10 problems

Let $\mathcal{D}_{n}^{(3)}$ be the space of polynomials in $x_1,\ldots,x_n,y_1,\ldots,y_n,z_1,\ldots,z_n$ annihilated by every polarized power-sum operator … where…

Let $f$ be any homogeneous symmetric polynomial, and let $\mathcal{M}_f({\bold q})$ be the Hilbert series of its polarization module. $h$-positivity conjecture. There exist…

Let $f$ be a homogeneous symmetric polynomial of degree $d$, and let $h_m$ denote the complete homogeneous symmetric function of degree $m$ as used in the source. Embedding conject…

Let $f$ be a homogeneous diagonally symmetric polynomial of degree $4$ or $5$, and let $\mathcal{M}_f$ be its polarization module. Completeness conjecture. Up to $n$-exceptions, th…

Let $\mathcal{T}_3$ be the family of all monomials of total degree $3$ in $x_{11},\ldots,x_{1n}$, and let $\mathcal{M}_{\mathcal{T}_3}$ be its associated polarization module. Frobe…

Let $\mathcal{T}_2$ be the family of all monomials of total degree $2$ in $x_{11},\ldots,x_{1n}$, and let $\mathcal{M}_{\mathcal{T}_2}$ be its associated polarization module. Frobe…

Let $\mathcal{A}=\{x_{1j}^d:1\leq j\leq n\}$, $\mathcal{B}=\{x_{1,i}^d-x_{1,j}^d:1\leq i<j\leq n\}$, and $\mathcal{C}=\{\prod_{a\in A}x_{1,a}:A\subseteq[n],\lvert A\rvert=d\}$. Let…

Let $m_{(2,1^{d-2})}$ be the monomial symmetric function indexed by $(2,1^{d-2})$, and let $s_j=s_j({\bold q})$ abbreviate the Schur function. For $d\geq3$ and $n\geq d$, consider…

Let $p_k$ denote the power-sum symmetric function of degree $k$, and let an $n$-exception be a homogeneous symmetric polynomial whose associated polarization module has the excepti…

Let $\mathcal{M}_{e_{d-1,1}}$ and $\mathcal{M}_{m_{(2,1^{d-2})}}$ be the polarization modules generated by the indicated symmetric functions, regarded as…