3 problems

Non-commutative $k$-Schur positivity conjecture. The coefficients $b_{w,v}$ are positive, namely

Let $\tilde{F}_w(X)$ be the affine Stanley symmetric function indexed by $w\text{,}$ and let $c^w_{u,v}$ be the affine Schubert structure constants defined by … For…

Lapointe–Lascoux–Morse conjecture. The $k$-Schur functions expand positively in terms of $(k+1)$-Schur functions; in particular, they expand positively in terms of ordinary Schur f…