127 problems

Let $xi_{\ell,r,c}$ denote the integer in the reduced $Gr_2$-simplex in layer $\ell$, row $r$, and NE-SW diagonal $c$, and set $\xi_{\ell,r,c}=0$ when there is no entry in that pos…

Arrange the entries in each layer of the $Gr_2$-simplex as coefficients of a polynomial in $a,b,c$ according to the arrangement described in the paper. Divisibility conjecture. Eac…

Consider the Grassmannian $\limfunc{Gr}_{n,N}$ and the ring … For partitions ${\Greekmath 0115}$ and ${\Greekmath 0116}$ indexing opposite Schubert varieties and fixed points, let…

Let a surface in ${\rm Gr}(2,4)$ be defined by generic polynomials of degrees $d_1,d_2$, and let a threefold in ${\rm Gr}(2,4)$ be defined by a generic polynomial $f$ of degree $d$…

Integral cohomology and non-isomorphism conjecture. One has an isomorphism of integral cohomology groups

Braid-group fixed-point conjecture. For every $i\in[d-1]$, if $(k,n)$ is $(3,9)$ or $(4,8)$, every unstable fixed point of $\sigma_i$ in $\mathbb{C}[\operatorname{Gr}(k,n)]$ is a c…

Let $\mathcal{A}_{n,k,m}(Z)$ be the amplituhedron in $\operatorname{Gr}_{k,k+m}(\mathbb{C})$. The amplituhedron positive geometry conjecture. The pair … should be a positive geomet…

Dense-orbit and finite-orbit equivalence. Unless

Compatibly split positroid conjecture. The only compatibly split irreducible subvarieties of $\mathrm{Gr}(k,n)$ are the closed positroid varieties.

Let $T\in\operatorname{SSYT}(k,[n])$ be a tableau whose distinct entries are $a_1<\ldots<a_r$. Let $n'\ge n$ and let $f:\{a_1,\ldots,a_r\}\to[n']$ satisfy $f(a_1)<\ldots<f(a_r)$; w…

For positive integers $k,n,r$, let $N_{k,n,r}$ denote the number of cluster variables of rank $r$ in $\mathbb{C}[\operatorname{Gr}(k,n)]$. Cluster-variable counting conjecture. The…

Let $\mathbb G(r,n)$ denote the Grassmannian of $r$-planes in projective $n$-space, and let $h$ be a positive integer. A projective variety is called $h$-defective when its $h$-sec…

Let $m,p>1$, let $\operatorname{Grass}(p,m+p)$ be the Grassmannian, and let $\alpha^1,\ldots,\alpha^n$ be Schubert data. A real upper-triangular matrix with diagonal entries $1$ is…

Mirror-family conjecture. The pair $M_y+W_y$ is the mirror family of $X_A+Y_A$, with $M_y$ and $W_y$ forming different parts of the B-model moduli space and $X_A$ and $Y_A$ forming…

Let ${\mathcal G}_k$ be the Grassmannian of $k$-dimensional singular subspaces of a building of type $\textsf{X}_{n}=\textsf{C}_{n},\textsf{D}_{n}$, and let ${\mathfrak G}_k$ be it…

Let $\mathbf{G}$ be the Grassmannian in its Plücker embedding, let $\mathbf{Pf}$ be its projective-dual Pfaffian variety, and write $\mathbf{P}^*$ for the dual projective space. A…

The Schubert calculus conjecture. For generic $z=(z_1,\ldots,z_n)$, the intersection of Schubert varieties $\mathcal{I}_{\{{\bf w}\}}(z)$ is transversal. Here generic means that…

Chow quotient and log canonical model conjecture. The inclusion $X(r,n)\subset \overline{X}(r,n)$ is the log canonical model precisely in the cases

Let $[1\dots n]$ be labeled cyclically, and let a non-crossing family be a family of pairwise non-crossing $k$-subsets, where non-crossing means that no chord with endpoints in…

The $e_m$-overkiller conjecture. For $3\leq m\leq k$,

The exact $e_m$-killer conjecture. For $1\leq m\leq k$,

The filtration-saturation conjecture. For $1\leq m\leq k$,

The filtered Hilbert-series conjecture. For $m\geq 0$,

For integers $k$ and $n$ in the range where the polynomial $\hat{E}_{k,n}(q)$ is defined, let $[q]\hat{E}_{k,n}(q)$ denote the coefficient of $q$ in $\hat{E}_{k,n}(q)$. Linear-coef…

Let $d\geq 1$ be an integer, and let $\lambda$, $\mu$, and $\nu$ be Young diagrams indexing Schubert classes in the Grassmannian. Denote by $\langle\lambda,\mu,\nu\rangle_d$ the co…