108 problems

Let $q$ be a prime power, let $\sigma$ be an operator with $\sigma^n=\operatorname{id}$, and let $W_1$ and $W_2$ be $\sigma$-invariant subspaces satisfying … Write $h_{\sigma}(V)$ for…

Let $V=\bigoplus_i V_i$ be a finite-dimensional vector space, with each $V_i$ equipped with a non-degenerate symmetric bilinear pairing, and let $K\subset V$ be a subspace such tha…

Let $C\in\mathbb{C}^{d\times d}$ be a scaling matrix of rank four. A principal $4$-minor is non-vanishing when the corresponding principal submatrix has nonzero determinant. Rank-f…

Let $V_1,V_2,V_3$ be the local qudit vector spaces with dimensions $d_1,d_2,d_3$, and let $n_a$ be the corresponding one-party nullities. For a tripartite state…

Let $V_1,V_2,V_3$ be the local qudit vector spaces with dimensions $d_1,d_2,d_3$, and let $n_a$ be the corresponding one-party nullities. For a tripartite state…

Let $V_1,V_2,V_3$ be the local qudit vector spaces with dimensions $d_1,d_2,d_3$, and let $n_a$ be the corresponding one-party nullities. For a tripartite state…

Let $S(2n,\mathbb{R})$ denote the space of real symmetric $2n\times 2n$ matrices, let $J$ be the standard symplectic matrix, and let $\rho$ be the symplectic-group action on…

Let $U$ and $V$ be finite-dimensional vector spaces over a field $\mathbb{F}$. Let $\mathcal{S}$ be a linear subspace of $\mathrm{Hom}(U,V)$. We say that it is intransitive when…

Let $p$ and $q$ be monic polynomials of degree $2$, let $V$ be a vector space, and let $(b,u)$ be a symplectic pair on $V$. A symplectic $(p,q)$-difference is a symplectic pair sat…

Alternating-form symmetric-operator equality conjecture. There exists a maximal partially complete $b$-singular flag $\mathcal{F}$ of $V$ such that

Classify, up to standard equivalence, all matrices $H\in\mathbb{C}^{6\times 6}$ satisfying $|H_{ij}|=1$ for every $i,j$ and $HH^{*}=6I_{6}$. The conjecture asserts that every such…

Under Assumption, let $D$ be the diameter and let ${\tilde V}^{\downarrow \downarrow}_{ij}$, ${\tilde V}^{\uparrow \uparrow}_{rs}$, ${\tilde V}^{\downarrow \uparrow}_{ij}$, and…

Let $L$ be an $(n-k)$-dimensional linear subspace of $K^n$ whose tropical Plücker coordinates are all finite, and let $M\in K^{m\times n}$, with $m\geq k$, have nonzero rows in the…

Let $M_{d,d}$ be the generic $d\times d$ matrix over a field $k$, and let $I$ be the ideal of $d\times d$ permanents of the $d\times2d$ matrix $(M_{d,d}\mid M_{d,d})$. De Loera's r…

Let $A,A^*$ be a tridiagonal pair on a finite-dimensional vector space $V$ over an algebraically closed field $\mathbb K$. Let $d$ be its diameter, and let…

Assume $K$ is algebraically closed, $q\in\mathbb K$ is not a root of unity, $V$ is a finite-dimensional nonzero $K$-vector space, and $R,L:V\to V$ are nilpotent linear maps satisfy…

Let $r\geq 2$ and let $A\in \operatorname{Mat}_r(\mathbf Z)$ be nonsingular. Define $\gcd(A-I)$ as the greatest common divisor of the entries of $A-I$, and call $A$ primitive when…

Let $\mathbb{F}$ be a field, let $M$ be a nonsingular matrix over $\mathbb{F}$, and let $x$ be a vector over $\mathbb{F}$. Alon–Jaeger–Tarsi conjecture. For any field $\mathbb{F}$…

Johnson's conjecture. For every $n\ge 2$ and every Toeplitz matrix $A\in M_n(\mathbb{R})$ satisfying $A+A^{\top}=2J_n$,

Let $\mathfrak{g}$ be a finite-dimensional minimal-graph-admissible Lie algebra associated with a minimal graph $G(V,E)$, with $\dim(\mathfrak{g})\geq 2$, and let…

Xin–Zhang's determinant conjecture. The determinant of $C$ is

Let $A_1,B_1,C_1,A_2,B_2,C_2$ be finite-dimensional vector spaces over a field $\mathbb{K}$, and let $T_1\in A_1\otimes B_1\otimes C_1$ and $T_2\in A_2\otimes B_2\otimes C_2$. Thei…

Bethe-vector basis conjecture. For every $|\mathbf{x}\rangle\in\mathbb{V}$,

Let $(\sigma,\pi)$ be a reduced pair of compositions, and let ${\sf RSK}_{\sigma,\pi}$ be the associated RSK operator. A pair is triangular in the sense used in the paper. Non-inte…

Let $(\sigma,\pi)$ be a pair of compositions, let $\ell(\sigma)$ and $\ell(\pi)$ denote their lengths, and let ${\sf RSK}_{\sigma,\pi}$ be the associated RSK operator. Non-real eig…