3 problems

For $1\leq k\leq n$, let $\mathbb{V}_{n,k}$ be the smallest subspace of $\Omega_n[y_1,\dots,y_n]$ containing the superspace Vandermonde $\delta_{n,k}$, closed under…

Let $\Omega_n[y_1,\dots,y_n]=\mathbb{C}[x_1,\dots,x_n]\otimes\mathbb{C}[y_1,\dots,y_n]\otimes\bigwedge\{\theta_1,\dots,\theta_n\}$, with $S_n$ acting triply diagonally, and let … b…

Let $F$ be a symmetric function with a positive monomial-basis expansion, let $G$ be a symmetric function with a positive Schur-basis expansion, and let $\Delta_F$ and $\Xi$ denote…