7 problems

Let $\Phi=G_2$. For a dominant weight $\theta$, let $\chi_\theta$ be the character of the irreducible representation $V_\theta$ of $G_2$ of lowest weight $-\theta$, shifted to be a…

Let $B_{\boldsymbol{\pi}'}$ and $B_{\boldsymbol{\pi}}$ be the crystals embedded as in equation, and let Theorem denote the stated agreement between the combinatorial realization an…

Let $C$ be a connected component of the graph given by the modified operators $\phi_i$, and call a hexagon broken by $\phi_i$ after $\phi_j$ when it has the broken-hexagon property…

No broken diamond in hexagon conjecture. The paths $E_i$ and $E_{i+1}$ in the graph induced by the modified crystal operators do not contain any edges that give way to Stembridge E…

Let $K=\mathbb{C}$ and let $\Pi=\Pi(C,D)$. Define the support of a constructible function $f\colon \operatorname{nil}_E(\Pi,d)\to\mathbb{C}$ by … Let…

Let $\lambda\in\mathcal{P}$, let $t\in\mathbb{N}$ satisfy $\lambda_{(2)}=\Delta_t$, and set $\mu=\bar{\lambda}^{(2)}$. Assume that $\ell$ is sufficiently large, that $e$ is odd, an…

Let $t$ be a non-negative integer, set $r=t(t+1)/2$ and $\iota=r\pmod 2$, and let $\widetilde{\mathcal{G}}_{\iota,q,\ell}^{t,\leq d}$ be the induced subgraph of the reindexed Haris…