11 problems

Vigneras's conjecture. This parametrisation depends only on $l$, and not on the coefficient ring $R$.

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…

Type $C_n$ charge conjecture. The identity $$ holds when $chin^C$ is replaced by $mathrm{ch}{Cn}$, for all partitions $lambda$ and $mu$.

Type $C_n$ cyclage-statistic conjecture. This statistic yields an analogue of Lascoux–Schützenberger's theorem.

Let $\lambda$ be a partition, and let $\mathrm{Skel}(\lambda)$ denote the skeleton associated with a connected crystal whose associated Young tableaux have shape $\lambda$. Let the…

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…

For an odd-degree cover of the split odd orthogonal group $SO_{2r+1}$, consider the $p$-power contributions to the Whittaker coefficients of the minimal parabolic Eisenstein series…