8 problems

Existence conjecture. Skew-symmetric conference matrices exist for every order divisible by four.

Let $X$ be a real skew-symmetric $n\times n$ matrix, and let $\mathcal{V}_{X^2}$ denote the variety of symmetric matrices arising as squares $X^2$. Generation-degree conjecture. Th…

Let $Z\in\mathbb{Q}^{n\times n}$ be a nonzero quaternion skew-symmetric matrix. For $n>2$, call $Z$ basic if there is no unitary matrix $U$ such that $UZU^*=\Sigma$, where $\Sigma$…

For $n>k\geq 1$ and $x\in\mathbb{R}$, let $A(n,k,x)$ be the $n\times n$ skew-symmetric Toeplitz matrix whose first $k$ superdiagonals have entries $1$ and whose remaining superdiag…

The colon ideal conjecture. If $n$ is odd, then $\Delta_{(n)n}\neq 0$. For every $n\geq 4$,

Let $A$ and $B$ be $n\times n$ skew-symmetric real matrices. Two such matrices have equal corresponding principal minors of all orders if, for every subset $X\subseteq [n]$, the pr…

Let $V$ be a vector space of dimension $2c+2$, and let $A\subseteq \wedge^2 V$ be a linear space of skew-symmetric matrices of constant rank $2c$, equivalently of co-rank $2$. In t…

Fomin–Shapiro–Thurston's conjecture. Besides adjacency matrices of triangulations of bordered two-dimensional surfaces and matrices mutation-equivalent to one of these nine types,…