180 problems

Geiss-Leclerc-Schröer's rank-vector conjecture. The vector $\alpha$ is a root of the Kac-Moody Lie algebra of type $C$ if and only if $\alpha$ is the rank vector of some $\tau$-loc…

Let $R$ be a Weyl root system of rank $n$, with Weyl denominator $\Delta$, and let $d_1,\ldots,d_n$ be the degrees of the fundamental invariants of its Weyl group. For a Fourier po…

Let $G$ be a split reductive group over $\x211a$ and let $k_G$ be the positive integer from the saturation theorem: for dominant integral weights $\lambda$, $\mu$, and $\nu$ such t…

Let $\Phi$ be an irreducible root system with Coxeter number $h$, and let $V$ be its ambient vector space. For integers $a\leq b$, define the truncated affine Weyl arrangement … wh…

Clean root system conjecture. All rank $3$ root systems are clean.

Let $k\in\mathbb{Z}_{>0}$, let $\Phi$ be a root system, let $\Phi^+$ be its set of positive roots, and let $\delta\in\Phi^+$. Write $\delta^c$ for the complement of $\delta$ in the…

Let $E_k$ denote the Dunkl kernel for the root system $A_n$, and let $E_k^W$ denote its Weyl-group symmetrization. The formulas for the type $A_2$ Dunkl kernel are denoted by and.…

Let $n$ be a positive integer. Consider the set of roots associated with non-self-intersecting curves in an $n$-punctured disc, and the set of real Schur roots of acyclic valued qu…

Let $W$ be a finite Weyl group with positive-root poset $\Phi^+$, rank $r$, Coxeter number $h$, and longest element $w_0\in W$. Let $\operatorname{NN}(W)=\mathcal{A}(\Phi^+)$ be th…

Chapoton's conjecture. The incidence algebras $k[J(\Phi^+)]$ are fractionally Calabi--Yau.

Let $W$ be a finite Weyl group of type $A$, and let $\widetilde{W}$ be its corresponding affine Weyl group. Consider the reduced random walk on the alcoves of $\widetilde{W}$ obtai…

Let $\Phi$ be a reduced root system with positive roots $\Phi^+$, and let $\mathcal I$ be an ideal in $\Phi^+$. The arrangement of ideal type $\mathscr A_{\mathcal I}$ is the subar…

Let $G$ be a reductive, connected, simply connected group of Lie type that is $F$-split, where $F$ is a Frobenius automorphism of $G$. Let $p$ be the corresponding good and regular…

Let $R$ be an irreducible root system in $\mathbb{R}^n$, and let $a$ and $b$ be integers with $a+b\geq 2$. Define the truncated $R$-affine arrangement ${\cal A}^{ab}(R)$ by the hyp…

Let $\Phi$ be an irreducible crystallographic root system in $\mathbb R^l$, not necessarily reduced, and fix a system of positive roots $\Phi^+$. For nonnegative integers $a,b$, no…

Suppose that $\tilde{\Phi}$ is not of type $\mathsf{B}\mathsf{C}$. Let $\tilde{\mathbf A}$ be the Cartan matrix whose realization is given by the vector space ${}^e\!{\mathfrak s}$…

Categorical McKay correspondence conjecture. There exists such a category and functor satisfying all of the following: $\mathcal{D}$ is 2-periodic, with $T^2=\operatorname{id}$; ev…

Equivariant-bijection conjecture. Assuming the $m=1$ case of the preceding conjecture, there exists a $\mathbb Z/2h\mathbb Z$-equivariant bijection

Panyushev's order-divisibility conjecture. The order of $C$ always divides $2h$.

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…

Athanasiadis's Fuss–Narayana h-vector conjecture. The $m$-generalized Narayana numbers form the $h$-vector of an $(n-1)$-dimensional Cohen–Macaulay simplicial complex.

Let $\Phi$ be a crystallographic root system of rank $\ell$, let $m\geq 1$, and let $\Delta^m_+(\Phi)$ be the positive part of the generalized cluster complex, with $N^+(\Phi,m-1)$…

Let $\Phi$ be an irreducible crystallographic root system of rank $\ell$, let $m\geq 1$, and let ${\mathcal A}^m_\Phi$ be its generalized Catalan hyperplane arrangement. Let…

The covering-polynomial conjecture. The covering polynomial of $\mathfrak{Ad}_0$ satisfies either, equivalently,

Let $p,q,t\in\mathbb C$ satisfy $M<|t|^{n+1}<1$. For fixed $w\in\mathbb T^n$, let $C_w$ be a contour inside the annulus … for infinitesimally small positive $\epsilon$, with the po…