17 problems

Let $X_w^v$ be a Richardson variety in the Grassmannian, let $G_{k,n,\ell}|_w^v$ be the corresponding restricted set of relations, and let…

Regularity conjecture for tilted Richardson cells. The CW-complex in the preceding stratification is regular.

Positive-part closure conjecture. The closure of the positive part equals the nonnegative part:

Let $W$ be the Weyl group of a Kac–Moody group, let $v,w\in W$ satisfy $v\leq w$, and let $\textnormal{\textbf{w}}$ be a reduced expression for $w$. Denote by…

Let $G/B$ be the flag variety of a reductive group $G$ over $\mathbb{C}$, with Weyl group $W$. For $u,v\in W$, let $X(u)^\circ$ and $Y(v)^\circ$ be the Schubert and opposite Schube…

Let $v\leq w$ and let $\tau_{v,w}:\mathring{\mathcal{R}}_{v,w}\to\mathring{\mathcal{R}}_{v,w}$ be the right twist map. A quasi-cluster transformation means a sequence of cluster mu…

Isomorphism conjecture. If $X_\theta\cong X'_{\theta'}$ as varieties, then $P(\theta)\sim P(\theta')$.

Let $G=\operatorname{SL}_n(\mathbb{C})$, let $v\leq w$ be elements of the Weyl group $W$, and let $Q_{\operatorname{Lec}}$ and $Q_{\operatorname{Ing}}$ be the quivers constructed f…

Let $G$ be a semisimple algebraic group, let $N$ be its maximal unipotent subgroup, and let $\Rich_v^w$ be the open Richardson variety associated to $v\leq w$ in the Weyl group. Fo…

Let $v,w\in\mathfrak{S}_{n+1}$ with $v\leq w$, let $X_w^v$ be the Richardson variety, and let ${\mathsf{Q}}_w^v$ be its Bruhat interval polytope with vertices indexed by the fixed…

Refined open Richardson conjecture. The open Richardson variety $R^u_v$ is a cluster variety satisfying the Louise property and of full rank.

Folklore cluster algebra conjecture. The coordinate ring $\mathcal{O}(R^u_v)$ is a cluster algebra.

Let $G$ be a semisimple algebraic group with Weyl group $W$, and let $v,w\in W$ satisfy $v\leqslant w$. Let $\mathcal{R}_{v,w}$ be the corresponding Richardson stratum, and let…

Type D Richardson-clan conjecture. The statement of the type $B$ Richardson-clan conjecture also holds in type $D$.

Type B Richardson-clan conjecture. With these definitions, in cases (1)–(3), $X_u^v$ is one of the two $L$-orbit closures associated to $\gamma(u,v)$ and $X_{w_0v}^{w_0u}$ is the o…

Trapa's conjecture. In type $B$, all such Schubert constants $c_{u,v}^w$ are $0$, $1$, or $2$, with $c_{u,v}^w=0$ only if $w\cdot\gamma(u,v)\ne\gamma_0$; $c_{u,v}^w=2$ only if…

Let $X_u^v=X_u\cap X^v$ be a Richardson variety in $GL_n/B$, let $wB\in X_u^v$, and let the tangent cone at $wB$ be defined by the associated graded local ring of $X_u^v$ at that p…