6 problems

Let $G$ be a simple simply connected complex algebraic group of simply-laced type $A$, $D$, or $E$, let $J$ be a subset of the simple roots, and let $K$ be its complementary subset…

Let ${\bf a}=1<a_1<a_2<\cdots<a_r<a_{r+1}=n$, and let $(\alpha^1,i_1),\ldots,(\alpha^s,i_s)$ be Grassmannian Schubert data for the partial flag manifold $\mathbb{F}\ell_{\bf a}$, w…

Let $(\mathbf n,\mathbf m)$ be as above, and let $\boldsymbol\alpha_{\mathbf n}$ and $\boldsymbol\beta_{\mathbf m}$ be positive sequences satisfying the stated conditions and summa…

Partial flag Toeplitz limit conjecture. There exists a sequence $u^{(n)}d4d9\text{in}X_{P_{d4c3}^{(n)}}(\mathbb R_{>0})$ converging to $u^{d4c4}$; conversely, every infinite totall…

Let $G$ be an algebraic group, let $P\supset B$ be a parabolic subgroup, and let $\mathsf{k}[(G/P)(\mathbb{F}_q)]$ be the permutation representation over $\mathsf{k}$. Let…

Cluster-structure conjecture. The homogeneous coordinate ring equals the lifted cluster algebra: