9 problems

Let $P$ be a $(\mathbf{3}+\mathbf{1})$-free poset, let $\lambda\vdash n$, and let $m_\lambda^P(\mathbf{u})$ be the relevant noncommutative elementary coefficient in…

Non-negativity conjecture. The coefficients

Lam's strengthened square-free positivity conjecture. For every partition $\lambda$, $\mathfrak{J}_\lambda(\mathbf{u})$ is $\mathbb{Z}$-monomial positive modulo…

Noncommutative Schur positivity conjecture. For every integer partition $\lambda$ and every positive integer $k$, $\mathfrak{J}_\lambda(\boldsymbol{u})$ is $\mathbb{Z}$-monomial po…

Let $\mathcal{U}_\varnothing$ be the subalgebra generated by the uncolored variables, let $\mathfrak{J}^\varnothing_\nu$ be the corresponding noncommutative Schur function for a pa…

Let $\mathcal{A}$ be the ordered colored alphabet, let $\mathcal{U}$ be the associated noncommutative algebra, let $\mathfrak{J}_\nu(\mathbf{u})$ be the noncommutative super Schur…

Hall–Littlewood immaculate positivity conjecture. The element $\mathcal{Q}'_\lambda$ expands in the immaculate basis with coefficients that are positive polynomials in $q$. More ex…

Immaculate product positivity conjecture. The coefficients ${\bf c}_{\alpha,\lambda}^\beta$ are non-negative integers.

The signed left Pieri rule. The product has the form