20 problems

Immaculate product positivity conjecture. In the cases under consideration, the product of two immaculate functions expands positively in the immaculate basis.

Let $K$ be a unital commutative ${\mathbb Q}$-algebra, let $z=(z_1,\ldots,z_n)$ be commuting variables, and let $H(z)\in K[z]^{\times n}$ be homogeneous of degree $d=3$. Let…

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…

Let $a(n)$ be the sequence defined by $a(0)=1$ and … For a composition $\alpha$, let $g^{\alpha}$ denote the number of standard immaculate tableaux of shape $\alpha$, and write…

Non-negativity conjecture. The coefficients

Yang–Baxter trace conjecture. For rectangular partitions, multi-$t$ Macdonald polynomials are obtained as equivariant traces of certain Yang–Baxter elements of Hecke algebras.

Sign conjecture. The coefficient $c_\sigma$ of $e_\sigma$ is positive if $n-1$ is even, and negative if $n-1$ is odd.

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 $\mathcal{U}$ be the noncommutative algebra and let $I_{\mathrm{Kron}}$ and $I_{\mathrm{Kron-K}}$ be the ideals defined in the paper. Let the flag Schur theorem refer to the th…

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…

Let $\mathcal{A}$ be the ordered colored alphabet, let $\mathcal{U}$ be the algebra generated by the corresponding noncommuting variables $u_x$, and let…

Dual immaculate Pieri conjecture. The dual quasisymmetric basis should satisfy a signed multiplicity-free Pieri rule.

Signed left Pieri conjecture. There is a statistic $\operatorname{sign}(\alpha,\beta)$ such that

Stability conjecture. If $\ell(\nu)\leq\ell(\alpha)$, then

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…

Dual immaculate Pieri conjecture. The dual immaculate basis has a multiplicity-free signed Pieri rule; explicitly,

The signed left Pieri rule. The product has the form

Baker–Richter's conjecture. The Hurewicz homomorphism is a homomorphism of Hopf algebras, with target the Novikov double of the Hopf algebra defined by the BFK diagonal on…