93 problems

Product formula conjecture. If the representation is bounded, then

Let $(V,c)$ be a braided vector space of diagonal type over an algebraically closed field of characteristic zero, and let $\mathfrak{B}(V)$ denote its Nichols algebra. The finite G…

Let $\Delta$ be a simply-laced root system with highest root $\theta$, and let $\theta$ be fundamental. For a simple root $\alpha$, let $\EuScript F_{\alpha}$ be the associated min…

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…

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…

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…

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…

Let $\Phi$ be a finite root system of rank $n$. Define … where $h_{k,\ell}$ counts antichains of cardinality $k$ containing $\ell$ simple roots. Let…

Let $\Phi$ be a finite root system of rank $n$, with positive roots ordered by root order. A full root is a positive root whose support is the full Dynkin diagram, and an antichain…

Let $\Phi$ be a finite root system of rank $n$, with positive roots $\Phi_{\geq 0}$ ordered by root order. An antichain is a set of pairwise incomparable positive roots. For an ant…

Edelman–Reiner conjecture. The following arrangements are free:

Let $N$ be a positive integer, let $t$ and $t_1,\dots,t_{n+3}$ be parameters, and suppose that … For an integer $\lambda$, write $\theta(a)_\lambda=\theta(a;p;q)_\lambda$. The term…

Let $n$ be a positive integer. Let $z_i$, $i=1,\dots,n$, $t_k$, $k=1,\dots,n+1$, and $f_j$, $j=1,\dots,n+2$, be independent complex variables, and set … With…

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 $\Delta$ be the root system under consideration, let $I$ index the complete flag $\mathcal{S}_\bullet^1$, and for each $i$ with $1\leq i\leq |I|$ define … The roots…

Colored-mutation root-count conjecture. The number of $\Gamma$-mutations in one period is the total number of roots in the root systems associated to the $\Gamma$-components,…

Let $(\Gamma, \Delta)$ be an admissible Dynkin biagram with $r$ vertices. Let $h_{\Gamma}$ and $h_{\Delta}$ denote the Coxeter numbers associated to the respective components, and…

Let $(W,S)$ be a Coxeter system with associated root system $\Phi$, positive roots $\Phi^+$, and reflection $s_\alpha$ corresponding to each $\alpha\in\Phi^+$. Let…

Indecomposable-root conjecture. The root $\check\eta_w(k)$ is indecomposable in $\check R^+(w)$. In particular, the set

Root-order conjecture. If $\check\alpha\in\check R^+(w)$ is decomposable, then there exists a decomposition

Let $\mathcal R=(\Delta,S^p,\Sigma,\Gamma)$ be a complete spherical skeleton, where $R$ is the underlying root system, $R^+$ is its set of positive roots, and $R^+_{S^p}$ is the se…

Optimality conjecture. In the theorem, all stated inequalities for orders of vanishing are actually equalities. The theorem establishes the displayed lower bounds for compact simpl…

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…