11 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 $G$ be a simple simply connected complex algebraic group of simply-laced type $A$, $D$, or $E$, and let $P$ be a parabolic subgroup with unipotent radical $N_P$. Let…

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…

Finite-sequence partial flag quantum parameter limit conjecture. The conjecture predicts the limiting ratios of quantum parameters at the terminal positions of finite flag sequence…

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…

Infinite partial flag Toeplitz approximation conjecture. There exists a sequence $u^{(n)}\in X_{P_{(\mathbf n,\mathbf m)}^{(n)}}(\mathbb R_{>0})$ converging to $u^{\infty}$. This c…

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 $T$) be a maximal torus in $\operatorname{GL}_n$. For $j=1,\dots,k$, let … be formal variables, and set $X^{(k+1)}=(T_1,\dots,T_n)$ to be the equivariant parameters in…

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…

Geiß–Leclerc–Schröer conjecture. The localization of ${\mathbb C}[G/P_K^-]$ by the non-minuscule minors equals the corresponding localization of $\widehat{\mathcal{A}}$:

Galashin–Karp–Lam conjecture. The totally nonnegative part of ${\mathcal P}_K$ is a regular CW complex, and $Q_K$ is its face poset.