19 problems

Let $\alpha$ and $\gamma$ be weak compositions, with $\alpha$ skew-symmetric. Say that $\kappa^\mathsf{P}_\alpha-\kappa_\gamma$ is key positive when it is a nonnegative linear comb…

Let $\alpha$ be a symmetric weak composition, and let a P-key polynomial mean a polynomial of the form $\kappa^\mathsf{P}_\eta$ for a skew-symmetric weak composition $\eta$. Conver…

Let $I_\infty$ denote the involutions of the infinite symmetric group, let $\mathfrak S^{\mathsf{O}}_z$ be the corresponding orthogonal involution Schubert polynomial, and let…

Let $\alpha$ be a skew-symmetric weak composition, let $n=\ell(\alpha)$, and let $\widetilde\rho(\alpha)$ and $\widetilde\gamma(\alpha)$ denote its strict sub-diagonal row and colu…

Let $\alpha$ and $\gamma$ be distinct symmetric weak compositions. Injectivity conjecture. Then … This conjecture is supported by computations, although the authors report no heuri…

Atom-positivity conjecture. The row operator applied to a key polynomial is atom-positive:

Key-polynomial operator conjecture. For any composition $\alpha$,

Poset-type conjecture. The multiplicity $m^{k,l}_{\eta}(w)$ depends only on the poset type of the set of sums $\alpha+\beta=\eta$, with $\alpha\in A_l(w)$ and $\beta\in A_k(w)$. Mo…

Let $n$ be a positive integer, let $w\in S_n$ be a permutation, let $\zeta$ be a weak composition, and let $\boldsymbol{x}=(x_1,\ldots,x_n)$. Write…

Vexillary involution crystal conjecture. There is an isomorphism of $\mathfrak q^+_n$-crystals

Involution Schubert expansion conjecture. Each involution Schubert polynomial $\mathfrak S^{\mathsf{Sp}}_z$ (respectively, $\mathfrak S^{\mathsf{O}}_z$) is an $\mathbb{N}$-linear c…

Let $b[?]$ be a valuation and $Q$ an abstract key polynomial (ABKP) for $b[?]$. For an associated diskoid $b?)$, let $b[?]_{b?)}$ denote the induced function on $b[?][X]$. An RT ex…

Let $b[?]$ be a valuation on $K[X]$ and let $Qb[?]K[X]$ be an abstract key polynomial for $b[?]$, with $b?b[?]b[?]b?$. A diskoid is a subset $b?b[?]b[?]$ whose polynomial-value min…

Let $\alpha\in\mathbb{N}^n$ be a composition, and let $\kappa_\alpha(x_1,\ldots,x_n)$ be its key polynomial. Key-polynomial conjecture. The normalized polynomial … is Lorentzian fo…

Let $a$ and $b$ be weak compositions, and let ${\mathfrak{D}}_a$ and ${\mathfrak{D}}_b$ denote the corresponding key polynomials. Let ${\mathfrak{A}}_c$ denote the Demazure atom in…

Positive key expansion conjecture.

For compositions $\gamma$ and $\alpha$, write $\gamma\preceq\alpha$ when $\lambda(\gamma)=\lambda(\alpha)$ and $w(\gamma)\leq w(\alpha)$ in Bruhat order, where $\lambda(\gamma)$ is…

Let $\alpha$ be a composition, let $t_{ij}$ swap positions $i$ and $j$, and let $m_{ij}(\alpha)=\alpha+e_i-e_j$. Define $\beta<_\kappa\alpha$ when $\beta$ can be generated from…

Let $\alpha=(\alpha_1,\alpha_2,\ldots)\in\mathbb Z_{\geq0}^{\infty}$ have finite size, and let $\kappa_\alpha$ be the key polynomial defined using Demazure operators. A polynomial…