Polynomiality of Subdimensions of Diagonal Harmonics and a Sharp Stability Bound

Abstract

A sequence of $S_n$-representations $\{V_n{\}}_{n\ge 1}$ is representation stable if, writing $V(\lambda )=(n-|\lambda |,{\lambda }_1,{\lambda }_2,\dots )$ for each partition $\lambda$, the multiplicity of the irreducible indexed by $V(\lambda )$ in $V_n$ is eventually independent of $n$. In particular, Church, Ellenberg and Farb \cite{Church_2015} found that if we fix $a$ and $b$, then the space of diagonal harmonics $D{H}_n^{a,b}$ exhibits this behavior, and its dimension stabilizes to a polynomial in $n$ eventually. Building on this result, we use the Schedules Formula by Haglund and Loehr \cite{HAGLUND2005189} to get an explicit combinatorial polynomial for the dimension of the bigraded spaces $D{H}_n^{a,b}$. This derivation not only yields the dimension formula but also produces a new stability bound of \( a + b \) which is sharp, and determines the exact degree of the dimension polynomial, which is also \( a + b \).