21 problems

Let $\eta$ be a braid word and let $X(\eta)$ denote the corresponding braid variety. A cluster algebra structure conjecture asserts that the coordinate ring of any braid variety…

A positroid is a matroid arising from a cell of the totally positive Grassmannian. A matroid base polytope is Ehrhart positive when all coefficients of its Ehrhart polynomial are n…

Maximum-size conjecture. Every 3-connected rank-$r$ positroid with no $U_{2,\ell+2}$-minor has at most

Realizability conjecture for orthopositroids. Every orthopositroid $\mathscr{M}$ of type $(k,n)$ is realizable: there exists $X\in\operatorname{OGr}_{+}(k,n)$ such that…

Positroid Ehrhart-positivity conjecture. Every positroid is Ehrhart positive.

Mcalmon–Oh–Xiang's conjecture. $M'\trianglelefteq_Q M$ if and only if every CCW-arrow of $M$ is the union of CCW-arrows of $M'$.

Let $Z_{\hat{G}(\mathcal{T})}$ be a positroid tile of the amplituhedron $\mathcal{A}_{n,k,2}(Z)$. A facet is a codimension-one boundary piece, and each facet lies on a hypersurface…

Cyclic c-equivalence conjecture. The set

Gorsky–Hogancamp–Simental–Rasmussen conjecture. The knots $\hat\beta(\mathbf{a}(f))$ and $\hat\beta_f$ are isotopic. Up to a monomial in $q$ and $t$,

Let $\Pit_{k,n}$ be the torus quotient and let $J_{k,n-k}$ be the compactified Jacobian of the plane curve singularity $x^k=y^{n-k}$. Their cohomology carries a weight filtration a…

Let $\sigma$ and $\pi$ be positroids on $[n]$ of ranks $k-1$ and $k$, respectively. For $A\subset[k]$, let $\overleftarrow{\rho_A}(\pi)$ denote the positroid obtained from $\pi$ by…

Let $P_n$ be the poset of positroids on $[n]$, ordered by the quotient relation. For $\tau1,\tau2 otin Pn$? Let $\tau1,\tau2 otin Pn$ be such that $\tau1 ottrianglelefttau2$. Let…

Positroid stratification conjecture. The positroid stratification of $Gr^{\geq 0}(k,n)$ is a regular CW-complex. In particular, the closure $\overline{\Pi}_{\mathcal{M}}$ of each p…

Let $f\in\tilde{S_n}(-k,n-k)$ be an affine permutation with $\operatorname{inv}(f)=k\ell$. A cell $f$ is $(n,k,m)$-admissible when it satisfies the source's admissibility condition…

Cluster structure conjecture. The coordinate ring of an open positroid variety has a cluster structure obtained from the quiver associated to any of its reduced plabic graphs.

Maximal-cell conjecture. Every nonempty Grassmann polytope has a maximal cell.

Facet equality conjecture.

Let $w\in S_n$. Let $DM_w$ be the diagram matroid associated to the permutation diagram of $w$, and let $\mathscr{P}_w$ be the corresponding positroid. Tutte-polynomial equality co…

Let $w\in S_n$, and let $DM_w$ be the diagram matroid of $w$, obtained from a generic matrix $[A\mid I_n]$ whose entries satisfy $A_{ij}=0$ whenever $(i,j)\in D(w)$, where … For a…

Let ${\mathcal{M}}$ be a positroid, and let an ordered pair $(I,J)$ of $k$-element subsets be ${\mathcal{M}}$-standard if $I=\pi_k(x)$ and $J=\pi_k(y)$ for some…

da Silva's conjecture. Every positively oriented matroid is realizable.