12 problems

Let $I_w$ be the relevant matrix Schubert ideal, let $\prec'$ be the diagonal term order, and write $V(\operatorname{init}_{\prec'}(I_w))$ for the variety defined by its initial id…

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;

Cohen–Macaulay conjecture. Each individual $E_\pi$ and each union

The Schubert-equation conjecture. The closure $\overline{E_\pi}$ is scheme-theoretically defined by the equations expressing the upper-triangularity of $XY,YX$, the diagonal relati…

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 $w$ be a permutation, and let $\underline\lambda$ and $\underline\mu$ be partition-tuples. For a positive integer $N$, write $N\underline\lambda=(N\lambda_1,N\lambda_2,\dots)$…

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. The CDG generators are a diagonal Gröbner basis for $I_w$ if and only if $w$ avoids all eight patterns

Primary decomposition conjecture. For every $y\in\mathcal{I}_n$, the ideal $\hat I_y$ is prime, and there is a primary decomposition of $\operatorname{\textsf{init}}(\hat I_y)$ who…