Lusztig polynomial. Gao–Xie's conjecture. For every matroid $\mathsf{M}$, all coefficients of…

Gedeon–Proudfoot–Young’s conjecture. For a non-degenerate matroid $M$, the roots of $P_M(t)$ and those of the contraction $M/e$ satisfy interlacing properties. The source does not…

Let $(W,S)$ and $(W',S')$ be Coxeter systems, with $J\subseteq S$ and $J'\subseteq S'$. For $u,v\in W^J$ and $u',v'\in (W')^{J'}$ with $u\leqslant v$ and $u'\leqslant v'$, write ……

For permutations $a,b,v,w$, let $P_{a,b}(q)$ and $P_{v,w}(q)$ denote Kazhdan–Lusztig polynomials, and let $[a,b]$ and $[v,w]$ be the corresponding Bruhat order intervals. Lusztig's…

Let $G$ be a simply-laced group with Weyl group $W$, let $u,v\in W$ satisfy $u\leqslant v$ in the Bruhat order, and let $m_{u,v}=\mathcal{A}_v\psi_u(1)$. Let $Q_{u,v}$ be the inver…

Negative-real-root conjecture. For any matroid $M$, all roots of $P_M(t)$ lie on the negative real axis.

Let $w\in \mathfrak{S}_n$ be a $321$-avoiding and $2$-repeating permutation. Let $u,v\in\mathfrak{S}_n$ satisfy $u\leq v\leq w$, and let $\tilde{R}_{u,v}(t)$ denote the correspondi…

Non-negativity conjecture. The coefficients of each polynomial $c^\nu_{\lambda^{(0)},\ldots,\lambda^{(r-1)}}(q)$ are non-negative. The polynomial $\tilde G_\lambda^{(r)}(X;q)$ is a…

Kazhdan's poset-invariance conjecture. The polynomial $P_{y,w}$ depends only on the partially ordered set of elements $z$ between $y$ and $w$.

Let $[u,v]$ and $[x,w]$ be intervals in the relevant Bruhat orders, and let $P_{u,v}(q)$ and $P_{x,w}(q)$ denote their Kazhdan–Lusztig polynomials. Interval-isomorphism invariance…

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…

The 0-1 conjecture. For $x,w\in S_n$,

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…

Positivity conjecture. For $f\in\mathbb Z^{m|n}$:

Let $n$ be a positive integer and let $w_{k,l}\in\mathfrak{S}_n$ have the reduced expression … where $2\leq k<l<n$. Haiman's conjecture. The displayed source says that the theorem…

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…