7 problems

Let $S_n$ be the symmetric group, let $[ij]$ be the generators of the Fomin–Kirillov algebra, and let $\theta_i$ be the Dunkl elements … Let $\mathfrak{S}_w$ be the Schubert polyno…

Let $\EuScript{E}_m$ be the Fomin–Kirillov algebra over a field $\mathbf{k}$ associated with the symmetric group $\mathbb{S}_m$. Fomin–Kirillov's conjecture. The algebra…

Let $n$ be a positive integer, let $\widetilde{S}_n$ be the affine symmetric group, and let $\widetilde{FK}_n$ be the affine Fomin–Kirillov algebra. Assuming the affine Schubert el…

Let $\mathcal{E}_n$ be the Fomin–Kirillov algebra generated by $\mathbf{x}_{ab}$ for $1\leq a<b\leq n$, and define the commuting Dunkl elements … Let $\mathfrak{S}_w$ denote the Sc…

Cyclotomic Hilbert-series conjecture. The Hilbert series $\mathcal H_G(t)$ is a product of cyclotomic polynomials.

Six-vertex infinite-dimensionality conjecture. The algebra $\mathcal E_G$ is infinite-dimensional if and only if $G$ contains a subgraph isomorphic to one of the graphs shown in Fi…

Melnikov–Sottile conjecture. The bilinear form