3 problems

Let $R$ be a strongly discrete coherent ring. For $n\a0\geq 1$, let $R[X_1,\ldots,X_n]$ be a polynomial ring, and let $\prec$ be a rational monomial order on its monomials. For a f…

Mascia et al.'s conjecture. $\mathcal{P}$ is prime if and only if $\mathcal{P}$ contains no zig-zag walk.

Let $v,w$ be permutations with $v\leq w$, and let $I_{v,w}$ denote the Kazhdan–Lusztig ideal. Suppose that $I_{v,w}$ is standardly homogeneous. If $ws_i<w$ but $vs_i>v$, then…