8 problems

Knutson's conjecture. The variety $\overline{E}_\pi$ is defined as a scheme by the following three sets of equations and rank conditions: $XY$ and $YX$ are upper triangular;

Let $\omega$ be a partial permutation, and let $X_\omega\subset\mathfrak{M}_{m,n}(\mathbb{R})$ denote the corresponding open dense regular part of the real matrix Schubert variety.…

Fix a permutation $w$. Identify each partition-tuple $(\underline\lambda|\underline\mu)$ with a point in ${\mathbb R}^{m+n}$, and let … A lattice point means a point of the ambient…

Fix a permutation $w$, and let … Here $\underline\lambda$ and $\underline\mu$ are partition-tuples and $c^w_{\underline\lambda|\underline\mu}$ is the associated coefficient. Semigr…

Let $\mathfrak X_w$ denote the relevant combinatorial object, let $\prec'$ be the chosen term order, and let $\mathcal T(\mathfrak X_w,\prec')$ be its tablet; let the support of a…

Let $I_w$ be the relevant matrix Schubert ideal, let $\prec'$ be the lex-diagonal term order, and let $\widetilde{J}$ be the ideal obtained by polarizing…

Hamaker–Pechenik–Weigandt's conjecture. If the Schubert polynomial of $w$ is a multiplicity-free sum of monomials, then the CDG generators of $I_w$ are a diagonal Gröbner basis.

Hamaker–Pechenik–Weigandt's conjecture. The CDG generators are a diagonal Gröbner basis for $I_w$ if and only if $w$ avoids all eight patterns