Let $\Pi_{ (b^a)}$ be the locus of unordered set partitions of $[ab]$ with $a$ blocks of size $b$. We embed unordered set partitions of $[n]$ into the affine space $\mathbb{ C}^{ \binom{ [n]}{ 2}}$ with coordinate ring $\mathbb{ C}\Big[\mathbf{ x}_{ \binom{ [n]}{ 2}}\Big]$. Then, we apply orbit harmonics to $\Pi_{ (2^a)}$ and $\Pi_{ (a^2)}$, yielding graded $\mathfrak{ S}_{ 2a}$-modules whose graded character formulae respectively refine the Schur expansions of $h_a[h_2]$ and $h_2[h_a]$ according to $\lambda_1$. We further extend this $\lambda_1$-separation phenomenon to quotients of $\mathbb{ C}^{ \binom{ [n]}{ 2}}$ where $n$ is odd. Combining $\Pi_{ (b^a)},\Pi_{ (a^b)}$ and orbit harmonics, we propose a conjecture related to Foulkes' conjecture, and we prove the special case $b=2$. We also apply orbit harmonics to the locus $\Pi_{ n,m}$ of unordered set partitions of $[n]$ without blocks of size greater than $m$, yielding a graded $\mathfrak{ S}_n$-module $R(\Pi_{ n,m})$. We determine the standard monomial basis of $R(\Pi_{ n,m})$ with respect to any monomial order, as well as its graded character formula.
Categories: math.CO math.AC math.RT