8 problems

Let $\lambda=(\lambda_1,\lambda_2)\vdash n$ with $\ell(\lambda)=2$, and set $l_\lambda=\max\{\lambda_1,\lambda_2+1\}$. For $l_\lambda\le m\le n$, let $\mathsf{Par}(\lambda;m)$ be t…

Let $\mathrm{Int}(n)$ be the set of nonempty left weak Bruhat intervals in $S_n$. For $I,J\in\mathrm{Int}(n)$, write $I\mathrel{\Deq}J$ if there exists a descent-preserving poset i…

Indecomposability conjecture. If $\sh(\tau_P)$ is disconnected and contains no disconnected ribbon, then the associated module $\sfM_P$ is indecomposable.

Module-isomorphism conjecture. For $P,Q\in\sfRP_n$, if

Let $\lambda$ be a partition with $\ell(\lambda)\le 2$, and let $\mathcal{E}_{\lambda;m}$, $\mathsf{Par}(\lambda;m)$, $\mathbf{G}_E$, and $X_\mu$ be the sets and $0$-Hecke modules…

Let $L$ be a meet semi-lattice with a fixed linear extension, and let $D$ range over the valid diagrams of $L$, meaning subsets containing the minimal elements and stable under joi…

Universal-word conjecture. The word $u_N=(1,2,\ldots,N-2,N-1,N-2,\ldots,2,1)$ is universal. This conjecture proposes a uniform, simple expression whose signed versions produce demi…

Universality conjecture. The word $u_N$ is universal.