9 problems

Let $u$ be a permutation. An affine Grothendieck--Lascoux expansion conjecture asserts that there exist positive integers $c_1,\dotsc,c_t$, nonnegative integers $e_1,\dotsc,e_t$, a…

Let $\mathbb N^n$ denote the set of compositions indexing the Lascoux polynomials $\mathfrak L_\alpha$, and let $i\in[n]$. Lascoux product-positivity conjecture. For every…

Reiner–Yong conjecture.

Set-valued tableau conjecture. We have

Let $lambda$ be any partition and let $w\in S_n^{\lambda}$. A flagged Grothendieck polynomial is understood in the sense used in the paper, and $w$ is 312-avoiding when it has no s…

Let $w$ be a permutation and $lambda$ a partition, and let $\operatorname{SLT}_{w\lambda}$ be the set of semistandard set-valued skyline tableaux of shape $w\lambda$. Define the La…

Let $a=(a_1,otsc,a_n)in\boldsymbol{Z}_{geq0}^n$. A K-Kohnert diagram is a subset $D$ of $\boldsymbol{Z}_{>0}^n$ together with a subset of marked boxes, obtained from the skyline di…

Let a K-crystal be the enriched crystal structure on semistandard set-valued tableaux whose Demazure-like subcrystals have Lascoux polynomials as characters. For a fixed rectangula…

Let $\alpha\in\mathbb Z_{\geq0}^{\infty}$, and let $\Omega_\alpha$ be the inhomogeneous analogue of the key polynomial obtained by replacing the Demazure operators with…