14 problems

Let $X$ and $Y$ be generic $n$ by $n$ matrices, let $I$ be the ideal generated by all entries of $XY-YX$, and let $J$ be generated by the off-diagonal entries. The determinants for…

Let $X$ and $Y$ be generic $n$ by $n$ matrices, let $J$ be the ideal generated by the off-diagonal entries of $XY-YX$, and consider determinants of matrices whose columns are the d…

Let $I$ be the ideal generated by the entries of $XY-YX$ for generic $n$ by $n$ matrices, and let $J$ be generated by the off-diagonal entries. Let $d_{\min}$ be the quantity from…

Let $X$ and $Y$ be generic $n$ by $n$ matrices, let $I$ be the ideal generated by the entries of $XY-YX$, let $J$ be the subideal generated by the off-diagonal entries, and let $E$…

Let $I$ be the ideal generated by the entries of $XY-YX$ for generic $n$ by $n$ matrices $X$ and $Y$, and let $k$ denote the syzygy degree in the displayed Betti-number convention.…

Let $X$ and $Y$ be generic $n$ by $n$ matrices, let $Z=XY-YX$, and write $I=(f_1,\ldots,f_{n^2})$ for the ideal generated by the entries of $Z$. A syzygy on $I$ is an $n^2$-tuple…

Let $R$ be the quotient ring associated with two generic commuting $n$ by $n$ matrices, as above. Domain conjecture. The ring $R$ is a domain. The source says this follows from Coh…

Let $Q=(u,u-r)$ be a stable partition, with $r\geq 2$ and $u-r\geq 1$, and let $\mathcal{Q}^{-1}(Q)$ denote the set of partitions having generic commuting partition $Q$. Oblak's co…

Let $A=k\langle x^{\pm1},y\rangle$ and consider the Koszul complex for the relation $[x,y]=0$, where $x$ is represented by an invertible matrix. Denote the matrix-entry variables b…

Let $A=k\langle x,y\rangle$ and let $\mathcal{K}_n(A)$ be the Koszul complex for the commuting-matrix relation $[x,y]=0$. Write $x_{uv},y_{uv}$ for the matrix-entry variables,…

Let the moduli space of pairs of commuting matrices over $\mathbb{C}$ be the scheme parametrizing pairs $(\Sigma,\Phi)$ of commuting complex matrices. Commuting-matrix conjecture.…

Let $K$ be a field, let $g$ be a nonzero polynomial in the noncommuting indeterminates $u_1,\ldots,u_{k+1}$ such that … for every $i\leq k$, the indeterminate $u_i$ appears exactly…

Let $P$ be a partition, let $\mathcal U_B$ be the relevant space of nilpotent matrices commuting with a fixed nilpotent matrix of Jordan type $P$, and let $A$ be an adequate matrix…

Let $A$ be a cyclic group of order $m$. For $\boldsymbol{\pi}=(\pi_1,\ldots,\pi_m)\in(\mathbf{Z}/m\mathbf{Z})^m$ with $\pi_1\ge\cdots\ge\pi_m$, an element $x\in A$ is…