36 problems

da Silva's conjecture. Every positively oriented matroid arises from a positroid in this way.

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…

Let $[n]=\{1,2,\ldots,n\}$, and let $M[(S_1,S_2,\dots,S_k)]$ be the transversal matroid presented by subsets $S_1,S_2,\dots,S_k\subseteq[n]$. A set system $(S_1,S_2,\dots,S_k)$ is…

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…

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…

For parameters $k,n,r,s$, define the cuspidal matroid polytope … Such polytopes are particular cases of positroids. A lattice polytope is Ehrhart positive when all coefficients of…

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 $M'$.

Fix $I\in\binom{[2,n-1]}{k}$, and let $Z_I$ be the closure of the region $Z_I^{\circ}$ in the amplituhedron $\mathcal{A}_{n,k,2}(Z)$ where the sequence of brackets…

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…

Let $\Delta_{k+1,n}$ be the hypersimplex, and let $\operatorname{Gr}_{k+1,n}^{\geq 0}$ and $\operatorname{Gr}_{k,n}^{\geq 0}$ be the totally nonnegative Grassmannians. A positroid…

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 ${\mathcal{M}}$ be an arbitrary positroid, let $J \in {\mathcal{M}}$, and let a cluster ${\mathcal{C}} \subset {\mathcal{M}}$. A Laurent polynomial is positive if every monomia…

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 $(\mathcal{P},[n])$ be a Wilson loop diagram, with associated positroid cells of dimension $3|\mathcal{P}|$, and let the Amplituhedron parametrize positroid cells of dimension…

Let $W$ be a Wilson loop diagram with associated positroid cell $\Sigma(W)$, and let $R(W)$ denote the denominator of its differential form $\Omega(W)$. The zeros of $R(W)$ produce…

Let $\mathcal{B}$ be a transversal positroid. For $a\in[n]$, write $<_a$ for the cyclic order beginning at $a$, and order equal-sized sets by the corresponding $a$-Gale order: if…

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.

A Wilson loop is a planar diagram associated with Grassmannians, and positivity refers to the relevant Grassmannians having positive Plücker coordinates. Positivity conjecture for…

Amplituhedron triangulation conjecture. The set $C(k,n)$ gives a triangulation of the amplituhedron.