6 problems

Let ${\mathcal{UZ}}_{n,m,r}$ be the upper rook placement locus and let $R({\mathcal{UZ}}_{n,m,r})$ be its orbit harmonics ring, with the natural…

Let $G$ be a group and $(V_1,\dots,V_N)$ a sequence of $G$-modules. The sequence is $G$-log-concave if there are $G$-module embeddings … for every $1<i<N$. For a graded $G$-module…

Let ${\mathfrak{D}}_n$ be the set of derangements in the symmetric group ${\mathfrak{S}}_n$, and let ${\mathbf{R}}({\mathfrak{D}}_n)$ be the associated graded module. For…

For integers $n$ and $m$, let $Pi{n,m}$ be the locus of unordered set partitions of $[n]$ with block sizes bounded by $m$, and let $R( Pi{n,m})$ be its orbit-harmonics graded…

Let $Pi{(b^a)}$ and $Pi{(a^b)}$ denote the loci of unordered set partitions whose block-size types are $(b^a)$ and $(a^b)$, respectively, and let ${rm gr}{bf I}( Pi)$ be the asso…

Let $Z\subset\mathbb{C}^n$ be any finite $S_n$-stable set such that $\mathbb{C}[x_1,\dots,x_n]/\operatorname{gr}\mathbf{I}(Z)$ is a graded $S_n$-module, where $\mathbf{I}(Z)$ is th…