30 problems

Panyushev's irreducibility conjecture. If the rank of the symmetric pair $(\mathfrak g,\mathfrak g_0)$ is greater than $1$, then $\mathfrak C(\mathfrak g_1)$ is irreducible.

GLS criterion. For a multisegment $\mathfrak{m}$, $Z(\mathfrak{m})$ is square-irreducible if and only if the GLS condition is satisfied.

Irreducibility and component-count conjecture. The varieties $P_q$ in the preceding proof are irreducible, and the number of irreducible components of $\mathfrak C(\mathfrak g_1)$…

Let $C_n$ be the commuting scheme and let $F$ be the actual flat limit of its degeneration described in the source. Let $F_+$ be the upper-bound scheme defined by the displayed tri…

Minor-equation conjecture. The equations given by the vanishing of every size-$n$ minor of this matrix define the other component or components of the diagonal commutator scheme.

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 ${\mathfrak g}$ be a reductive Lie algebra over ${\mathbb C}$, let ${\mathfrak h}$ be a Cartan subalgebra, and define the diagonal commutator scheme … The two-component conject…

Radicality implication. The conjecture that the ideal $J$ coincides with its radical $\sqrt{J}$ at the level of $\mathfrak{g}$-invariants would imply

Let $\mathcal{A}'_c$ and $\mathcal{A}_c$ be the cuspidal and Catalan dg modules, respectively, and let $p'$ and $p$ be the corresponding maps to the commuting variety and its Hilbe…

Let $(m,n)\in\mathbb{Z}\times\mathbb{N}$ be coprime. Let $\mathcal{H}_{m,n}$ and $\mathcal{H}'_{m,n}$ be the derived objects obtained from the ordinary and eccentric flag commuting…

Let $n\in\mathbb{N}$, let $(m,n)\in\mathbb{Z}\times\mathbb{N}$ be coprime, and let $d\geq 1$. For the equivariant derived category…

Single-relation conjecture. The coordinate rings satisfy

Let $\alpha=(K(a_1),\dots,K(a_d))$ be a pure-$K$ leading term datum, and let $k$ be any field. Consider the affine variety … defined by … and by the vanishing conditions $X_{ij}=0$…

Let the partial resolution of the commuting variety constructed in the paper be the variety under consideration. Local terminalization conjecture. In general, although this variety…

Let $k$ be a field, let $n\geq 4$ and $r\geq 4$, and write $\mathfrak{gl}_n$ for the Lie algebra of $n\times n$ matrices. Let $C_r(\mathcal{N}(\mathfrak{gl}_n))$ denote the variety…

Let $\mathfrak{m}$ and $\mathfrak{m}'$ be multisegments satisfying the condition referred to in the source as … , (1) $\operatorname{SA}(\mathfrak{m},\mathfrak{m}')$ and…

Let $V$ and $V'$ be finite-dimensional graded complex vector spaces, and let $\mathfrak{C}_{\mathfrak{m}}\subset\mathfrak{X}(V)$ and…

Let $V$ and $V'$ be finite-dimensional graded complex vector spaces, and let $\mathbb{G}(V)$ and $\mathbb{G}(V')$ be their groups of grading-preserving automorphisms. For multisegm…

Let $G$ be a reductive group with Lie algebra ${\mathfrak g}$, let ${\mathfrak t}$ be a Cartan subalgebra, let $W$ be the Weyl group, and let ${\mathfrak C}^d_G$ be the scheme of c…

Let $\mathfrak g$ be a complex semisimple Lie algebra, and consider the variety of commuting nilpotent pairs in $\mathfrak g$. Baranovsky's equidimensionality conjecture. This vari…

Monotonicity conjecture. If $C_r(\mathcal{N}_n)$ is reducible, then $C_r(\mathcal{N}_{n+1})$ is reducible.

Parabolic component conjecture. The variety $V_P$ is an irreducible component of $C_r(\mathcal{N}_n)$ for all $n,r\ge 4$.

Irreducibility implication. If $C_r(\mathfrak{gl}_n)$ is irreducible, then $C_r(\mathcal{N}_n)$ is irreducible.

Analogous reducibility conjecture. The reducibility behaviors of $C_r(\mathfrak{gl}_n)$ and $C_r(\mathcal{N}_n)$ should be similar.